The Structure of the 2-Sylow Subgroups of the Ideal Class Groups of Imaginary Bicyclic Biquadratic Fields

Ramona Renee Ranalli

Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in Mathematics

Charles Parry, Chair Ezra Brown Robert McCoy Monte Boisen John Rossi

December, 1997 Blacksburg, Virginia

Keywords: Biquadratic Fields, Bicyclic Fields, Class Groups Copyright 1997, Ramona Renee Ranalli

The Structure of the 2-Sylow Subgroups of the Ideal Class Groups of Imaginary Bicyclic Biquadratic Fields

Ramona Renee Ranalli

(ABSTRACT)

In this dissertation class groups of imaginary bicyclic biquadratic fields are considered. In chapter 1 we develop a method for determining the structure of the 2-class group of K. In chapters 3, 4 and 5 this method is applied determine all imaginary bicyclic biquadratic extensions of ℚ with class number 4, 8 and 16, as well as determine the specific structure of each.

Dedication

In memory of my father, who never doubted this day would come. I only wish he were here to celebrate with me.

Acknowledgments

Whenever one manages to reach a milestone in ones life there is a myriad of people whose contributions made the achievement possible. It is impossible to list them all in these few words. The greatest of my thanks goes to God who set me on this path and provided me with the necessary support and ability needed to complete it. Without Him I would not be. He blessed me with parents who have believed in me and supported me when I did not believe in myself. My thanks to both my mother and father as well as to the rest of my family for their support. In addition I would like to thank Dr. Parry. Asking him to be my advisor was one of the best choices I ever made. I thank Frank Taylor for his friendship, for influencing me towards a computational problem and, afterwards, his willingness to help me learn Mathematica. My questions must have tried his patience at times and I am grateful for his forbearance. Finally, I thank David Alger who helped me find and keep my sanity for this last year so that I might finish.

Contents

1 Introduction 1 2 Notation 2 3 Class Group Structure of K 4 4 Examples 9 5 Fields with 2-Class Group ℤ_{2} × ℤ_{2}. 22 6 Fields with Class Numbers 4, 8 and 16. 31

List of Tables

6.1 Fields where H ≃ ℤ_{2} × ℤ_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2 Fields where H ≃ ℤ_{4} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.3 Fields where H ≃ ℤ_{8} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.4 Fields where H ≃ ℤ_{4} × ℤ_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.5 Fields where H ≃ ℤ_{2} × ℤ_{2} × ℤ_{2} . . . . . . . . . . . . . . . . . . . . . . . . 56 6.6 Fields where H ≃ ℤ_{16} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.7 Fields where H ≃ ℤ_{8} × ℤ_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.8 Fields where H ≃ ℤ_{4} × ℤ_{4} . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.9 Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2} . . . . . . . . . . . . . . . . . . . . . . . . 85 6.10 Fields where H ≃ ℤ_{2} × ℤ_{2} × ℤ_{2} × ℤ_{2} . . . . . . . . . . . . . . . . . . . . . 90

Chapter 1

Introduction

Given a bicyclic biquadratic field K, we wish to be able to determine the structure of the class group of K whenever the structure of the class groups of the three quadratic subfields is known. It is easy to show that the odd part of the class group of K is the direct product of the odd parts of the class groups of the quadratic subfields. However, the structure of the 2-Sylow subgroup, H, of the class group of K is much more difficult to determine.

In the first chapter we will develop an algorithm for determining H when the structure of the 2-class groups of the quadratic subfields is known. This process will include computing the rank of H by computing the rank of a matrix in ℤ_{2} as well as the ranks of \frac{H^{2^{i+1}}}{H^{2^{i+2}}} for certain values of i. In the second chapter we will look at specific examples and discuss some of the twists and turns which can occur. The third chapter develops theory to classify all K with 2-class group ℤ_{2} ×ℤ_{2}. In the final chapter we discuss we go through the computations required to develop complete lists of all imaginary bicyclic biquadratic fields with class numbers 4, 8 and 16, respectively. These lists are given in the various tables.

Notation

The following notation will be used for the remainder of this dissertation.

K : An imaginary bicyclic biquadratic extension of ℚ. k_{1}, k_{2}, k_{3} : The quadratic subfields of K with k_{2} real. d_{1}, d_{2}, d_{3} : Square free integers with k_{i} = ℚ(\sqrt{d_{i}}) for i = 1, 2, 3. f : The conductor of K. H, H_{1}, H_{2}, H_{3} : The ideal class groups of K, k_{1}, k_{2} and k_{3}, respectively. h, h_{1}, h_{2}, h_{3} : The class numbers of K, k_{1}, k_{2} and k_{3}, respectively. Ĥ_{i} : The group of quadratic character values on the group H_{i}. Each element of Ĥ_{i} corre-

sponds to a genus in H_{i} for i = 1, 2, 3. Ã: The ideal class determined by the ideal A. Ŝ: The subgroup of Ĥ_{1} ×Ĥ_{2} ×Ĥ_{3} consisting of those character values which are consistent

on each pair of H_{1}, H_{2} and H_{3}. S : The subgroup of H_{1} × H_{2} × H_{3} with character group Ŝ.

Ramona R Ranalli  Chapter 2. Notation  3

Amb : The subgroup of Ĥ_{1} × Ĥ_{2} × Ĥ_{3} consisting of the ambiguous classes. θ : The homomorphism H_{1} × H_{2} × H_{3} → H defined by θ(C_{1}, C_{2}, C_{3}) = C_{1}C_{2}C_{3}. ker : The kernel of θ. H_{0} : The image of θ. t : The positive integer determined such that 2^{t} is the product of the ramification indices

of all primes for the extension K/ℚ. The number of rational primes which ramify in K is t − 1 or t according as 2 is totally ramified in K or not. Hence t will be referred to as the total number of discrimintal divisors.

t_{i} : The number of rational primes ramified in k_{i} for i = 1, 2, 3. r_{a} : The rank of H_{1} × H_{2} × H_{3}. r_{H} : The rank of H. l, q, r, s : Distinct prime numbers. (l, q, r) : An element of H_{1} × H_{2} × H_{3} determined by the ideal classes of prime divisors of

l, q and r in k_{1}, k_{2} and k_{3}, respectively. ψ : The isomorphism from the multiplicative group {±1} to the additive group Z_{2}. (\frac{a}{b}) : The Kronecker symbol using the convention (\frac{b}{2}) = (\frac{2}{b}) for all odd positive integers. u, v, w, x, y, z : ψ(\frac{q}{s}), ψ(\frac{r}{s}), ψ(\frac{q}{r}), ψ(\frac{l}{q}), ψ(\frac{l}{r}), ψ(\frac{l}{s}) respectively. M : A Z_{2}-matrix determined by Ŝ · k̂er. G^{(2)} : The 2-Sylow subgroup of the group G.

Class Group Structure of K

The structure of the odd part of the class group of K is easily shown to be the direct product of the class groups of its subfields. While the structure of the 2-Sylow subgroup of H depends on the structures of the 2-Sylow subgroups of H_{1}, H_{2} and H_{3}, the relation is more complicated. In this chapter we describe a method for determining the 2-class group of H.

Lemma 1. Let (C_{1}, C_{2}, C_{3}) ∈ S. Then there is a prime p of ℚ which has a prime divisor P_{0} in K such that 𝔓_{i} = P_{0} ∩ k_{i} = P_{0}P_{i} with 𝔓i and C_{i} in the same genus and where (p) = P_{0}P_{1}P_{2}P_{3} in K.

Proof Since the characters on C_{i} in Ĥi are consistent with one another for i = 1, 2, 3, there is a prime p of ℚ which satisfies these character values. Now p splits completely in K and so (p) = P_{0}P_{1}P_{2}P_{3} in K. Since the Galois group of K|ℚ is transitive on the primes P_{0}, P_{1}, P_{2}, and P_{3} there is a σ_{i} in G(K|ℚ) such that σ_{i}(P_{0}) = P_{i}. Number the primes P_{1}, P_{2}, and P_{3} so that σ_{i} fixes the subfield k_{i} of K. Let 𝔓_{i} = P_{0} ∩ k_{i} for i = 1, 2, 3. Then 𝔓_{i} = P_{0}P_{j} ∩ k_{i} for some j. Since the Galois group of K|k_{i} is transitive on the factors of 𝔓_{i} and G(K|k_{i}) = {1, σ_{i}} we have σ_{i}(P_{0}) = P_{i} and so, by the definition of σ_{i}, 𝔓_{i} = P_{0} ∩ k_{i} = P_{0}P_{i} ∩ k_{i}. Since P̃_{i} and C_{i} have the same character values they are in the same genus.

Ramona R Ranalli  Chapter 3. Class Group Structure of K  5

Theorem 1. The homomorphism θ induces an isomorphism \frac{S^{2^{i}}}{S^{2^{i}}∩Ker} ≃ H^{2^{i+1}} for any integer i ≥ 0.

Proof Let (C_{1}^{2^{i}}, C_{2}^{2^{i}}, C_{3}^{2^{i}}) ∈ S^{2^{i}} with (C_{1}, C_{2}, C_{3}) ∈ S. By Lemma 1 we have a prime p in ℚ which splits completely in K and has a prime divisor P_{0} such that 𝔓_{i} = P_{0} ∩ k_{i} = P_{0}P_{i} ∩ k_{i} where (p) = P_{0}P_{1}P_{2}P_{3} in K with 𝔓i and C_{i} in the same genus. Note that (𝔓̃_{1}^{2^{i}}, 𝔓̃_{2}^{2^{i}}, 𝔓̃_{3}^{2^{i}}) ∈ S^{2^{i}} with 𝔓̃_{i} and C_{i} being in the same genus of k_{i}. Now

θ(𝔓̃_{1}^{2^{i}}, 𝔓̃_{2}^{2^{i}}, 𝔓̃_{3}^{2^{i}}) = 𝔓̃_{1}^{2^{i}} 𝔓̃_{2}^{2^{i}} 𝔓̃_{3}^{2^{i}} = (𝔓̃_{1}𝔓̃_{2}𝔓̃_{3})^{2^{i}} = (P̃_{0}^{2}p̃)^{2^{i}} = P̃_{0}^{2^{i+1}} ∈ H^{2^{i+1}}.

Since 𝔓̃_{i}C_{i}^{−1} is in the principal genus of k_{i}, 𝔓̃_{i}C_{i}^{−1} = B_{i}^{2} for some class B_{i} of k_{i}. Hence

(𝔓̃_{1}C_{1}^{−1}, 𝔓̃_{2}C_{2}^{−1}, 𝔓̃_{3}C_{3}^{−1}) = (B_{1}^{2}, B_{2}^{2}, B_{3}^{2}),

so

B_{1}^{2}B_{2}^{2}B_{3}^{2} = (𝔓̃_{1}𝔓̃_{2}𝔓̃_{3})(C_{1}C_{2}C_{3})^{−1} = P̃_{0}^{2}(C_{1}C_{2}C_{3})^{−1}.

Therefore

(B_{1}B_{2}B_{3})^{2^{i+1}} = P̃_{0}^{2^{i+1}}(C_{1}^{2^{i}}C_{2}^{2^{i}}C_{3}^{2^{i}})^{−1}

and C_{1}^{2^{i}}C_{2}^{2^{i}}C_{3}^{2^{i}} ∈ H^{2^{i+1}}.

Conversely, let C^{2^{i+1}} ∈ H^{2^{i+1}} and P_{0} ∈ C be a prime ideal of degree 1 and index 1 over ℚ. Let 𝔓_{i} = P_{0} ∩ k_{i} for i = 1, 2, 3. Then 𝔓_{1} = P_{0}P_{1}, 𝔓_{2} = P_{0}P_{2} and 𝔓_{3} = P_{0}P_{3} where P_{0} ∩ ℚ = (p) = P_{0}P_{1}P_{2}P_{3}. Now (𝔓̃_{1}, 𝔓̃_{2}, 𝔓̃_{3}) ∈ S and 𝔓̃_{1}𝔓̃_{2}𝔓̃_{3} = P̃_{0}^{2} = C^{2}. Thus 𝔓̃_{1}^{2^{i}} 𝔓̃_{2}^{2^{i}} 𝔓̃_{3}^{2^{i}} = P̃_{0}^{2^{i+1}} = C^{2^{i+1}}. Therefore \frac{S^{2^{i}}}{S^{2^{i}}∩Ker} ≃ H^{2^{i+1}}.

Theorem 2. \frac{H^{2^{i+1}}}{H^{2^{i+2}}} ≃ \frac{S^{2^{i}}∩Amb}{S^{2^{i}}∩Ker} · (S^{2^{i+1}} ∩ Ker).

Proof Let S = ⟨C_{1}, C_{2}, C_{3}, …, C_{k}⟩ and T = S^{(2)}. Arrange the C_{i} so that o(C_{1}) ≥ o(C_{2}) ≥ o(C_{3}) ≥ … ≥ o(C_{k}). Then, for any i where 2^{i} < o(C_{1}), there exists an m_{i} such that o(C_{m_{i}}) > 2^{i} and o(C_{m_{i+1}}) ≤ 2^{i}. Thus T^{2^{i}} = ⟨C_{1}^{2^{i}}, C_{2}^{2^{i}}, C_{3}^{2^{i}}, …, C_{m_{i}}^{2^{i}}⟩. Then we see that \frac{S^{2^{i}}}{S^{2^{i+1}}} ≃ ℤ_{2} × ℤ_{2} × … × ℤ_{2} where the rank of the right-hand side is m_{i} and T^{2^{i}} ∩ Amb ≃ ℤ_{2} × ℤ_{2} × … × ℤ_{2} with the rank of the right side again being m_{i}. Now, by Theorem 1,

\frac{H^{2^{i+1}}}{H^{2^{i+2}}} ≃ \frac{\frac{S^{2^{i}}}{S^{2^{i}}∩Ker}}{\frac{S^{2^{i+1}}}{S^{2^{i+1}}∩Ker}} .

Using a simple order argument we now see that

\frac{|S^{2^{i}}|}{|S^{2^{i}} ∩ Ker|} · \frac{|S^{2^{i+1}} ∩ Ker|}{|S^{2^{i+1}}|}

= \frac{|\frac{S^{2^{i}}}{S^{2^{i+1}}} |}{|S^{2^{i}} ∩ Ker|} · |S^{2^{i+1}} ∩ Ker|

= |\frac{S^{2^{i}} ∩ Amb}{S^{2^{i}} ∩ Ker}| · |S^{2^{i+1}} ∩ Ker|.

The isomorphism follows since both groups are elementary.

Lemma 2. The order of Ŝ is 2^{t−2}.

Proof Each odd prime which ramifies in K determines a character. If 2 ramifies in K then it determines either 3 or 1 characters depending on whether 2 totally ramifies or not. In the former case only two of these characters are independent. These t characters and their products are only restricted by the conditions ∏_{χ∈Ĥi} χ = +1, for i = 1, 2, 3. However,

any two product conditions determine the third. Therefore Ŝ has 2^{t−2} elements.

Lemma 3. The number t is determined by t_{1} + t_{2} + t_{3} = 2t − τ where τ = 1 if 2 is totally ramified and τ = 0 otherwise. Moreover,

r_{a} = \begin{cases}t_{1} + t_{2} + t_{3} − 4 & if a prime q ≡ 3 (mod 4) divides d_{2}. \\ t_{1} + t_{2} + t_{3} − 3 & otherwise. \end{cases}  (3.1)

Consequently, r_{a} = \begin{cases}2t − 5 & if 2 is totally ramified and q|d_{2} for some prime q ≡ 3 (mod 4). \\ 2t − 3 & if 2 is not totally ramified and no prime q ≡ 3 (mod 4) divides d_{2}. \\ 2t − 4 & otherwise. \end{cases}

(3.2)

Proof Each odd prime which ramifies in K, ramifies in exactly two of the subfields k_{i} for i = 1, 2, 3. The same is true for 2 unless it is totally ramified in K. The 2-rank of H_{i} is t_{i} − 1 unless i = 2 and some prime q ≡ 3 (mod4) divides d_{2}. In that case the 2-rank of H_{2} is t_{2} − 2. The final expressions for r_{a} are immediate.

Lemma 4. The order of S is given by |S| = \frac{h_{1}h_{2}h_{3}}{2^{ra}} |Ŝ| = \frac{h_{1}h_{2}h_{3}}{2^{so}} where

s_{o} = \begin{cases}t − 3 & if r_{a} = 2t − 5. \\ t − 2 & if r_{a} = 2t − 4. \\ t − 1 & if r_{a} = 2t − 3. \end{cases}  (3.3)

Proof The order of Ĥ_{1} × Ĥ_{2} × Ĥ_{3} is 2^{r_{a}}. Now the same number of classes of H_{1} × H_{2} × H_{3} belong to each character value of Ĥ_{1} × Ĥ_{2} × Ĥ_{3}. Since there are 2^{t−2} character values in Ŝ, the first result follows. Since s_{o} = r_{a} − t + 2, the final result is immediate from the previous lemma.

In the remainder of this chapter we will show that the rank of H is given by the rank of a ℤ_{2}-matrix.

Theorem 3. The 2-rank of H is given by

r_{H} = t − 2 + log_{2}[H_{1} × H_{2} × H_{3} : S · Ker]

Proof Now

r_{H} = log_{2}[H : H^{2}]
= log_{2}[H : H_{0}] + log_{2}[H_{0} : H^{2}]
= log_{2}[H_{0} : H^{2}] + t − 2

where the last equality follows from Kubota [9]. But H_{0}/H^{2} ≃ \frac{H_{1}×H_{2}×H_{3}/Ker}{S/S∩Ker} and S/S ∩ Ker ≃ S · Ker/Ker so [H_{0} : H^{2}] = [H_{1} × H_{2} × H_{3} : S · Ker]. The result follows.

Corollary 1. If the unit index is 2 then s_{o} ≤ r_{H} ≤ r_{a}. In any case t − 2 ≤ r_{H} ≤ r_{a}.

Proof From Kubota [9], |Ker| = 2^{t−2} or 2^{t−1} according as the unit index of K is 2 or 1. Let 2^{e} = |Ker|. Then

[H_{1} × H_{2} × H_{3} : S · Ker] = \frac{|H_{1} × H_{2} × H_{3}|}{|S||Ker|}|S ∩ Ker|
= 2^{s_{o}−e}|S ∩ Ker|.

The results now follow from Lemma 4 and Theorem 3.

Corollary 2. In Theorem 3, the term [H_{1} × H_{2} × H_{3} : S · Ker] can be replaced with [(H_{1} × H_{2} × H_{3})^{(2)} : S^{(2)} · Ker].

Proof Note that all elements of odd order in H_{1} × H_{2} × H_{3} are in S since they belong to the principal genus of each k_{i}. Thus [(H_{1} ×H_{2} ×H_{3})^{(2)} ·S ·Ker] = H_{1} ×H_{2} ×H_{3}. Moreover, since Ker is a 2-group, (H_{1}×H_{2}×H_{3})^{(2)}∩S ·Ker = S^{(2)}·Ker. Hence H_{1}×H_{2}×H_{3}/S ·Ker ≃ (H_{1} × H_{2} × H_{3})^{(2)}/S^{(2)} · Ker.

Theorem 4. Let m denote the rank of Ŝ · K̂er. Then

r_{H} = r_{a} + t − 2 − m = \begin{cases}3t − 7 − m & if r_{a} = 2t − 5. \\ 3t − 6 − m & if r_{a} = 2t − 4. \\ 3t − 5 − m & if r_{a} = 2t − 3. \end{cases}  (3.4)

Proof Let ϕ : (H_{1} × H_{2} × H_{3})^{(2)} → Ĥ_{1} × Ĥ_{2} × Ĥ_{3} be the mapping determined by taking

a class C_{i} of H_{i} to its character system in Ĥi. Then \begin{matrix}(H_{1}×H_{2}×H_{3}) & ^{(2)} \\ (Kerϕ(S·Ker)) & ^{(2)} \end{matrix} ≃ \frac{Ĥ_{1}×Ĥ_{2}×Ĥ_{3}}{ϕ(S^{(2)}·Ker)} . But

ϕ(S^{(2)} · Ker) = ϕ(S^{(2)}) · ϕ(Ker) = Ŝ · K̂er. Moreover, Kerϕ is the direct product of the 2-Sylow subgroups of the principal genera of k_{1}, k_{2} and k_{3} which is clearly contained in S^{(2)}. Thus \frac{(H_{1}×H_{2}×H_{3})^{(2)}}{S^{(2)}·Ker} ≃ \frac{Ĥ_{1}×Ĥ_{2}×Ĥ_{3}}{Ŝ·K̂er} . The result now follows from Theorem 3 and Corollary 2.

In order to determine r_{H} all that remains to be done is to compute m = rank (Ŝ · K̂er). This will be discussed and illustrated in the next chapter.

Examples

In this section we show how the formula of the last Theorem can be used to commute r_{H}. Recall that rank Ŝ = t − 2 and rank K̂er = t − 1 or t − 2 according as the unit index of K is 1 or 2. Let n = rank Ŝ + rank K̂er and M be a n × r_{a} ℤ_{2}-matrix whose rows correspond to generators of Ŝ · K̂er by means of the isomorphism ψ. Then m is the rank of M.

Note that the characters of the real field will often have to be normalized in order to ensure that −1 lies in the principal genus. This is done by multiplying one column where −1 has a − character by each of the other columns where −1 also has a − character. The initial choice of column does not matter. The following example illustrates this as well as the technique for finding H.

Example Let k_{1} = ℚ(\sqrt{−lqrs}), k_{2} = ℚ(\sqrt{lq}) and k_{3} = ℚ(\sqrt{−rs}) with l ≡ q ≡ r ≡ 3 (mod 4) and s ≡ 1 (mod 4). Here the unit index is 1, t = 4 and r_{a} = 5. Normalizing the character of the real field, the table of consistent character systems is:

\begin{matrix}l & q & r & s & lq & r & s \\ + & + & + & + & + & + & + \\ − & − & + & + & + & + & + \\ + & + & − & − & + & − & − \\ − & − & − & − & + & − & − \end{matrix} .

Here Ŝ is generated by (1, 1, 0, 0, 0, 0, 0) and by (0, 0, 1, 1, 0, 1, 1) representing a + by a 0 and a − by a 1. Moreover, Ker is generated by {(l, 1, 1), (q, 1, 1), (r, 1, r)}. Since in k_{2} lq is always + it will just generate a column of zeroes in our matrix and so can be deleted. Thus our columns will correspond to l, q, and r in k_{1} and r in k_{3}. So our matrix is

M = (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ ψ((\frac{l}{q}) (\frac{l}{r}) (\frac{l}{s})) & ψ (\frac{l}{q}) & ψ (\frac{l}{r}) & 0 \\ 1 + ψ (\frac{l}{q}) & ψ((\frac{q}{l}) (\frac{q}{s}) (\frac{q}{r})) & ψ (\frac{q}{r}) & 0 \\ 1 + ψ (\frac{l}{r}) & 1 + ψ (\frac{q}{r}) & ψ((\frac{l}{r}) (\frac{r}{s}) (\frac{q}{r}) & ψ (\frac{r}{s}) \end{matrix})

= (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ x + y + z & x & y & 0 \\ 1 + x & 1 + x + w + u & w & 0 \\ 1 + y & 1 + w & y + v + w & v \end{matrix})

where the first two rows corresponds to the generators of Ŝ and the last three rows correspond to generators of Ker. We have deleted one character from each subfield since the product of the characters for a quadratic field is +1. The first three columns correspond to characters for k_{1}, determined by l, q and r, and the last column to a character for k_{3}, determined by r. Performing elementary row operations we find:

(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & z + y & y & 0 \\ 0 & w + u & w & 0 \\ 0 & w + y & w + y + v & v \end{matrix}) ∼ (\begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & y & y + z \\ 0 & 0 & w & w + u \\ 0 & v & y + v + w & y + w \end{matrix})

∼ (\begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & y & y + z \\ 0 & 0 & w & w + u \\ 0 & 0 & w + y & w + y \end{matrix}) ∼ (\begin{matrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & y & z \\ 0 & 0 & w & u \\ 0 & 0 & y + w & 0 \end{matrix})

So we see that

m = \begin{cases}2 & if y = w = u = z = 0. \\ 3 & if y = z = w = u = 1. \\ & if exactly one of u, w, y, z is non-zero. \\ 4 & if y = z and w = u. \\ & if y = u and w = z. \\ & if exactly one of u, w, y, z is zero. \end{cases}  (4.1)

Now r_{a} = 4 and t = 4 so r_{H} = 6 − m.

Now we are ready to compute the structure of H for any K. The following proposition will be helpful in that it states a basic fact which will be used many times in our computations.

Proposition 1. Let b = 2^{j} for some j > 1 where b|h_{i} for some i = 1, 2, 3. Also, let T = S^{(2)}. If (C_{1}^{b/2}, C_{2}^{b/2}, C_{3}^{b/2}) is in T^{b/2} ∩ Amb then (C_{1}^{b/2}, 1, 1), (1, C_{2}^{b/2}, 1) and (1, 1, C_{3}^{b/2}) are in T^{b/4} ∩ Amb.

Proof Let (C_{1}, C_{2}, C_{3})^{b/2} = (C_{1}^{b/2}, C_{2}^{b/2}, C_{3}^{b/2}) be in T^{b/2} ∩Amb. Then C_{i}^{b/2} is ambiguous for each i. Since C_{1}^{2} is in the principal genus of k_{1}, (C_{1}^{2}, 1, 1) is in T, so (C_{1}^{b/2}, 1, 1) = (C_{1}^{2}, 1, 1)^{b/4} is in T^{b/4} ∩ Amb. A similar argument is true for (1, C_{2}^{b/2}, 1) and (1, 1, C_{3}^{b/2}).

Now we will compute H for a variety of cases in order to get a feel for the algorithm. In order to find h we use the following formula due to Kubota [9]:

h = \frac{1}{2}[unit index]h_{1}h_{2}h_{3}.

Example For our first example let d_{1} = −402, d_{2} = 3953 and d_{3} = −354. Here H_{1}^{(2)} ≃ H_{3}^{(2)} ≃ ℤ_{8} ×ℤ_{2} and H_{2} has odd class number. First note that the Ker is {(1, 1, 1), (1, 1, 6), (2, 1, 2), (2, 1, 3), (3, 1, 2), (3, 1, 3), (6, 1, 1), (6, 1, 6)}. The table of consistent characters is

\begin{matrix}−2 & 3 & 67 & 59 & 67 & −2 & 3 & 59 \\ + & + & + & + & + & + & + & + \\ + & − & − & − & − & + & − & − \\ − & − & + & + & + & − & − & + \\ − & + & − & − & − & − & + & − \end{matrix}

Normalizing the characters in the real field we get

\begin{matrix}−2 & 3 & 67 & 59 · 67 & −2 & 3 & 59 \\ + & + & + & + & + & + & + \\ + & − & − & + & + & − & − \\ − & − & + & + & − & − & + \\ − & + & − & + & − & + & − \end{matrix}

Now our goal here is to use Theorem 2 to find H. To do this we need to determine the structure of S^{2^{i}} ∩ Amb and S^{2^{i}} ∩ Ker for various i. As a first step we will determine which of these lines of characters belong to ambiguous classes. This is equivalent to determining which genera can be generated by the Jacobi symbol of a product of ramified primes of a field over the respective prime divisors. In k_{1} we find that 2 and 3 have the character lines + − − and that 6 is in the principal genus . In k_{3} it is 2 which corresponds to + − − and 3 which is in the principal genus. Thus, in both k_{1} and k_{3}, the character line + − − represents a genus with elements of order two and the other non-principal genera contain only classes with elements of order eight. In these latter genera, the fourth power of any element is in the principal genus and is ambiguous and so can be represented by 6 in k_{1} and by 3 in k_{3} respectively. The following table reflects this information.

\begin{matrix}−2 · 3 · 67 & 59 · 67 & −2 · 3 · 59 \\ (1, 1) & (1, 1) & (1, 1) \\ (1, 1) & (1, 1) & (1, 1) \\ (4, 6) & (1, 1) & (4, 3) \\ (4, 6) & (1, 1) & (4, 3) \end{matrix}

In this table each (a, b) means that in that genus there is an element of order 2a whose ath power is in the class represented by b. Thus, in this example, we see that

S^{4} ∩ Amb = ⟨(6, 1, 3)⟩ and that S^{4} ∩ Ker = ⟨(1, 1, 1)⟩.

Thus

\frac{S^{4} ∩ Amb}{S^{4} ∩ Ker} ≃ ℤ_{2}

and

\frac{H^{8}}{H^{16}} ≃ ℤ_{2}.

Hence there is a factor of Z_{16} in H. Now, by the Proposition,

S^{2} ∩ Amb = ⟨(6, 1, 3), (6, 1, 1), (1, 1, 3)⟩

and

S^{2} ∩ Ker = ⟨(6, 1, 1)⟩.

And so,

\frac{S^{2} ∩ Amb}{S^{2} ∩ Ker} ≃ ℤ_{2}

making

\frac{H^{4}}{H^{8}} ≃ ℤ_{2}.

This is satisfied by the factor of ℤ_{16} in H^{(2)} that we already have.

Using the methods already described, we can determine that the rank of H is three and the order is 2^{7}. Thus we have two more factors of combined order 2^{3}. Clearly then,

H^{(2)} ≃ ℤ_{16} × ℤ_{4} × ℤ_{2}.

Example For our next example let d_{1} = −406, d_{2} = 182 and d_{3} = −377. Here H_{1}^{(2)} ≃ H_{3}^{(2)} ≃ ℤ_{8} × ℤ_{2} and H_{2}^{(2)} ≃ ℤ_{2}. First note that the Ker is {(1, 1, 1), (1, 1, 13), (1, 2, 2), (1, 2, 26), (2, 1, 2), (2, 1, 26), (2, 2, 1), (2, 2, 13), (7, 1, 2), (7, 1, 26), (7, 2, 1), (7, 2, 13), (14, 1, 1), (14, 1, 13), (14, 2, 2), (14, 2, 26)}. The table of consistent characters is

\begin{matrix}2 & 7 & 29 & −2 · 7 & 13 & −1 & 13 & 29 \\ + & + & + & + & + & + & + & + \\ − & − & + & + & + & + & + & + \\ + & − & − & + & + & − & + & − \\ − & + & − & + & + & − & + & − \\ + & − & − & − & − & + & − & − \\ − & + & − & − & − & + & − & − \\ + & + & + & − & − & − & − & + \\ − & − & + & − & − & − & − & + \end{matrix}

Note that we have normalized the characters of the real field and that in k_{3} −1 denotes the character determined by the ramified prime 2. In k_{1} we find that 2 has the character line − + − and that 7 is in the principal genus . Thus the character line − + − represents a genus with elements of order two and the other non-principal genera lines contain only classes with elements of order eight. In these latter genera, the fourth power of any element in the principal genus and is ambiguous and so can be represented by 7. In k_{3} 2 has the character line + − − and 13 is in the principal genus. Thus + − − represents a genus with elements of order two and the other non-principal genera lines contain only classes with elements of order eight. In these latter genera, the fourth power of any element in the principal genus and is ambiguous and so can be represented by 13. The following table reflects this information.

\begin{matrix}2 · 7 · 29 & −2 · 7 · 13 & −1 · 13 · 29 \\ (1, 1) & (1, 1) & (1, 1) \\ (4, 7) & (1, 1) & (1, 1) \\ (4, 7) & (1, 1) & (4, 13) \\ (1, 1) & (1, 1) & (4, 13) \\ (4, 7) & (1, 1) & (1, 1) \\ (1, 1) & (1, 1) & (1, 1) \\ (1, 1) & (1, 1) & (4, 13) \\ (4, 7) & (1, 1) & (4, 13) \end{matrix}

Again, in this table each (a, b) means that in that class there is an element of order 2a whose ath power is in the class represented by b. Thus, in this example, we see that

S^{4} ∩ Amb = ⟨(7, 1, 1), (1, 1, 13)⟩ and that S^{4} ∩ Ker = ⟨(1, 1, 13)⟩.

Thus

\frac{S^{4} ∩ Amb}{S^{4} ∩ Ker} ≃ ℤ_{2}

and

\frac{H^{8}}{H^{16}} ≃ ℤ_{2}.

Hence there is a factor of Z_{16} in H. Now we see that for the second power everything remains the same. So

S^{2} ∩ Amb = ⟨(7, 1, 1), (1, 1, 13)⟩

and

S^{2} ∩ Ker = ⟨(1, 1, 13)⟩.

And so, since

\frac{S^{2} ∩ Amb}{S^{2} ∩ Ker} ≃ ℤ_{2}

, Theorem 2 shows that

\frac{H^{4}}{H^{8}} ≃ ℤ_{2} × ℤ_{2}.

The one ℤ_{2} is satisfied by the factor of ℤ_{16} that we already have, so the other one must indicate a factor of ℤ_{8} in H^{(2)}.

Using the methods already described, we can determine that the rank of H^{(2)} is three and the order is 2^{8}. Thus we have one more factor of order 2. Clearly then,

H^{(2)} ≃ ℤ_{16} × ℤ_{8} × ℤ_{2}.

Now, things worked out in a very straight-forward manner in these two examples because each H_{i} had at most one non-elementary factor. This meant that, once we identified the genera that contained the ambiguous classes, we knew the order of the classes of the other genera. If there are two or more non-elementary factors of unequal order in some H_{i} then this will not be the case. To determine S^{n} ∩Amb for any n = 2^{j} where j is a whole number, we must know not only the minimum order of a class in each genus but also the ambiguous class in the principal genus which it generates.

Let q be an unramified prime whose character values place its prime divisors in a genus that does not contain ambiguous classes. Say 𝔮, a prime divisor of q in k_{i}, belongs to a class of order m. Then 𝔮^{m/2} belongs to an ambiguous class in the principal genus. Assume that 𝔭 is an ideal in this class (this is always the case when k_{i} is imaginary). Then 𝔮^{m/2}𝔭 ∼ (1) is a principal ideal of k_{i}. If 𝔮^{m/2}𝔭 = (x + \sqrt{d_{i}}y) where x and y are both integers or half of integers then x^{2} − d_{i}y^{2} = 𝔮^{m/2}p. Conversely, assume x^{2} − d_{i}y^{2} = 2^{l}q^{m/2}p, where l = 0, unless d_{i} ≡ 1 (mod 4) and then l = 0 or 2 and gcd(x, 𝔮p) = 1. If p ramifies in the field k_{i} and if the prime divisor 𝔭 in k_{i} is not principal then the prime divisors of 𝔮 in k_{i} belong to classes of order m.

So the key to solving such problems will be to find an x and y in ℤ and a prime 𝔮 in a non-principal genus such that x^{2} −d_{i}y^{2} = p𝔮^{m/2} for each ramified (p) in the principal genus. The genus which contains the divisors of 𝔮 contains the classes of order m.

A slight complication occurs if the H_{i} contains one or more odd factors. In this case the right-hand side of the equation may be 𝔮^{gm} where g is a divisor of the product of the orders of the odd factors.

Example For our next problem let d_{1} = −18761, d_{2} = 2482 and d_{3} = −8738. Here H_{1}^{(2)} ≃ ℤ_{32} × ℤ_{4}, H_{2}^{(2)} ≃ ℤ_{4} × ℤ_{2} and H_{3}^{(2)} ≃ ℤ_{16} × ℤ_{4}. First note that the Ker is ((1, 1, 1), (1, 2, 2), (1, 17, 17), (1, 34, 34), (2, 1, 2), (2, 2, 1), (2, 17, 34), (2, 34, 17), (73, 1, 34), (73, 2, 17), (73, 17, 2), (73, 34, 1), (146, 1, 17), (146, 2, 34), (146, 17, 1), (146, 34, 2)). The table of consistent characters is

\begin{matrix}−1 & 73 & 257 & 2 & 17 & 73 & −2 & 17 & 257 \\ + & + & + & + & + & + & + & + & + \\ − & + & − & + & + & + & − & + & − \\ + & − & − & + & − & − & + & − & − \\ − & − & + & + & − & − & − & − & + \\ + & − & − & − & + & − & − & + & − \\ − & − & + & − & + & − & + & + & + \\ + & + & + & − & − & + & − & − & + \\ − & + & − & − & − & + & + & − & − \end{matrix}

Note that in this case the real field did not have to be normalized. Finding the ambiguous classes in the principal genus, we see that in k_{1} the divisors of 2 and of 73 are both in the principal genus, in k_{2} it is the divisors of 2 and, in k_{3}, the divisors of 2 and of 17.

It is necessary in this case, for both of the imaginary fields, to find the quadratic representations of a prime in the non-principal genera. It will be sufficient to find two such primes (perhaps even in the same genus) where some power of each is in the same ambiguous class as the divisors of 2, 73 or 146 in k_{1} and as 2, 17 or 34 in k_{3}. For k_{1} we get:

(73 · 4)^{2} + 18761 · 3^{2} = 73 · 59^{2}
141^{2} + 18761 = 2 · 139^{2}.

Since we find that both 59 and 139 have characters - + -, we know this genus contains classes of order four which square to an ambiguous class containing divisors of 2. The other non-ambiguous genera must have classes of order thirty-two whose sixteenth power is in an ambiguous class containing divisors of 146 (since 73 was also seen to be a square).

For k_{3} we get:

108^{2} + 8738 = 2 · 101^{2}
51^{2} + 8738 · 2^{2} = 17 · 47^{2}.

Here 101 and 47 have characters - + - and so we know this genus contains classes of order four which square to an ambiguous class containing divisors of 2 (and 17). So the other non-principal genera must have classes of order sixteen whose eighth power is in an ambiguous class containing divisors of 34. Thus, using the quadratic representations, we get the following table.

\begin{matrix}−1 · 73 · 257 & 2 · 17 · 73 & −2 · 17 · 257 \\ (1, 1) & (1, 1) & (1, 1) \\ (16, 146) & (1, 1) & (2, 2) \\ (16, 146) & (1, 1) & (8, 34) \\ (2, 2) & (1, 1) & (8, 34) \\ (16, 146) & (2, 2) & (2, 2) \\ (2, 2) & (2, 2) & (1, 1) \\ (1, 1) & (2, 2) & (8, 34) \\ (16, 146) & (2, 2) & (8, 34) \end{matrix}

Again, in this table each (a, b) means that in that class there is an element of order 2a whose ath power is in the class represented by b. Thus, in this example, we see that

S^{16} ∩ Amb = ⟨(146, 1, 1)⟩ and that S^{16} ∩ Ker = ⟨(1, 1, 1)⟩.

Thus

\frac{S^{16} ∩ Amb}{S^{16} ∩ Ker} ≃ ℤ_{2}

and

\frac{H^{32}}{H^{64}} ≃ ℤ_{2}.

Hence there is a factor of Z_{64} in H. Now we see that for the eighth power we have

S^{8} ∩ Amb = ⟨(145, 1, 1), (1, 1, 34)⟩

and

S^{8} ∩ Ker = ⟨(1, 1, 1)⟩.

And so,

\frac{S^{8} ∩ Amb}{S^{8} ∩ Ker} ≃ ℤ_{2} × ℤ_{2}

making

\frac{H^{16}}{H^{32}} ≃ ℤ_{2} × ℤ_{2}.

The one ℤ_{2} is satisfied by the factor of ℤ_{64} that we already have, so the other one must indicate a factor of ℤ_{32} in H. Clearly, since there are no terms (a, b) in the above table with a = 4,

S^{4} ∩ Amb ≃ S^{8} ∩ Amb

so we move on to S^{2}. Since

S^{2} ∩ Amb = ⟨(146, 1, 1), (1, 1, 34), (2, 2, 1)⟩

and

S^{2} ∩ Ker = ⟨(2, 2, 1)⟩

we have

\frac{S^{2} ∩ Amb}{S^{2} ∩ Ker} ≃ ℤ_{2} × ℤ_{2}

and

\frac{H^{4}}{H^{8}} ≃ ℤ_{2} × ℤ_{2}.

Both of these are accounted for by the factors of H already obtained. So all other factors of H are of order 2 or 4. Since it can be determined by previous methods that H has rank 5 and h = 2^{15} we see that

H^{(2)} ≃ ℤ_{64} × ℤ_{32} × ℤ_{4} × ℤ_{2} × ℤ_{2}.

In our final example we will examine a special case for the real field: weak ambiguous classes. While these do not happen very often, they do occur and so must be dealt with. If in the real field the characters of −1 are all positive and the norm of the fundamental unit is +1 then there exists at least one ambiguous class which does not contain an ambiguous ideal. Such a class is referred to as a weak ambiguous class. The number of such classes occurring is the same as the number of classes generated by taking the product of all strong ambiguous classes with any one of the weak ambiguous classes. In other words, half of the ambiguous classes will be weak. When this occurs we just know that for some w_{i} in H_{2} the divisors of w_{i}^{2} are in the principal genus. It is not necessary to know exactly what w_{i} is, only that it exists.

Example For our next problem let d_{1} = −1326, d_{2} = 14722 and d_{3} = −16887. Here H_{1} ≃ ℤ_{2} × ℤ_{2} × ℤ_{2} × ℤ_{5} , H_{2} ≃ ℤ_{4} × ℤ_{4} and H_{3} ≃ ℤ_{4} × ℤ_{8} × ℤ_{3}. Recall that the odd factor H_{3} puts a slight twist on the calculations for the quadratic representations. Next note that the Ker is {(1, 1, 1), (1, 17, 39), (1, 34, 39), (2, 1, 1), (2, 17, 39), (2, 34, 39), (3, 1, 3), (3, 17, 13), (3, 34, 13), (6, 1, 3), (6, 17, 13), (6, 34, 13), (13, 1, 13), (13, 17, 3), (13, 34, 3), (17, 1, 39), (17, 17, 1), (17, 34, 1), (26, 1, 13), (26, 17, 3), (26, 34, 3), (34, 1, 39), (34, 17, 1), (34, 34, 1)}.

The table of consistent characters is

\begin{matrix}2 & 3 & 13 & 17 & 2 & 17 & 433 & 3 & 13 & 433 \\ + & + & + & + & + & + & + & + & + & + \\ + & − & − & + & + & + & + & − & − & + \\ + & + & − & − & + & − & − & + & − & − \\ + & − & + & − & + & − & − & − & + & − \\ − & + & − & + & − & + & − & + & − & − \\ − & − & + & + & − & + & − & − & + & − \\ − & + & + & − & − & − & + & + & + & + \\ − & − & − & − & − & − & + & − & − & + \end{matrix}

Note that again in this case the real field did not have to be normalized. Finding the ambiguous classes in the principal genus, we see that in k_{1} all genera contain ambiguous classes, and in k_{2} the divisors of 17 are in the principal genus. Since the norm of the fundamental unit here is +1 and the characters of −1 are all positive, there must be at least one weak ambiguous class. We find that 1517^{2} − 14722 · 2^{2} = 17 · 19^{2}. The characters for 19 in k_{2} are -, +, - . Thus the square of any class in this genus is the ambiguous class containing a divisor of 17. So there must be two weak ambiguous classes. Let w_{1} and w_{2} be their representatives whose divisors are in the principal genus. Then 17w_{1} ∼ w_{2}. Now, in k_{3} the divisors of 3 and of 13 are in the principal genus. Using the quadratic representations we get the following table.

\begin{matrix}2 · 3 · 13 · 17 & 2 · 17 · 433 & 3 · 13 · 433 \\ (1, 1) & (1, 1) & (1, 1) \\ (1, 1) & (1, 1) & (1, 1) \\ (1, 1) & (2, w_{2}) & (4, 13) \\ (1, 1) & (2, w_{2}) & (2, 3) \\ (1, 1) & (2, 17) & (4, 13) \\ (1, 1) & (2, 17) & (2, 3) \\ (1, 1) & (2, w_{1}) & (1, 1) \\ (1, 1) & (2, w_{1}) & (4, 13) \end{matrix}

Again, in this table each (a, b) means that in that class there is an element of order 2a whose ath power is in the class represented by b. Thus, in this example, we see that S^{4} ∩ Amb is given by ⟨(1, 1, 13)⟩ and that S^{4} ∩ Ker is ⟨(1, 1, 1)⟩. Thus \frac{S^{4}∩Amb}{S^{4}∩Ker} ≃ ℤ_{2} and \frac{H^{8}}{H^{16}} ≃ ℤ_{2}. Hence there is a factor of Z_{16} in H. Now we see that for the second power we have S^{2} ∩ Amb = ⟨(1, 1, 13), (1, w_{2}, 3), (1, 17, 3)⟩ and S^{2} ∩ Ker = ⟨(1, 17, 3)⟩. And so, \frac{S^{2}∩Amb}{S^{2}∩Ker} ≃ ℤ_{2} × ℤ_{2} making \frac{H^{4}}{H^{8}} ≃ ℤ_{2} × ℤ_{2}. The one ℤ_{2} is satisfied by the factor of ℤ_{16} that we already have, so the other two must indicate two factors of ℤ_{8} in H. So all other factors of H are of order 2 or 4. Since it can be determined by previous methods that H has rank 4 and r_{a} = 2^{11}, we see that H ≃ ℤ_{16} × ℤ_{8} × ℤ_{4} × ℤ_{4}.

Fields with 2-Class Group ℤ_{2} × ℤ_{2}.

In this chapter we determine all imaginary bicyclic biquadratic fields which have 2-class group ℤ_{2} × ℤ_{2}. We have previously determined all such fields with 2-class group Z_{4} by different techniques [10]. For the first case, Corollary 1 of Theorem 3 shows that the total number of discrimintal divisors, t, is at most 4.

When t = 2, r_{a} ≤ 1 and the 2-Sylow subgroup of H is cyclic. In this chapter we determine r_{H} when t = 3 and when t = 4. Theorem 5. Let 2 be totally ramified in K, t = 3, r_{a} = 2 and l be the unique odd prime which ramifies in K. Then r_{H} = 1 or 2 according as l ≡ 5 (mod 8) or l ≡ 1 (mod 8).

Proof Since 2 is totally ramified in K, the only possible values for d_{1}, d_{2} and d_{3} are −l, 2l, −2; −2l, 2l, −1 and −2l, 2, −l with l ≡ 1 (mod 4). In each case Ŝ has the form ±±. Moreover Ker = ⟨ (2, 1, 1), (1, 2, 1), (1, 1, 2)⟩.

Thus Ŝ · K̂er is generated by character values corresponding to the matrix M = (\begin{matrix}1 & 1 \\ x & 0 \\ 0 & x \end{matrix})

Ramona R Ranalli  Chapter 4. Fields with 2-Class Group ℤ_{2} × ℤ_{2}  23

over ℤ_{2} which reduces to (\begin{matrix}1 & 0 \\ 0 & x \\ 0 & 0 \end{matrix}). Now the rank Ŝ · K̂er = rank M, so Theorem 4 shows r_{H} = 1 or 2 according as x = 1 or 0; i.e. according as l ≡ 5 or 1 (mod 8). Theorem 6. Let t = 3, r_{a} = 2 and 2 not be totally ramified in K. Assume the 2-Sylow subgroup of each H_{i} is cyclic. If the unit index of K is 1 then r_{H} = 1 or 2 according as h_{i} ≡ 2 (mod 4) for some i or not. If the unit index of K is 2 then r_{H} = 2.

Proof Since r_{a} = 2, (H_{1} ×H_{2} ×H_{3})^{(2)} ≃ ℤ_{2^{a}} ×ℤ_{2^{b}}. When the unit index is 1, |Ker| = 4 and Ker ≃ ⟨ (2^{a−1}, 0), (0, 2^{b−1})⟩. Here S^{(2)} ≃ ⟨(2i, 2j), (2i + 1, 2j + 1)⟩. Thus if either a = 1 or b = 1 then S^{(2)} · Ker ≃ ℤ_{2^{a}} × ℤ_{2^{b}}, Theorem 3 and Corollary 2 show r_{H} = 1. If a > 1, b > 1 then S^{(2)} · Ker = S^{(2)}, Theorem 3 and Corollary 2 show r_{H} = 2. If the unit index is 2 then S^{(2)} is the same as above, but |Ker| = 2. In order for the unit index to be 2, either d_{1} and d_{3} are principal divisors of k_{2} or d_{3} = −1 and 2 is a principal divisor of k_{2}. In the first case Ker ≃ ⟨(2^{a−1}, 2^{b−1})⟩. In the second case, since 2 is not totally ramified in K, it must be unramified in k_{1}. Hence Ker is the same as above. Theorem 3 and Corollary 2 show r_{H} = 2 if and only if Ker ⊆ S^{(2)} if and only if a = b = 1 or a > 1, b > 1. To complete the proof we show a = 1 if and only if b = 1. Since the unit index is 2, the discriminant of k_{2} is divisible by all three primes which ramify in K. Using the symmetry of k_{1} and k_{3} we may assume H_{1}^{(2)} ≃ ℤ_{2^{a}} and H_{2}^{(2)} ≃ ℤ_{2^{b}}. Thus the discriminant of k_{1} is divisible by exactly two primes l and r with l ≢ r (mod 4), where we choose l to be odd. Now a > 1 if and only if (\frac{r}{l}) = +1. From above we see r is not a principal divisor of k_{2}, so b > 1 if and only if (\frac{r}{l}) = +1.

Corollary 3. For each of the following values of d_{1}, d_{2} and d_{3}, r_{H} = 2. Moreover, if r_{H} = 2 and K satisfies the hypothesis of Theorem 5 or 6 then K is listed below. Here l, q, and r denote primes with l ≡ 1, q ≡ r ≡ 3 (mod 4).

d_{1}  d_{2} d_{3}  conditions
−2l 2  −l  l ≡ 1 (mod 8)
−2l 2l −1  l ≡ 1 (mod 8)
−l 2l −2 l ≡ 1 (mod 8)
−lq qr −lr (\frac{l}{q}) = (\frac{l}{r}) = +1
−lq lqr −r  (\frac{l}{r}) = +1
−2q qr −2r (\frac{2}{q}) = (\frac{2}{r}) = +1
−lq q  −l l ≡ 1 (mod 8), (\frac{l}{q}) = +1
−lq lq −1  l ≡ 1 (mod 8)
−l lq −q (\frac{l}{q}) = +1
−2l 2q −lq l ≡ 1 (mod 8), (\frac{l}{q}) = +1
−2l 2lq −q  (\frac{l}{q}) = +1
−lq 2lq −2  (\frac{2}{l}) = +1
−2q 2qr −r  (\frac{2}{r}) = +1

Proof The results follow from Theorems 5 and 6 and conditions for h_{i} ≡ 0 (mod 4), see Brown [3, 4, 2].

Theorem 7. If 2 is not totally ramified in K, t = 3, r_{a} = 2, and exactly one quadratic subfield of K has even class number then r_{H} = 2 for precisely the values listed below where l, q and r are primes with l ≡ q ≡ r ≡ 3 (mod 4):

d_{1} d_{2} d_{3} conditions
−lqr qr −l (\frac{l}{q}) = (\frac{l}{r}) = −1
−2lq lq −2 (\frac{2}{l}) = (\frac{l}{q}) = −1
−2lq 2q −l (\frac{2}{l}) = +1, (\frac{l}{q}) = −1
−lq q −l (\frac{2}{l}) = +1, (\frac{l}{q}) = −1

When these conditions are not satisfied r_{H} = 1.

Proof Since r_{a} = 2 and 2 is not totally ramified in K, q|d_{2} for some prime q ≡ 3 (mod 4).

Since t = 3, 2-rank H_{2} ≤ 1, so 2-rank H_{2} = 0. The only possible values for d_{1}, d_{2} and d_{3} other than those listed above are −lq, lq, −1. In all cases Ker = ⟨(l, 1, 1), (r, 1, 1)⟩ where r = 2 when it is not explicitly defined. In the first two lines the generating matrix for Ŝ ·K̂er

is M = (\begin{matrix}0 & 1 \\ x + y & x \\ y + 1 & w + 1 \end{matrix}), where the columns correspond to l and q respectively. The matrix

reduces to (\begin{matrix}0 & 1 \\ y + 1 & 0 \\ x + 1 & 0 \end{matrix}). In the last two lines the generating matrices are (\begin{matrix}0 & 1 \\ x + y + 1 & x \\ y & w \end{matrix}),

and (\begin{matrix}0 & 1 \\ x + 1 & x \\ y & w \end{matrix}), where the columns correspond to l and q. Both of these reduce to

(\begin{matrix}0 & 1 \\ x + 1 & 0 \\ y & 0 \end{matrix}). The results follow. In the case −lq, lq, −1, Ker = ⟨(l, 1, 1), (2, 1, 1)⟩ and

the generating matrix is (\begin{matrix}0 & 1 \\ 1 & x + 1 \\ y + w & y \end{matrix}), with columns corresponding to −1 and l. The

matrix reduces to (\begin{matrix}0 & 1 \\ 1 & 0 \\ y + w & 0 \end{matrix}). Thus r_{H} = 1 in this case.

Next we turn to the case where t = 3 and r_{a} = 3. Here 2 is not totally ramified in K and d_{2} is divisible by no prime congruent to 3 modulo 4. We consider separately the cases where each H_{i} is cyclic and where some H_{i} is noncyclic for i = 1, 2, 3. In the following theorems k_{1}, k_{2} and k_{3} will be defined by two ordered n-tuples where the first n-tuple defines d_{1}, d_{2} and d_{3} and the second n-tuple specifies congruence conditions on l, q, r etc. modulo 4.

Theorem 8. If t = 3 and r_{a} = 3 with each H_{i} cyclic then K and r_{H} are determined as follows: a (−lq, lr, −qr), (1, 2 or 3, 1 or 2), r_{H} = 3 − w − x − y + wxy.

b (−l, lr, −r), (1, 2, 1), r_{H} = 3 − w − x − y + wx + wxy.

Proof Since each H_{i} is cyclic and d_{2} is divisible by no prime congruent to 3 modulo 4, the discriminant of each k_{i} is divisible by exactly two primes. Since 2 is not totally ramified the only possible values are those listed above. In case (a) Ker is generated by ⟨(l, l, 1), (q, 1, q)⟩

and the matrix M reduces to (\begin{matrix}1 & 0 & 0 \\ 0 & y & w \\ 0 & x & x + w \end{matrix}). In case (b) Ker = ⟨(1, l, 1), (2, 1, 2)⟩ and M

reduces (\begin{matrix}1 & 0 & 0 \\ 0 & y & 0 \\ 0 & x & x + w \end{matrix}). The results follow.

Theorem 9. If t = 3 and r_{a} = 3 with some H_{i} noncyclic then K and r_{H} are determined as follows:

a (−lqr, lr, −q), (1, 2 or 3, 1 or 2), r_{H} = 3 − x − w.

b (−lr, lr, −1), (1, 2, 1), r_{H} = 3 − x − w + xw.

c (−lr, l, −r), (1, 2, 1) or (−lqr, l, −qr), (1 or 2, 2 or 3, 1 or 2), r_{H} = 3 − x − y.

Proof Since 2 is not totally ramified in K and no prime congruent to 3 (modulo 4) divides d_{2}, it is seen that H_{2} is cyclic and we may assume H_{1} is noncyclic. It follows that only the three cases listed above can occur. In each case we list generators for Ker and a matrix which M reduces to. The results follow immediately.

a ⟨(l, l, 1), (q, 1, 1)⟩, (\begin{matrix}1 & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & w \end{matrix})

b ⟨(l, l, 1), (2, 1, 1)⟩, (\begin{matrix}1 & 0 & 0 \\ 0 & x & w \\ 0 & 0 & 0 \end{matrix})

c ⟨(l, 1, 1), (q, 1, q)⟩, (\begin{matrix}1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & x \end{matrix})

When t = 4, 3 ≤ r_{a} ≤ 5 and 2 ≤ r_{H} ≤ r_{a}. We now determine r_{H} for all fields K with t = 4. The notation here is similar to that used in Theorems 8 and 9. Theorem 10. When r_{a} = 3, the fields K and the values of r_{H} are listed below:

a (−2lq, q, −2l), (1, 3), r_{H} = 3 − x − y + xy. b (−2lq, q, −2l), (3, 3), r_{H} = 3 − x̄ − y + x̄y. c (−2lq, 2q, −l), (1, 3), r_{H} = 3 − x − y + xy. d (−lq, 2q, −2l), (3, 3), r_{H} = 3 − x̄ − y + x̄y. e (−2lq, lq, −2), (1, 3), r_{H} = 3 − y. f (−2q, lq, −2l), (1, 3), r_{H} = 3 − x − y + xy. g (−2lq, 2lq, −1), (1, 3), r_{H} = 3 − y. h (−2lq, 2lq, −1), (3, 3), r_{H} = 2. i (−lq, 2lq, −2), (3, 3), r_{H} = 3 − w̄ − ȳ + w̄ȳ.

Ramona R Ranalli  Chapter 4. Fields with 2-Class Group ℤ_{2} × ℤ_{2}  28

j (−2q, 2lq, −l), (1, 3), r_{H} = 3 − x − y + xy.

Proof In order that r_{a} = 3, 2 must be totally ramified in K and there exists a prime q ≡ 3 (mod 4) which divides d_{2}. Hence there is exactly one other odd prime l which ramifies in K. The only possible fields are those listed above. In part (a), Ker = ⟨(2, 1, 1), (l, 1, 1), (1, 1, 2)⟩

and the matrix M reduces to (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & y \\ 0 & 0 & x \\ 0 & 0 & 0 \end{matrix}). Thus m = 2 + x + y − xy and the formula for

r_{H} follows from Theorem 4. The remaining parts are done similarly. Also, each case has been computer checked. We now will determine all fields K with 2-class group ℤ_{2} × ℤ_{2}. Theorem 11. The fields with 2-class group ℤ_{2} × ℤ_{2} are exactly those listed below:

a (−2lq, q, −2l), (1, 3), (\frac{2}{l}) = −1 and either (\frac{l}{q}) = −1 or (\frac{2}{q}) = −1. b (−2lq, q, −2l), (3, 3), (\frac{2}{l}) = (\frac{l}{q}) = −1, (\frac{2}{q}) = +1. c (−2lq, 2q, −l), (1, 3), (\frac{2}{l}) = −1 and either (\frac{l}{q}) = −1 or (\frac{2}{q}) = −1. d (−lq, 2q, −2l), (3, 3), (\frac{2}{l}) = −1 and either (\frac{l}{q}) = −1 or (\frac{2}{q}) = −1. e (−2lq, lq, −2), (1, 3), (\frac{2}{l}) = −1 and either (\frac{l}{q}) = −1 or (\frac{2}{q}) = −1. f (−2q, lq, −2l), (1, 3), (\frac{l}{q}) = +1, (\frac{2}{l}) = (\frac{2}{q}) = −1. g (−2lq, 2lq, −1), (1, 3), (\frac{2}{l}) = −1 and either (\frac{l}{q}) = −1 or (\frac{2}{q}) = −1. h (−2lq, 2lq, −1), (3, 3), (\frac{2}{l}) = (\frac{l}{q}) = −(\frac{2}{q}).

Ramona R Ranalli  Chapter 4. Fields with 2-Class Group ℤ_{2} × ℤ_{2}  29

i (−lq, 2lq, −2), (3, 3), (\frac{2}{l}) = (\frac{l}{q}) = −(\frac{2}{q}). j (−2q, 2lq, −l), (1, 3), (\frac{2}{l}) = (\frac{2}{q}) = −1, (\frac{l}{q}) = +1. k (−lqr, qr, −l), (3, 3, 3), (\frac{l}{q}) = (\frac{l}{r}) = −1, h_{1} ≡ 8 (mod 16). l (−2lq, 2q, −l), (3, 3), (\frac{2}{l}) = +1, (\frac{l}{q}) = −1, h_{1} ≡ 8 (mod 16). m (−2lq, lq, −2), (3, 3), (\frac{2}{l}) = (\frac{2}{q}) = −1, h_{1} ≡ 8 (mod 16). n (−lq, q, −l), (3, 3), (\frac{2}{l}) = +1, (\frac{l}{q}) = −1, h_{1} ≡ 8 (mod 16). o (−l, lr, −r), (1, 1), (\frac{l}{r}) = (\frac{2}{l}) = (\frac{2}{r}) = −1. p (−lqr, lr, −q), (1, 2 or 3, 1 or 2), (\frac{l}{r}) = −1, (\frac{q}{r}) ≠ (\frac{q}{l}). q (−lr, lr, −1), (1, 1), (\frac{l}{r}) = −1, (\frac{2}{l}) ≠ (\frac{2}{r}). r (−lr, l, −r), (1, 1), (\frac{2}{r}) = (\frac{l}{r}) = −1, (\frac{2}{l}) = +1. s (−lqr, l, −qr), (1, 2 or 3, 1 or 2), (\frac{q}{r}) = −1, (\frac{l}{q}) ≠ (\frac{l}{r}). t (−2qr, 2, −qr), (1, 3), (\frac{q}{r}) = −1, (\frac{2}{q}) ≠ (\frac{2}{r}). u (−lq, lqr, −r), (1, 3, 3), (\frac{l}{q}) = −1, (\frac{l}{r}) = +1. v (−lq, lq, −1), (1, 3), (\frac{2}{l}) = +1, (\frac{l}{q}) = −1. w (−l, lq, −q), (1, 3), (\frac{2}{l}) = −1, (\frac{l}{q}) = +1. x (−2l, 2lq, −q), (1, 3), (\frac{2}{l}) = −1, (\frac{l}{q}) = +1. y (−lq, 2lq, −2), (1, 3), (\frac{2}{l}) = +1, (\frac{l}{q}) = −1. z (−2l, 2lq, −q), (3, 3), (\frac{2}{l}) = −1, (\frac{2}{q}) = +1.

Proof Since h ≡ 4 (mod 8) and h = h_{1}h_{2}h_{3} or \frac{1}{2} h_{1}h_{2}h_{3} according as the unit index of K is 2 or 1, it follows that r_{a} = 2 or 3. The only possible fields come from Corollary 3 to Theorem 6 and Theorems 7, 8, 9 and 10 where r_{H} = 2. The results follow by using conditions from Brown [3, 4, 2, 5] to determine when h ≡ 4 (mod 8).

Fields with Class Numbers 4, 8 and

16.

Complete lists of all imaginary bicyclic biquadratic fields with class number 1 and 2 have already been obtained [5, 7]. In this chapter we calculate all imaginary bicyclic biquadratic fields with class groups of order four, eight and sixteen as well as compute the actual structure of each class group and give the results accordingly.

In order to determine all fields of class number 2^{n}, we use the well known class number formula

h = \frac{q_{o}}{2} h_{1}h_{2}h_{3}

where q_{o} = 1 or 2 is the index in the unit group of K of the subgroup of units contained in the quadratic subfields. Hence, if h = 2^{n}, then h_{1}h_{3} ≤ 2^{n+1}. Since we wish to find all fields K with h = 2^{n} where n ≤ 4, we need to know complete lists of imaginary quadratic fields of class number 2^{n} for n ≤ 5. Stark [12, 11] has shown that there are 9 such fields of class number 1,

ℚ(\sqrt{−d}) : d = 1, 2, 3, 7, 11, 19, 43, 67, 163

and 18 fields of class number 2,

ℚ(\sqrt{−d}) : d = 5, 6, 10, 13, 15, 22, 35, 37, 51, 58, 91, 115, 123, 187, 235, 267, 403, 427. Moreover, Arno [1] has shown that

ℚ(\sqrt{−d}) : d = 14, 17, 21, 30, 33, 34, 39, 42, 46, 55, 57, 70, 73,

78, 82, 85, 93, 97, 102, 130, 133, 142, 155, 177, 190, 193, 195, 203, 219,
253, 259, 291, 323, 355, 435, 483, 555, 595, 627, 667, 715, 723, 763, 795,
955, 1003, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555.

are the only imaginary quadratic fields of class number 4. While complete lists of imaginary quadratic fields of class number 8, 16 and 32 have not been determined, Buell [6] and again in more recent calculations has computed the class numbers of all imaginary quadratic fields with discriminant 2.2 billion. In fact, Hoffstein’s [8] bounds on the L-series show there can exist at most one imaginary field of class number 8, 16 or 32 with discriminant greater than 2.2 million. Therefore, Buell’s calculations almost certainly provide us with complete lists of necessary imaginary quadratic fields.

Now the unit index is 2 if and only if \sqrt{ωε} ∈ K where ω is a root of unity in a quadratic subfield of K and ε is the fundamental unit of the real quadratic subfield. Since K is an imaginary field \sqrt{ωε} can not be real so we can assume ω = −1 or i. If ε has norm −1 then the unit index is always 1. If ε has norm 1 then

\sqrt{ε} = \frac{(\sqrt{N(ε + 1)} + \sqrt{−N(ε − 1)}}{2}

where N is the norm function from k_{2} to ℚ. Now

−N(ε + 1)N(ε − 1) = −N(ε^{2} − 1) = u^{2}d_{2}

for some rational number u. The square free kernels of N(ε + 1) and −N(ε − 1) are called the principal divisors of k_{2}. Thus the unit index is 2 only when −d_{1} and −d_{3} are the principal divisors or d_{1} = −1 and 2 is a principal divisor. Moreover, the product of the two principal divisors is d_{2} or 4d_{2} where the latter value occurs only when d_{2} ≡ 3(mod4). Hence if some prime divides the discriminant of K but not that of k_{2} the unit index is always 1.

Since we may assume h_{1} ≤ h_{3}, we first let k_{1} and k_{3} run through all fields of class number 1, 2 and 4. If h_{3} = 8, 16 or 32 we only need to let k_{1} run through those fields with h_{1}h_{3} ≤ 32. We first estimate h_{2} using genus theory since this gives a lower bound for h_{2}, if the class formula shows h > 16 the field is discarded. Otherwise, h_{2} is determined to give the exact value of h.

When h = 4 or 8, a simple computation of the rank determines the structure of the class group. For fields with class number 16 the basic procedure is the same except that the structure of the class group is not always completely determined by the rank. If the class group has rank 2 there are two possible group structures. The final determination requires that we use Theorem 2. These computations were done on computer. Assuming that Buell’s lists [6] of imaginary quadratic fields of class number 8, 16 and 32 are complete, all fields K with class number 16 are listed in Table’s 6.6, 6.7, 6.8 , 6.9 and 6.10. Specifically, fields with class group ℤ_{16} are listed in Table 6.6, with class group ℤ_{8} × ℤ_{2} in Table 6.7, with class group ℤ_{4} × ℤ_{4} in Table 6.8, with class group ℤ_{4} × ℤ_{2} × ℤ_{2} in Table 6.9, and with class group ℤ_{2} × ℤ_{2} × ℤ_{2} × ℤ_{2} in Table 6.10. In all but Table 6.1 the fields are listed by their conductor, f.

Table 6.1: Fields where H ≃ ℤ_{2} × ℤ_{2}
d_{1}  d_{2} d_{3}  d_{1}  d_{2} d_{3}
1,a  −30  3 −10  36 −1947 177 −11
2  −70  7 −10  37 −2163 721 −3
3 −190 19 −10  38 −2667 889 −3
4,b  −42  7 −6  39 −5467 781 −7
5,c  −30  6 −5  40,l −154 22 −7
6  −70 14 −5  41 −322 46 −7
7  −78  6 −13  42  −66 33 −2
8 −190 38 −5  43,m −114 57 −2
9,d  −21 14 −6  44 −258 129 −2
10  −33 22 −6  45 −418 209 −2
11  −33  6 −22  46 −498 249 −2
12  −57 38 −6  47  −77 11 −7
13  −93 62 −6  48,n −301 43 −7
14 −177 118 −6  49  −5 65 −13
15 −253 46 −22  50,o  −5 185 −37
16,e  −30 15 −2  51  −13 481 −37
17  −70 35 −2  52 −195 65 −3
18  −78 39 −2  53,p −555 185 −3
19 −190 95 −2  54 −715 65 −11
20,f  −22 55 −10  55  −70 10 −7
21,g  −30 30 −1  56  −78 26 −3
22  −70 70 −1  57 −190 10 −19
23  −78 78 −1  58  −85 85 −1
24 −190 190 −1  59  −85 17 −5
25  −42 42 −1  60 −195 13 −15
26,h  −21 42 −2  61 −435 29 −15
27,i  −93 186 −2  62 −555 37 −15
28 −133 266 −2  63 −1435 41 −35
29  −22 110 −5  64  −78 13 −6
30  −6 78 −13  65 −102 17 −6
31  −6 222 −37  66  −70  2 −35
32  −22 814 −37  67,t −102  2 −51
33 −651 217 −3  68  −15 165 −11
34,k −1659 553 −3  69,u  −15 285 −19
35 −1771 253 −7  70  −35 385 −11

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Fields where H ≃ ℤ_{2} × ℤ_{2}

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d_{1}  d_{2}  d_{3}  d_{1}  d_{2}  d_{3}
71 −35  665 −19  92,v −123 123 −1
72 −51  969 −19  93 −187 187 −1
73 −51 2193 −43  94 −267 267 −1
74 −51 3417 −67  95  −5  55 −11
75 −91  273  −3  96,w  −5  95 −19
76 −91 3913 −43  97 −13  39 −3
77 −115 1265 −11  98 −13 559 −43
78 −115 2185 −19  99 −37 111 −3
79 −123 5289 −43  100 −37 259 −7
80 −123 20049 −163  101 −37 407 −11
81 −187 3553 −19  102 −37 2479 −67
82 −187 8041 −43  103 −10 110 −11
83 −187 12529 −67  104,x −10 190 −19
84 −235 2585 −11  105 −58 406 −7
85 −235 4465 −19  106 −58 3886 −67
86 −267 2937 −11  107 −51 102 −2
87 −403 1209 −3 108,y −123 246 −2
88 −403 17329 −43  109 −187 374 −2
89 −427 1281  −3  110 −267 534 −2
90 −427 69601 −163  111  −6  42 −7
91 −51  51  −1  112 −22 154 −7

Table 6.2: Fields where H ≃ ℤ_{4}

f  d_{1} d_{2}  d_{3}  f  d_{1} d_{2}  d_{3}
56 −2  7 −14  385 −35 77 −55
56 −1 14 −14  385 −7 385 −55
95 −19 5 −95 408 −34 6 −51
111 −3 37 −111  408 −3 34 −102
120 −15 10  −6  408 −3 102 −34
120 −15  2 −30  429 −11 429 −39
120 −6  5  30  440 −10 22 −55
120 −3 10 −30  440 −2 110 −55
156 −13 3 −39 452 −1 113 −113
156 −1 39 −39  455 −35 65 −91
164 −1 41 −41 456 −19 6 −114
165 −15 33 −55  456 −3 38 −114
165 −3 165 −55  465 −15 93 −155
168 −6 21 −14  465 −3 465 −155
168 −3 42 −14  476 −7 119 −17
183 −3 61 −183  520 −10 13 −130
184 −2 23 −46  520 −2 65 −130
184 −1 46 −46  548 −1 137 −137
195 −39  5 −195  552 −3 46 −138
204 −17  3 −51  552 −6 69 −46
204 −3 51 −17  552 −3 138 −46
220 −5 11 −55  552 −2 69 −138
220 −1 55 −55  564 −1 141 −141
255 −15 85 −51  564 −3 47 −141
264 −11 6 −66 568 −2 71 −142
264 −3 22 −66  579 −3 193 −579
273 −91 21 −39  583 −11 53 −583
273 −7 273 −39  595 −35 17 −595
276 −1 69 −69  595 −7 85 −595
276 −3 23 −69  609 −3 609 −203
280 −7 10 −70  615 −15 205 −123
308 −11 7 −77 616 −11 14 −154
308 −1 77 −77  616 −22 77 −14
312 −2 78 −39  616 −11 154 −14
371 −7 53 −371  616 −2 77 −154

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Fields where H ≃ ℤ_{4}

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f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
620 −5  31 −155  1128  −2 141 −282
620 −1 155 −155  1204  −1 301 −301
651 −7  93 −651  1204 −43  7 −301
712 −2  89 −178  1209 −403  93  −39
741 −19 741  −39  1240 −10  62 −155
748 −17  11 −187  1240  −2 310 −155
748 −11 187  −17  1252  −1 313 −313
795 −15  53 −795  1265 −115 253  −55
795 −3 265 −795  1288  −2 161 −322
812 −1 203 −203  1299  −3 433 −1299
868 −1 217 −217  1348  −1 337 −337
816 −51  34  −6  1420  −5  71 −355
852 −3  71 −213  1420  −1 355 −355
868 −7  31 −217  1435  −7 205 −1435
904 −2 113 −226  1496 −187  34  −22
939 −3 313 −939  1496 −34  22 −187
952 −7 238  −34  1496 −11 374  −34
969 −51  57 −323  1533  −7 1533 −219
969 −3 969 −323  1624 −58  14 −203
979 −11  89 −979  1624  −2 406 −203
984 −82  6 −123  1659  −7 237 −1659
984 −3 246  −82  1672 −19  22 −418
987 −7 141 −987  1672 −11  38 −418
987 −3 329 −987  1704  −6 213 −142
1032 −43  6 −258  1704  −3 426 −142
1032 −3  86 −258  1771 −11 161 −1771
1036 −37  7 −259  1803  −3 601 −1803
1036 −1 259 −259  1828  −1 457 −457
1043 −7 149 −1043  1864  −2 233 −466
1064 −19 266  −14  1939  −7 277 −1939
1065 −15 213 −355  1947  −3 649 −1947
1065 −3 1065 −355  1992  −3 166 −498
1085 −35 217 −155  2001  −3 2001 −667
1085 −7 1085 −155  2024 −22 253  −46
1128 −3  94 −282  2037  −7 2037 −291

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Fields where H ≃ ℤ_{4}

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f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
2044  −7 511  −73  3507  −3 1169 −2507
2072  −2 518 −259  3553 −187 209 −323
2135 −35 305 −427  3553 −11 3553 −323
2136 −267 178  −6  3565 −115 713 −155
2139  −3 713 −2139  3608 −11 902  −82
2163  −7 309 −2163  3685 −67 3685  −55
2212  −1 553 −553  3729  −3 3729 −1243
2248  −2 281 −562  3752 −67 938  −14
2261  −7 2261 −323  3787  −7 541 −3787
2307  −3 769 −2307  3820  −5 191 −955
2365 −43 2365  −55  3820  −1 955 −955
2408 −43 602  −14  3857 −19 3857 −203
2409 −11 2409 −219  3883 −11 353 −3883
2451 −43  57 −2451  3963  −3 1321 −3963
2451 −19 129 −2451  4108  −1 1027 −1027
2485 −35 497 −355  4123 −19 217 −4123
2485  −7 2485 −355  4123  −7 589 −4123
2585 −235 517  −55  4233 −51 249 −1411
2611  −7 373 −2611  4233  −3 4233 −1411
2613 −67 2613  −39  4323 −11 393 −4323
2632  −7  94 −658  4323  −3 1441 −4323
2632  −2 329 −658  4521  −3 4521 −1507
2667  −7 381 −2667  4539 −51 1513 −267
2668  −1 667 −667  4665 −15 933 −1555
2716  −7 679  −97  4665  −3 4665 −1555
2840  −2 710 −355  4921 −19 4921 −259
2865 −15 573 −955  5061  −7 5061 −723
2865  −3 2865 −955  5336 −58  46 −667
2947  −7 421 −2947  5336  −2 1334 −667
3009 −51 177 −1003  5467 −11 497 −5467
3009  −3 3009 −1003  5947 −19 313 −5947
3052  −1 763 −763  6104  −2 1526 −763
3212 −11 803  −73  6220  −5 311 −1555
3243  −3 1081 −3243  6222  −1 1555 −1555
3507  −7 501 −3507  6232 −19 1558  −82

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Fields where H ≃ ℤ_{4}

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f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
6248 −22  781 −142  12440  −2 3110 −1555
6248 −11 1562 −142  12556 −43 3139  −73
6357 −163 6357  −39  12673 −19 12673 −667
6665 −43 6665 −155  13497 −11 13497 −1227
6685 −35 1337 −955  13737 −19 13737 −723
7021  −7 7021 −1003  14497 −19 14497 −763
7189 −91  553 −1027  15521 −187  913 −1411
7189  −7 7189 −1027  16685 −235 3337 −355
7285 −235 1457 −155  19497 −67 19497 −291
7337 −11 7337 −667  19513 −19 19513 −1027
7372 −19 1843  −97  21965 −115 4393 −955
7640 −10  382 −955  21976 −67 5494  −82
7640  −2 1910 −955  23313 −19 23313 −1227
7912 −43 1978  −46  23785 −67 23785 −355
7953 −11 7953 −723  24424 −43 6106 −142
8165 −115 1633 −355  25265 −163 25265 −155
8216  −2 2054 −1027  28681 −43 28681 −667
8393 −11 8393 −763  29992 −163 7498  −46
8492 −11 2123 −193  31089 −43 31089 −723
8589 −7 8589 −1227 31837 −403 2449 −1027
8729 −43 8729 −203  33089 −163 33089 −203
8965 −163 8965  −55  35697 −163 35697 −219
9128 −163 2282  −14  35765 −115 7153 −1555
9417 −43 9417 −219  38056 −67 9514 −142
9709  −7 9709 −1387  41065 −43 41065 −955
9877  −7 9877 −1411  42217 −163 42217 −259
10385 −67 10385 −155  44885 −235 8977 −955
10792 −19 2698 −142 47596 −163 11899 −73
10947 −123 3649 −267  51121 −67 51121 −763
11033 −187 649 −1003 52649 −163 52649 −323
11033 −11 11033 −1003  52761 −43 52761 −1227
11084 −163 2771 −17 53449 −43 53449 −1243
11297 −11 11297 −1027  57865 −163 57865 −355
12328 −67 3082  −46  59641 −43 59641 −1387
12440 −10  622 −1555  64801 −43 64801 −1507

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Fields where H ≃ ℤ_{4}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
66865 −43 66865 −1555  124369 −163 124369 −763
68809 −67 68809 −1027  155665 −163 155665 −955
73085 −235 14617 −1555  163489 −163 163489 −1003
82209 −67 82209 −1227  167401 −163 167401 −1027
83281 −67 83281 −1243  226081 −163 226081 −1387
100969 −67 100969 −1507  229993 −163 229993 −1411
104185 −67 104185 −1555  245641 −163 245641 −1507
108721 −163 108721 −667  253465 −163 253465 −1555

Table 6.3: Fields where H ≃ ℤ_{8}

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
195 −15 65 −39  728 −91  26  −14
248  −2 31 −62  732  −1 183 −183
248  −1 62 −62  744  −6  93  −62
260 −13  5 −65  744  −3 186  −62
260  −5 13 −65  760  −2 190  −95
280 −35 10 −14  885 −15 177 −295
280 −14  5 −70  885  −3 885 −295
285 −15 57 −95  888  −2 222 −111
285  −3 285 −95  903 −43  21 −903
312 −39  2 −78  903  −7 129 −903
312  −6 26 −39  920 −115  10  −46
340 −17  5 −85  935 −187  85  −55
340  −5 85 −17  984  −6  82 −123
376  −2 47 −94  1016  −2 127 −254
376  −1 94 −94  1023  −3 341 −1023
380  −5 19 −95  1047  −3 349 −1047
380  −1 95 −95  1060  −5  53 −265
399 −19 21 −399  1095 −15 365 −219
399  −7 57 −399  1113  −3 1113 −371
407 −11 37 −407  1128  −3 282  −94
435  −3 145 −435  1139 −67  17 −1139
440 −22 10 −55  1148  −7 287  −41
444 −37  3 −111  1159 −19  61 −1159
444  −1 111 −111  1160 −10 145  −58
471  −3 157 −471  1180  −1 295 −295
492 −123  3 −41  1185 −15 237 −395
492  −3 123 −41  1185  −3 1185 −395
559 −43 13 −559  1196 −13  23 −299
584  −2 73 −146  1196  −1 299 −299
632  −1 158 −158  1281 −427  21 −183
644  −1 161 −161  1281  −7 1281 −183
665 −35 133 −95  1295 −35 185 −259
665  −7 665 −95  1335 −15 445 −267
680 −10 85 −34  1356  −3 339 −113
715 −55 13 −715  1379  −7 197 −1379

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
1412  −1 353 −353  2211 −67  33 −2211
1416  −3 118 −354  2211  −3 737 −2211
1455 −15 485 −291  2292  −3 191 −573
1460  −5 365  −73  2292  −1 573 −573
1464  −2 366 −183  2332  −1 583 −583
1484  −1 371 −371  2356 −19  31 −589
1528  −2 191 −382  2356  −1 589 −589
1528  −1 382 −382  2360 −10 118 −295
1580  −1 395 −395  2360  −2 590 −295
1599 −123 533  −39  2392  −2 598 −299
1608 −67  6 −402  2516 −37 629  −17
1608  −3 134 −402  2568  −3 214 −642
1636  −1 409 −409  2712  −3 678 −226
1644  −3 411 −137  2728 −22 341  −62
1749  −3 1749 −583  2728 −11 682  −62
1804 −11 451  −41  2739 −11 249 −2739
1876 −67  7 −469  2739  −3 913 −2739
1896  −6 237 −158  2765  −7 2765 −395
1896  −3 474 −158  2811  −3 937 −2811
1912  −2 239 −478  2868  −3 239 −717
1912  −1 478 −478  2868  −1 717 −717
1940  −5 485  −97  2884  −7 103 −721
1983  −3 661 −1983  2884  −1 721 −721
2004  −3 167 −501  2937 −267  33 −979
2004  −1 501 −501  2937  −3 2937 −979
2013 −11 2013 −183  2968  −2 742 −371
2015 −403  65 −155  2985 −15 597 −995
2019  −3 673 −2019  2985  −3 2985 −995
2065 −35 413 −295  3063  −3 1021 −3063
2065  −7 2065 −295  3116 −19 779  −41
2093 −91 161 −299  3144  −3 262 −786
2093  −7 2093 −299  3160 −10 158 −395
2109 −19 2109 −111  3160  −2 790 −395
2136 −267  6 −178  3171  −7 453 −3171
2136  −3 534 −178  3289 −11 3289 −299

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
3336  −3 278 −834  4172 −1 1043 −1043
3363 −19 177 −3363  4179 −7 597 −4179
3363 −3 1121 −3363 4228 −1 1057 −1057
3367 −91 481 −259  4324 −1 1081 −1081
3416 −427 122 −14 4427 −19 233 −4427
3443 −11 313 −3443  4584 −6 573 −382
3448  −2 431 −862  4584 −3 1146 −382
3448  −1 862 −862  4596 −3 383 −1149
3471 −267 1157  −39  4596 −1 1149 −1149
3531  −3 1177 −3531  4612 −1 1153 −1153
3556  −1 889 −889  4664 −2 1166 −583
3585 −15 717 −1195  4683 −7 669 −4683
3603  −3 1201 −3603  4687 −43 109 −4687
3723 −51 1241 −219  4712 −19  62 −1178
3732  −3 311 −933  4712 −19 1178  −62
3732  −1 933 −933  4712 −2 589 −1178
3752 −67  14 −938  4773 −43 4773 −111
3784 −43  22 −946  4780 −5 239 −1195
3784 −11  86 −946  4780 −1 1195 −1195
3796 −13 949  −73  4804 −1 1201 −1201
3819 −67  57 −3819  4899 −3 1633 −4899
3819 −19 201 −3819  4939 −11 449 −4939
3832  −2 479 −958  4947 −51 1649 −291
3832  −1 958 −958  4971 −3 1657 −4971
3860  −5 965 −193  4984 −7 1246 −178
3944 −58 493  −34  5044 −13 1261  −97
3976  −2 497 −994  5356 −13 103 −1339
3980  −5 199 −995  5385 −3 5385 −1795
3980  −1 995 −995  5403 −3 1801 −5403
4008  −3 334 −1002  5428 −1 1357 −1357
4008  −2 501 −1002  5551 −91 793 −427
4085 −43 4085  −95  5572 −7 199 −1393
4136 −22 517  −94  5592 −3 1398 −466
4136 −11 1034  −94  5681 −19 5681 −299
4171 −43  97 −4171  5736 −6 717 −478

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
5736  −3 1434 −478  7257  −3 7257 −2419
5752  −2 719 −1438  7288  −2 911 −1822
5752  −1 1438 −1438  7288  −1 1822 −1822
5896 −67  22 −1474  7368  −3 614 −1842
5896 −11 134 −1474  7491 −11 681 −7491
5907  −3 1969 −5907  7491  −3 2497 −7491
5908  −1 1477 −1477  7588  −7 271 −1897
5992  −2 749 −1498  7756  −1 1939 −1939
6099  −3 2033 −6099  7819  −7 1117 −7819
6177  −3 6177 −2059  7828 −19 103 −1957
6312  −3 526 −1578  7869 −43 7869 −183
6312  −2 789 −1578  7923 −19 417 −7923
6365 −67 6365  −95  7960  −2 1990 −995
6369 −11 6369 −579  8043  −7 1149 −8043
6532  −1 1633 −1633  8043  −3 2681 −8043
6573  −7 6573 −939  8236  −1 2059 −2059
6604 −13 127 −1651  8283  −3 2761 −8283
6604  −1 1651 −1651  8299 −43 193 −8299
6628  −1 1657 −1657  8308 −67  31 −2077
6744  −3 1686 −562  8308  −1 2077 −2077
6785 −115 1357 −295  8344  −2 2086 −1043
6792  −3 566 −1698  8365 −35 1673 −1195
6853  −7 6853 −979  8365  −7 8365 −1195
6923 −43 161 −6923  8452  −1 2113 −2113
6952 −22 869 −158  8481  −3 8481 −2827
6952 −11 158 −1738  8548  −1 2137 −2137
6952 −11 1738 −158  8588 −19 2147 −113
6952  −2 869 −1738  8643 −67 129 −8643
6965  −7 6965 −995  8643 −43 201 −8643
6969  −3 6969 −2323  8683 −19 457 −8683
7051 −11 641 −7051  8764  −7 2191 −313
7144 −19 1786  −94  8859  −3 2953 −8859
7180  −1 1795 −1795  8968 −19 118 −2242
7185  −3 7185 −2395  8985  −3 8985 −2995
7257 −123 177 −2419  9085 −115 1817 −395

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
9093  −7 9093 −1299  11512  −2 1439 −2878
9208  −2 1151 −2302  11523  −3 3841 −11523
9219  −7 1317 −9219  11557 −91  889 −1651
9269 −403  713  −299  11557  −7 11557 −1651
9292  −1 2323 −2323  11572 −11  263 −2893
9373 −91  721 −1339  11572  −1 2893 −2893
9373  −7 9373 −1339  11649  −3 11649 −3883
9507  −3 3169 −9507  11788  −1 2947 −2947
9508  −1 2377 −2377  11848  −2 1481 −2962
9512 −58 1189  −82  11931 −123 3977  −291
9560  −2 2390 −1195  11980  −5  599 −2995
9580  −5  479 −2395  11980  −1 2995 −2995
9580  −1 2395 −2395  12008 −19 3002  −158
9919 −91 1417  −763  12027 −19  633 −12027
10036 −13 2509  −193  12117  −7 12117 −1731
10203 −19  537 −10203  12261 −67 12261  −183
10203  −3 3401 −10203  12383 −427 1769  −203
10209 −123  249 −3403  12387  −3 4129 −12387
10209  −3 10209 −3403  12523  −7 1789 −12523
10227  −7 1461 −10227  12565  −7 12565 −1795
10312  −2 1289 −2578  12585  −3 12585 −4195
10344  −3 2586  −862  12621  −7 12621 −1803
10348 −13  199 −2587  12747  −7 1821 −12747
10348  −1 2587 −2587  12772  −1 3193 −3193
10444  −1 2611 −2611  12801 −51  753 −4267
10563  −3 3521 −10563  12801  −3 12801 −4267
10587  −3 3529 −10587  12859 −11 1169 −12859
10785  −3 10785 −3595  12859  −7 1837 −12859
10843  −7 1549 −10843  12868  −1 3217 −3217
10988 −67 2747  −41  13123 −11 1193 −13123
11001 −19 11001  −579  13161  −3 13161 −4387
11077 −19 11077  −583  13208  −2 3302 −1651
11092  −1 2773 −2773  13288 −11  302 −3322
11179  −7 1597 −11179  13288  −2 1661 −3322
11473 −11 11473 −1043  13363  −7 1909 −13363

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
13528 −19 3382  −178  18584  −2 4646 −2323
13827  −3 4609 −13827  18601 −19 18601  −979
13865 −235 2773  −295  18988  −1 4747 −4747
14241  −3 14241 −4747  19160 −10  958 −2395
14356 −37 3589  −97  19160  −2 4790 −2395
14360 −10  718 −1795  19436 −43 4859  −113
14371  −7 2053 −14371  19765 −67 19765  −295
14380  −5  719 −3595  19789  −7 19789 −2827
14403  −3 4801 −14403  19833 −11 19833 −1803
14707 −11 1337 −14707  19987 −11 1817 −19987
14729 −11 14729 −1339  20108 −11 5027  −457
14828 −11 3707  −337  20504 −11 5126  −466
15148  −1 3787 −3787  20696  −2 5174 −2587
15387  −3 5129 −15387  20888  −2 5222 −2611
15485 −163 15485  −95  20965  −7 20965 −2995
15512  −2 3878 −1939  21032 −22 2629  −478
15547  −7 2221 −15547  21032 −11 5258  −478
15799 −427 2257  −259  21131 −187 1921 −1243
16149  −7 16149 −2307  21329 −11 21329 −1939
16156  −7 4039  −577  22348 −37  151 −5587
16168 −43 4042  −94  22348  −1 5587 −5587
16261  −7 16261 −2323  22649 −11 22649 −2059
16347  −3 5449 −16347  22828  −1 5707 −5707
16472  −2 4118 −2059  23564 −43 5891  −137
16616 −67 4154  −62  23576  −2 5894 −2947
16723  −7 2389 −16723  23821  −7 23821 −3403
16765  −7 16765 −2395  23960 −10 1198 −2995
16780  −1 4195 −4195  23960  −2 5990 −2995
16808 −11 4202  −382  24681 −19 24681 −1299
16933  −7 16933 −2419  24728 −11 6182  −562
16985 −43 16985  −395  24857 −67 24857  −371
17347 −11 1577 −17347  25165  −7 25165 −3595
18093 −163 18093  −111  25192 −67 6298  −94
18161 −11 18161 −1651  25267 −11 2297 −25267
18403 −11 1673 −18403  25441 −19 25441 −1339

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
25553 −11 25553 −2323  41629  −7 41629 −5947
25612 −19 6403  −337  41657 −11 41657 −3787
26465 −67 26465  −395  42097 −43 42097 −979
26609 −11 26609 −2419  42344 −67 10586 −158
27176 −43 6794  −158  42785 −43 42785 −995
27307  −7 3901 −27307  44849 −43 44849 −1043
27741  −7 27741 −3963  45961 −19 45961 −2419
28457 −11 28457 −2587  46937 −187 2761 −4267
28564 −37 7141 −193 46937 −11 46937 −4267
28721 −11 28721 −2611  48085 −163 48085 −295
28760 −10 1438 −3595  48257 −11 48257 −4387
28760 −2 7190 −3595 48737 −163 48737 −299
28963 −11 2633 −28963  49153 −19 49153 −2587
29032 −19 7258  −382  51385 −43 51385 −1195
29365  −7 29365 −4195  52217 −11 52217 −4747
29869  −7 29869 −4267  53273 −11 53273 −4843
30296  −2 7574 −3787  53713 −19 53713 −2827
30616 −43 7654  −178  53836 −43 13459 −313
30709  −7 30709 −4387  55993 −19 55993 −2947
33229  −7 33229 −4747  60568 −67 15142 −226
33560  −2 8390 −4195  61288 −163 15322  −94
34069  −7 34069 −4867  62777 −11 62777 −5707
36328 −19 9082  −478  64347 −267 21449 −723
36716 −67 9179  −137  64385 −163 64385 −395
37433 −11 37433 −3403  64657 −19 64657 −3403
37928 −11 9482  −862  65512 −19 16378 −862
37976  −2 9494 −4747  66665 −67 66665 −995
38744 −58  334 −4843  75297 −19 75297 −3963
38744  −2 9686 −4843  77185 −43 77185 −1795
38936  −2 9734 −4867  78604 −43 19651 −457
39121 −19 39121 −2059  80065 −67 80065 −1195
39949 −91 3073 −5707  83353 −19 83353 −4387
39949  −7 39949 −5707  83884 −67 20971 −313
40377 −43 40377  −939  87033 −67 87033 −1299
41509 −403 3193 −1339  89324 −163 22331 −137

continued on following page

Fields where H ≃ ℤ_{8}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
89713 −67 89713 −1339  218257 −163 218257 −1339
90316 −67 22579 −337  231016 −67 57754 −862
92017 −19 92017 −4843  240865 −67 240865 −3595
94377 −163 94377 −579  249064 −163 62266 −382
99244 −43 24811 −577  253729 −67 253729 −3787
102376 −67 25594 −382  255721 −43 255721 −5947
102985 −43 102985 −2395  260161 −67 260161 −3883
103016 −163 25754 −158  265521 −67 265521 −3963
106153 −19 106153 −5587  269113 −163 269113 −1651
110617 −67 110617 −1651  292585 −163 292585 −1795
112273 −43 112273 −2611  293889 −163 293889 −1803
115977 −67 115977 −1731  303832 −163 75958 −466
116056 −163 29014 −178  316057 −163 316057 −1939
120265 −67 120265 −1795  335617 −163 335617 −2059
121561 −43 121561 −2827  376041 −163 376041 −2307
124888 −67 31222 −466  378649 −163 378649 −2323
126721 −43 126721 −2947  382369 −67 382369 −5707
128104 −67 32026 −478  390385 −163 390385 −2395
128785 −43 128785 −2995  398449 −67 398449 −5947
148264 −43 37066 −862  421681 −163 421681 −2587
150616 −67 37654 −562  488185 −163 488185 −2995
154569 −67 154569 −2307  562024 −163 140506 −862
154585 −43 154585 −3595  585985 −163 585985 −3595
154636 −67 38659 −577  617281 −163 617281 −3787
155641 −67 155641 −2323  632929 −163 632929 −3883
159577 −163 159577 −979  645969 −163 645969 −3963
162073 −67 162073 −2419  683785 −163 683785 −4195
162185 −163 162185 −995  695521 −163 695521 −4267
170009 −163 170009 −1043  789409 −163 789409 −4843
173329 −67 173329 −2587  793321 −163 793321 −4867
174937 −67 174937 −2611  910681 −163 910681 −5587
194785 −163 194785 −1195  930241 −163 930241 −5707

Table 6.4: Fields where H ≃ ℤ_{4} × ℤ_{2}
f  d_{1} d_{2}  d_{3}  f  d_{1} d_{2}  d_{3}
120 −6 15 −10  616 −22 14 −77
120 −5 30  −6  616 −2 154 −77
136 −17  2 −34  616 −1 154 −154
136 −2 34 −17  644 −7 23 −161
136 −1 34 −34  660 −15 11 −165
168 −14  3 −42  660 −15 55 −33
168 −14  6 −21  660 −5 33 −165
260 −1 65 −65  660 −5 165 −33
264 −22 3 −66 660 −3 55 −165
264 −6 11 −66  660 −1 165 −165
328 −1 82 −82  663 −51 221 −39
399 −3 133 −399  680 −10 34 −85
408 −34  3 −102  696 −6 87 −58
408 −17  6 −102  776 −2 194 −97
408 −6 51 −34  777 −3 777 −259
408 −6 102 −17  780 −13 195 −15
420 −35 3 −105 780 −5 39 −195
420 −35 15 −21  820 −1 205 −205
420 −15 7 −105 840 −35 6 −210
420 −15 35 −21  840 −35 30 −42
420 −7 15 −105  840 −35 42 −30
420 −5 21 −105  840 −30 21 −70
420 −3 35 −105  840 −15 14 −210
420 −1 105 −105  840 −15 42 −70
456 −6 19 −114  840 −15 70 −42
520 −13 10 −130  840 −10 21 −210
520 −10 130 −13  840 −10 105 −42
520 −5 26 −130  840 −7 30 −210
552 −6 23 −138  840 −6 105 −70
552 −6 46 −69  840 −6 210 −35
552 −2 138 −69  840 −3 70 −210
552 −1 138 −138  840 −3 210 −70
580 −5 29 −145  840 −2 105 −210
584 −2 146 −73  876 −73  3 −219
616 −22  7 −154  876 −3 219 −73

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
876 −1 219 −219  1240 −10  31 −310
903 −3 301 −903  1240  −5  62 −310
915 −3 305 −915  1240  −2 155 −310
924 −21  77  −33  1240  −1 310 −310
924 −11 231  −21  1288 −14 161 −46
952 −7  34 −238  1288  −7 322 −46
984 −6 123  −82  1292 −19 323 −17
1020 −51  15  −85  1292 −17  19 −323
1020 −15  51  −85  1292  −1 323 −323
1020 −5 255  −51  1320 −22 165 −30
1020 −3 255  −85  1320 −15  22 −330
1023 −11  93 −1023  1320 −15 330 −22
1032 −6  43 −258  1320  −3 110 −330
1045 −19 1045  −55  1365 −91 105 −195
1060 −1 265 −265  1365 −35 273 −195
1064 −14  38 −133  1365 −15 1365 −91
1092 −91  3 −273  1365  −7 1365 −195
1092 −91  39  −21  1380 −15  23 −345
1092 −13  21 −273  1380  −3 115 −345
1092 −13 273  −21  1380  −1 345 −345
1092 −7  39 −273  1416  −2 177 −354
1092 −1 273 −273  1428 −51  7 −357
1128 −6  47 −282  1428 −51 119 −21
1128 −2 282 −141  1428  −7  51 −357
1128 −1 282 −282  1428  −3 119 −357
1131 −3 377 −1131  1480 −10 370 −37
1140 −5 285  −57  1496 −22 187 −34
1144 −13 286  −22  1496 −22 374 −17
1155 −35  33 −1155  1540  −5  77 −385
1155 −15  77 −1155  1540  −1 385 −385
1155 −7 165 −1155  1544  −2 386 −193
1155 −3 385 −1155  1560 −130  6 −195
1160 −5 290  −58  1560 −15  78 −130
1164 −97  3 −291  1560 −10  78 −195
1164 −3 291  −97  1560  −2 390 −195

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
1596 −57 21 −133  2184 −91  42  −78
1608  −2 201 −402  2184 −91  78  −42
1624 −58 203  −14  2211 −11 201 −2211
1635  −3 545 −1635  2220 −15 555  −37
1640 −10 205  −82  2220  −5 111 −555
1672 −22 19 −418  2244 −187  51  −33
1704  −2 426 −213  2244 −51 187  −33
1705 −55 341 −155  2280 −30  57 −190
1716 −13 429  −33  2280 −15 114 −190
1740  −5 87 −435  2280  −6 285 −190
1752  −6 146 −219  2280  −3 570 −190
1752  −3 146 −438  2289  −3 2289 −763
1752  −2 438 −219  2328  −6 194 −291
1768  −2 221 −442  2328  −2 582 −291
1785 −51 105 −595  2380 −85  7 −595
1785 −15 357 −595  2380 −35 119  −85
1820 −13 455  −35  2380  −7 595  −85
1820  −5 455  −91  2380  −5 119 −595
1848 −22 21 −462  2392 −13  46 −598
1848 −11 42 −462  2392 −13 598  −46
1848 −11 462  −42  2392  −2 299 −598
1848  −6 77 −462  2392  −1 598 −598
1848  −3 154 −462  2408  −2 602 −301
1860  −5 465  −93  2415 −115 105 −483
1876  −7 67 −469  2415 −35 345 −483
1932  −1 483 −483  2415 −15 805 −483
1992  −6 83 −498  2508 −57  11 −627
1995 −35 57 −1995  2508 −33  19 −627
1995  −7 285 −1995  2508 −33 209  −57
2020  −5 101 −505  2508 −19 627  −33
2024 −46 22 −253  2508 −11 627  −57
2040 −10 510  −51  2552 −22 319  −58
2072 −37 518  −14  2584 −34  38 −323
2145 −195 33 −715  2604  −7 651  −93
2145 −15 429 −715  2639 −91 377 −203

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
2660 −35  95 −133  3540 −15 295 −177
2660  −5 665 −133  3540  −5 885 −177
2760 −115 138  −30  3556  −7 127 −889
2760  −6 690 −115  3608 −22 451  −82
2805 −15 2805 −187  3612 −43 903  −21
2840 −10 355 −142  3615 −15 1205 −723
2840 −5 710 −142 3640 −91 70 −130
2849 −11 2849 −259  3640 −35 182 −130
2856 −51 238  −42  3740 −187  55  −85
2860 −13  55 −715  3752  −7 134 −938
2892  −1 723 −723  3815 −35 545 −763
2924 −43 731  −17  3819  −3 1273 −3819
2945 −19 2945 −155  3864  −2 966 −483
2964 −13 741  −57  3905 −55 781 −355
2968  −2 371 −742  3905 −11 3905 −355
2968  −1 742 −742  3976 −14 497 −142
3003 −91  33 −3003  4012 −17  59 −1003
3003 −11 273 −3003  4092 −33 341  −93
3003  −7 429 −3003  4161 −219  57 −1387
3016 −13 754  −58  4228  −7 151 −1057
3036  −3 759 −253  4268 −11 1067  −97
3080 −22 385  −70  4305 −123 105 −1435
3081 −39 237 −1027  4305 −15 861 −1435
3144  −2 393 −786  4389  −7 4389 −627
3171  −3 1057 −3171  4440 −10 222 −555
3172 −13  61 −793  4440  −2 1110 −555
3180  −5 159 −795  4488 −187  66 −102
3192 −19 798  −42  4488 −22 561 −102
3336  −2 417 −834  4669 −203 161 −667
3355 −11 305 −3355  4683  −3 1561 −4683
3432 −22 429  −78  4760 −10 238 −595
3432 −11 858  −78  4836 −403  39  −93
3444 −123 287  −21  4836 −13 1209  −93
3480 −58  30 −435  4872 −58 609  −42
3480 −10 174 −435  4956 −21 413 −177

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
4972  −1 1243 −1243  6135 −15 2045 −1227
5005 −91 385 −715  6324 −51 527  −93
5005 −35 1001 −715  6360 −10 318 −795
5005 −7 5005 −715 6424 −22 1606 −73
5016 −22 114 −627  6545 −187 385 −595
5016  −6 418 −627  6545 −35 6545 −187
5060 −115  55 −253  6745 −19 6745 −355
5060  −5 1265 −253  6792  −2 849 −1698
5124 −427 183  −21  6916 −91 247 −133
5180 −35 1295  −37  6923  −7 989 −6923
5320 −70 133 −190  7068 −19 1767  −93
5320 −35 266 −190  7347  −3 2449 −7347
5336 −58 667  −46  7368  −2 921 −1842
5412 −123 451  −33  7476 −267 623  −21
5548 −73  19 −1387  7707  −3 2569 −7707
5548 −19 1387  −73  7820 −115 391  −85
5565 −35 1113 −795  7843 −11 713 −7843
5628 −67 1407  −21  7923  −3 2641 −7923
5640 −235 282  −30  8008 −22 2002  −91
5640  −6 1410 −235  8024 −34 118 −1003
5644 −17  83 −1411  8024  −2 2006 −1003
5676 −43 1419  −33  8060  −5 2015 −403
5720 −130  22 −715  8211 −51 2737 −483
5720 −10 286 −715  8283 −11 753 −8283
5720  −2 1430 −715  8436 −37 2109  −57
5784  −6 482 −723  8540  −5 2135 −427
5848 −43 1462  −34  8584 −37 2146  −58
5852 −11 1463 −133  8643  −3 2881 −8643
5896  −2 737 −1474  8968  −2 1121 −2242
5907 −11 537 −5907  9112 −67 2278  −34
5908  −7 211 −1477  9165 −235 1833 −195
5928 −19 1482  −78  9204 −13 2301 −177
5992  −7 214 −1498  9219  −3 3073 −9219
6028  −1 1507 −1507  9348 −123 779  −57
6045 −403 465 −195  9672 −403 186  −78

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
9816  −2 2454 −1227  16472 −58 2059  −142
9944  −2 2486 −1243  16984 −22 4246  −193
10227  −3 3409 −10227  17020 −37 4255  −115
10248 −427  366  −42  17105 −55 3421 −1555
10465 −91 10465  −115  17105 −11 17105 −1555
10659 −187  969  −627  17204 −187  391  −253
10659 −51 3553  −627  17227  −7 2461 −17227
11005 −155 2201  −355  17347 −19  913 −17347
11193 −91 11193  −123  18285 −115 3657  −795
11288 −34  166 −1411  18403  −7 2629 −18403
11288  −2 2822 −1411  18760 −67 4690  −70
11832 −51 2958  −58  19057 −323 1121 −1003
11960 −115  598  −130  19564 −67 4891  −73
11985 −51 11985  −235  19684 −37 4921  −133
12040 −43 3010  −70  20292 −267 1691  −57
12220 −13 3055  −235  20445 −235 4089  −435
12513 −43 12513  −291  20769 −43 20769  −483
12747  −3 4249 −12747  20904 −67 5226  −78
12765 −115 2553  −555  21016 −37 5254  −142
13064 −46 1633  −142  21777 −51 21777  −427
13065 −67 13065  −195  21948 −93 1829  −177
13156 −13 3289  −253  22204 −13 5551  −427
13452 −57 1121  −177  22876 −43 5719  −133
13468 −37 3367  −91  23865 −43 23865  −555
13601 −67 13601  −203  24440 −235 1222  −130
13685 −115 2737  −595  24924 −67 6231  −93
13692 −163 3423  −21  25619 −187 2329 −1507
13764 −37 3441  −93  25899 −267 8633  −291
14105 −35 14105  −403  26809 −323 1577 −1411
14673 −67 14673  −219  26809 −19 26809 −1411
14952 −267 1246  −42  26961 −43 26961  −627
16027 −11 1457 −16027  27965 −235 5593  −595
16120 −403  310  −130  28536 −58 7134  −123
16120 −10 4030  −403  28633 −19 28633 −1507
16445 −115 3289  −715  29463 −427 4209  −483

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
29545 −19 29545 −1555  83249 −1003  4897 −1411
29605 −155 5921 −955  86716 −163 21679 −133
30444 −43 7611 −177 94537 −67 94537 −1411
32361 −67 32361 −483  94705 −235 94705 −403
32452 −427 1159 −133 99064 −58 24766 −427
32680 −43 8170 −190  107601 −267 107601 −403
32809 −43 32809 −763  109203 −267 36401 −1227
33108 −267 2759 −93 110405 −355 22081 −1555
33605 −235 6721 −715  114009 −267 114009 −427
34780 −37 8695 −235  116545 −163 116545 −715
35464 −22 8866 −403  164956 −163 41239 −253
35644 −67 8911 −133  297005 −955 59401 −1555

Table 6.5: Fields where H ≃ ℤ_{2} × ℤ_{2} × ℤ_{2}
f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
168 −21  2  −42  1848 −7  66 −462
264  −2 66  −33  1860 −15 155  −93
408  −2 51 −102  1932 −21  23 −483
408  −1 102 −102  1995 −19 105 −1995
456  −2 114  −57  1995 −15 133 −1995
520  −1 130 −130  1995 −3 665 −1995
660 −11 15 −165  2035 −11 185 −2035
680 −2 170 −85 2145 −11 2145 −195
780 −13 15 −195  2145 −3 2145 −715
780  −1 195 −195  2184 −7 546  −78
820  −5 41 −205  2184 −6 546  −91
840  −7 210  −30  2220 −37  15 −555
915 −15 61 −915  2220 −1 555 −555
924  −7 231  −33  2280 −19 570  −30
1092  −3 91 −273  2380 −1 595 −595
1140 −15 95  −57  2460 −5 615 −123
1155 −11 105 −1155  2604 −21 217  −93
1320 −11 30 −330  2715 −15 181 −2715
1320 −11 330 −30 2856 −7 714 −102
1320 −10 33 −330  2860 −5 143 −715
1320  −2 165 −330  2860 −1 715 −715
1380 −115  3 −345  2968 −7 106 −742
1416  −2 354 −177  3003 −3 1001 −3003
1428  −1 357 −357  3036 −33  69 −253
1443  −3 481 −1443  3045 −35 609 −435
1540 −35 11 −385  3045 −7 3045 −435
1540 −11 35 −385  3080 −22 770  −35
1540  −7 55 −385  3080 −11 770  −70
1560  −3 390 −130  3180 −1 795 −795
1596 −21 57 −133  3705 −19 3705 −195
1596 −21 133 −57 3740 −11 935 −85
1635 −15 109 −1635  3740 −5 935 −187
1740  −1 435 −435  3864 −42  46 −483
1752  −6 73 −438  3885 −7 3885 −555
1780  −5 89 −445  4760 −2 1190 −595

continued on following page

Fields where H ≃ ℤ_{2} × ℤ_{2} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
4785 −11 4785 −435  14145 −115 14145 −123
4920 −10 1230 −123 14620 −43 3655 −85
5313 −11 5313 −483  15105 −19 15105 −795
5320 −19 1330  −70  15785 −11 15785 −1435
5320  −7 1330 −190  17544 −43  4386 −102
5340  −5 1335 −267  20553 −51 20553 −403
5565  −7 5565 −795  20680 −22  5170 −235
5720 −11 1430 −130  21112 −58  5278  −91
5740  −5 287 −1435  22360 −43  5590 −130
5740  −1 1435 −1435  22780 −67  5695  −85
6045 −15 6045 −403  23496 −22  5874 −267
6405 −15 6405 −427  25585 −43 25585 −595
6545 −11 6545 −595  27265 −19 27265 −1435
7068 −57 589 −93 27676 −37 6919 −187
7480 −10 1870 −187  29145 −67 29145 −435
7548 −37 1887 −51 30745 −43 30745 −715
7752 −19 1938 −102  34185 −43 34185 −795
7788 −33 649 −177  37185 −67 37185 −555
8120 −35 2030 −58 39516 −37 9879 −267
8265 −19 8265 −435  39865 −67 39865 −595
8360 −11 2090 −190  43945 −187 43945 −235
8385 −43 8385 −195  49569 −123 49569 −403
9177 −19 9177 −483  49929 −187 49929 −267
9672 −6 2418 −403 52521 −123 52521 −427
9724 −13 2431 −187  61705 −43 61705 −1435
9880 −19 2470 −130  100345 −235 100345 −427
10005 −115 2001 −435  233905 −163 233905 −1435

Table 6.6: Fields where H ≃ ℤ_{16}

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
555 −555  5 −111  3297 −7 3297 −471
555 −15 185 −111  3417 −51 201 −1139
740 −37  5 −185  3417 −3 3417 −1139
740  −5  37 −185  3460 −5 173 −865
760 −190  2 −95  3580 −5 179 −895
888  −6  74 −111  3768 −2 942 −471
1464  −6 122 −183  3832 −1 958 −958
1495 −115  65 −299  3913 −91 301 −559
1628 −37  11 −407  3913 −7 3913 −559
1628  −1 407 −407  4088 −7 1022 −146
1855 −35 265 −371  4137 −3 4137 −1379
1880 −235  10 −94  4188 −1 1047 −1047
1884  −1 471 −471  4236 −3 1059 −353
1887 −51 629 −111  4472 −2 1118 −559
1924 −37  13 −481  4520 −10 565 −226
1924 −13  37 −481  4636 −1 1159 −1159
2132 −13 533 −41  4664 −22 106 −583
2236 −13  43 −559  4695 −15 1565 −939
2236  −1 559 −559  5516 −1 1379 −1379
2260  −5 565 −113  5593 −7 5593 −799
2397 −51 141 −799  5668 −13 109 −1417
2397 −3 2397 −799 5752 −1 1438 −1438
2685 −15 537 −895  5860 −5 293 −1465
2685  −3 2685 −895  6149 −11 6149 −559
2740  −5 685 −137  6168 −3 1542 −514
2829 −123  69 −943  6260 −5 1565 −313
2829  −3 2829 −943  6265 −35 1253 −895
2895 −15 965 −579  6265 −7 6265 −895
3048  −6 381 −254  6285 −3 6285 −2095
3048  −3 762 −254  6424 −11 1606 −146
3084  −3 771 −257  6495 −15 2165 −1299
3111 −51 1037 −183  6585 −15 1317 −2195
3140  −5 157 −785  6585 −3 6585 −2195
3224 −403  26 −62  6740 −5 1685 −337
3256  −2 814 −407  7124 −13 1781 −137

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
7160 −10  358 −895  13137  −3 13137 −4379
7160  −2 1790 −895  13281  −3 13281 −4427
7196  −7 1799 −257  13881  −7 13881 −1983
7288  −1 1822 −1822  14665 −35 2933 −2095
7329  −7 7329 −1047  14665  −7 14665 −2095
7733 −19 7733 −407  14817  −3 14817 −4939
7932  −1 1983 −1983  15169 −11 15169 −1379
7973  −7 7973 −1139  15365 −35 3073 −2195
8113 −427  133 −1159  15365  −7 15365 −2195
8376  −2 2094 −1047  15585 −15 3117 −5195
8380  −5  419 −2095  15585  −3 15585 −5195
8380  −1 2095 −2095  15963 −51 5321 −939
8655 −15 2885 −1731  16089  −3 16089 −5363
8780  −1 2195 −2195  16172 −13  311 −4043
8789 −187  517 −799  16172  −1 4043 −4043
9140  −5 2285 −457  16287 −267 5429 −183
9272  −2 2318 −1159  16724 −37 4181 −113
9320 −10 1165 −466  16760 −10  838 −2095
9695 −35 1385 −1939  16760  −2 4190 −2095
9879 −267 3293 −111  17256  −6 2157 −1438
9884  −7 2471 −353  17256  −3 4314 −1438
10373 −11 10373 −943  17329 −403 1333 −559
10621 −19 10621 −559  17501 −43 17501 −407
11176 −22 1397 −254  17516  −1 4379 −4379
11176 −11 2794 −254  17560 −10  878 −2195
11452  −7 2863 −409  17560  −2 4390 −2195
11496  −6 1437 −958  17917 −19 17917 −943
11496  −3 2874 −958  17996 −11 4499 −409
11517 −11 11517 −1047  18748  −1 4687 −4687
11540  −5 2885 −577  19185 −15 3837 −6395
12252  −1 3063 −3063  19185  −3 19185 −6395
12529 −187  737 −1139  19304 −19 4826 −254
12529 −11 12529 −1139  19532 −19 4883 −257
12749 −11 12749 −1159  19677  −7 19677 −2811
13055 −35 1865 −2611  19785  −3 19785 −6595

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
20253 −43 20253 −471  30004 −13 7501  −577
20585 −115 4117 −895  30585 −15 6117 −10195
20648 −58 2581 −178  30585  −3 30585 −10195
20780  −5 1039 −5195  30651 −51 10217 −1803
20780  −1 5195 −5195  30921 −11 30921 −2811
21009  −3 21009 −7003  30936  −3 7734 −2578
21153  −3 21153 −7051  30988  −1 7747 −7747
21441  −7 21441 −3063  31084 −19 7771  −409
21452  −1 5363 −5363  31180  −5 1559 −7795
21585 −15 4317 −7195  31180  −1 7795 −7795
21585  −3 21585 −7195  31276  −1 7819 −7819
21864  −6 2733 −1822  32284  −7 8071 −1153
21864  −3 5466 −1822  32344  −2 8086 −4043
22209 −11 22209 −2019  33609 −51 1977 −11203
22631 −427 3233 −371  33609  −3 33609 −11203
23385 −15 4677 −7795  34536  −6 4317 −2878
23385  −3 23385 −7795  34536  −3 8634 −2878
23739 −123 7913 −579  34797  −7 34797 −4971
23764 −13 5941 −457  35032 −58  302 −4379
24101  −7 24101 −3443  35032  −2 8758 −4379
24249  −3 24249 −8083  35544  −3 8886 −2962
24504  −2 6126 −3063  35985 −15 7197 −11995
25112 −43 6278 −146  35985  −3 35985 −11995
25580  −5 1279 −6395  36365  −7 36365 −5195
26216 −58 3277 −226  37496  −2 9374 −4687
26380  −5 1319 −6595  37677 −19 37677 −1983
26380  −1 6595 −6595  38311 −91 5473 −2947
27032  −2 6758 −3379  38361 −19 38361 −2019
27624  −6 3453 −2302  38485 −43 38485  −895
27624  −3 6906 −2302  38499 −123 12833  −939
28012  −1 7003 −7003  39052 −13  751 −9763
28301  −7 28301 −4043  39052  −1 9763 −9763
28780  −5 1439 −7195  39369  −3 39369 −13123
28780  −1 7195 −7195  40780  −5 2039 −10195
29197  −7 29197 −4171  40780  −1 10195 −10195

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
41560 −10 2078 −5195  51969  −3 51969 −17323
41560  −2 10390 −5195  52760  −2 13190 −6595
42065 −235 8413  −895  52844 −11 13211 −1201
42904  −2 10726 −5363  54028 −13 1039 −13507
43372  −1 10843 −10843  54028  −1 13507 −13507
43688 −43 10922  −254  54229 −427  889 −7747
44204 −43 11051  −257  54229  −7 54229 −7747
44428  −1 11107 −11107  54565 −35 10913 −7795
44473 −11 44473 −4043  54565  −7 54565 −7795
44716  −1 11179 −11179  54681 −11 54681 −4971
44765 −35 8953 −6395  55276 −13 1063 −13819
44765  −7 44765 −6395  56024  −2 14006 −7003
44985 −15 8997 −14995  56649  −3 56649 −18883
44985  −3 44985 −14995  57441 −123 1401 −19147
45021 −43 45021 −1047  57441  −3 57441 −19147
46165 −35 9233 −6595  57484  −1 14371 −14371
46165  −7 46165 −6595  58197 −19 58197 −3063
46324 −37 11581  −313  58993 −11 58993 −5363
46396  −7 11599 −1657  59164  −7 14791 −2113
47980  −1 11995 −11995  59433 −11 59433 −5403
48169 −11 48169 −4379  59965 −67 59965  −895
48185 −115 9637 −2095  59980  −5 2999 −14995
48697 −11 48697 −4427  59980  −1 14995 −14995
49231 −91 7033 −3787  61185 −15 12237 −20395
49676 −11 12419 −1129  61185  −3 61185 −20395
49837 −43 49837 −1159  61976  −2 15494 −7747
50092  −1 12523 −12523  62360 −10 3118 −7795
50365 −35 10073 −7195  62360  −2 15590 −7795
50365  −7 50365 −7195  62552  −2 15638 −7819
50485 −115 10097 −2195  63181 −67 63181  −943
51160 −10 2558 −6395  63272 −22 7909 −1438
51160  −2 12790 −6395  63272 −11 15818 −1438
51531 −267 17177  −579  64201 −19 64201 −3379
51557 −11 51557 −4687  64209 −51 3777 −21403
51969 −51 3057 −17323  64209  −3 64209 −21403

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
66341 −163 66341  −407  85804 −19 21451 −1129
66549  −7 66549 −9507  86009 −11 86009 −7819
67371 −51 22457 −3963  86744  −2 21686 −10843
67636 −37 16909  −457  86817 −43 86817 −2019
68072 −67 17018  −254  86889  −3 86889 −28963
68341 −91 5257 −9763  87628 −19 21907 −1153
68341  −7 68341 −9763  88856 −58  766 −11107
68529  −3 68529 −22843  88856  −2 22214 −11107
70185 −15 14037 −23395  89053 −19 89053 −4687
70185 −3 70185 −23395 89432 −2 22358 −11179
70971 −123 23657 −1731  90085 −43 90085 −2095
71365 −35 14273 −10195  91117 −163 91117  −559
71365  −7 71365 −10195  91289 −11 91289 −8299
72808 −19 18202  −958  91372  −1 22843 −22843
73209  −3 73209 −24403  91861  −7 91861 −13123
73923 −123 24641 −1803  92393 −67 92393 −1379
75081  −3 75081 −25027  92504  −2 23126 −11563
75801  −3 75801 −25267  93261  −7 93261 −13323
76741  −7 76741 −10963  93580  −5  4679 −23395
76773 −163 76773  −471  93580  −1 23395 −23395
76817 −19 76817 −4043  94348  −1 23587 −23587
77033 −11 77033 −7003  94385 −43 94385 −2195
78104  −2 19526 −9763  94549  −7 94549 −13507
78421  −7 78421 −11203  94604 −67 23651  −353
80168 −22 10021 −1822  95192 −163 23798  −146
80168 −11 20042 −1822  95513 −11 95513 −8683
81560 −10 4078 −10195  95960 −10  4798 −11995
81560  −2 20390 −10195  95960  −2 23990 −11995
81580  −5 4079 −20395  96733  −7 96733 −13819
81580  −1 20395 −20395  97612  −1 24403 −24403
83571 −267 27857  −939  100108  −1 25027 −25027
83965 −35 16793 −11995  100184  −2 25046 −12523
83965  −7 83965 −11995  101288 −22 12661 −2302
85217 −11 85217 −7747  101288 −11 25322 −2302
85396 −37 21349  −577  101897 −19 101897 −5363

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
102657 −19 102657 −5403  137753 −11 137753 −12523
103165 −235 20633 −2195  138472 −19 34618 −1822
104965 −35 20993 −14995  140365 −67 140365 −2095
104965  −7 104965 −14995  142765 −35 28553 −20395
107821  −7 107821 −15403  145885 −163 145885  −895
108056  −2 27014 −13507  146553 −11 146553 −13323
109288 −19 27322 −1438  147065 −67 147065 −2195
109612 −67 27403  −409  147085 −115 29417 −6395
110552  −2 27638 −13819  148049 −43 148049 −3443
111148 −37  751 −27787  148561 −19 148561 −7819
111148  −1 27787 −27787  148577 −11 148577 −13507
113432 −11 28358 −2578  149821  −7 149821 −21403
114429  −7 114429 −16347  151064  −2 37766 −18883
114968  −2 28742 −14371  151685 −115 30337 −6595
115611 −267 38537 −1299  152009 −11 152009 −13819
115717 −7 115717 −16531 154059 −267 51353 −1731
118279 −427 16897 −1939  157681 −19 157681 −8299
119960 −10  5998 −14995  158081 −11 158081 −14371
119960  −2 29990 −14995  159271 −427 22753 −2611
120873 −43 120873 −2811  163160 −10  8158 −20395
121261  −7 121261 −17323  163160  −2 40790 −20395
122177 −11 122177 −11107  163765 −35 32753 −23395
122969 −11 122969 −11179  163765  −7 163765 −23395
123233 −11 123233 −11203  164776 −43 41194  −958
124376  −2 31094 −15547  165109  −7 165109 −23587
124972  −1 31243 −31243  167564 −163 41891  −257
125333 −403  9641 −4043  168321 −19 168321 −8859
126632 −22 15829 −2878  169433 −11 169433 −15403
126632 −11 31658 −2878  170661 −163 170661 −1047
127193 −11 127193 −11563  171017 −11 171017 −15547
130328 −11 32582 −2962  174952 −19 43738 −2302
131709 −43 131709 −3063  179285 −115 35857 −7795
132248  −2 33062 −16531  179767 −427 25681 −2947
133784  −2 33446 −16723  180633 −19 180633 −9507
134029  −7 134029 −19147  180652 −19 45163 −2377

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
181841 −11 181841 −16531  283585 −43 283585 −6595
182744 −2 45686 −22843 296609 −67 296609 −4427
183953 −11 183953 −16723  300565 −235 60113 −6395
185497 −19 185497 −9763  301129 −43 301129 −7003
187160 −10 9358 −23395 302572 −67 75643 −1129
188337 −67 188337 −2811  302653 −403 23281 −9763
190361 −43 190361 −4427  309385 −43 309385 −7195
190553 −11 190553 −17323  309965 −235 61993 −6595
195224  −2 48806 −24403  310593 −19 310593 −16347
200216 −58  1726 −25027  313384 −43 78346 −1822
200216  −2 50054 −25027  314029 −67 314029 −4687
206017 −19 206017 −10843  317737 −19 317737 −16723
207713 −11 207713 −18883  333121 −43 333121 −7747
210617 −11 210617 −19147 335128 −163 83782 −514
211033 −19 211033 −11107  335185 −43 335185 −7795
212377 −43 212377 −4939  338165 −235 67633 −7195
218701 −7 218701 −31243 341485 −163 341485 −2095
218728 −19 54682 −2878  344885 −115 68977 −14995
219697 −19 219697 −11563  347569 −43 347569 −8083
222296 −2 55574 −27787 348065 −67 348065 −5195
223385 −43 223385 −5195  352707 −267 117569 −3963
226393 −67 226393 −3379  357785 −163 357785 −2195
230156 −163 57539 −353 363793 −19 363793 −19147
231007 −427 33001 −3787  366365 −235 73273 −7795
235433 −187 13849 −21403  367564 −43 91891 −2137
237937 −19 237937 −12523  373369 −43 373369 −8683
244165 −235 48833 −5195  385384 −67 96346 −1438
247336 −43 61834 −1438  395944 −43 98986 −2302
249337 −19 249337 −13123  408801 −43 408801 −9507
253137 −19 253137 −13323  408844 −43 102211 −2377
256633 −19 256633 −13507  418717 −403 32209 −13507
256744 −67 64186 −958 434017 −19 434017 −22843
270881 −67 270881 −4043  438385 −43 438385 −10195
275297 −11 275297 −25027  443416 −43 110854 −2578
275885 −115 55177 −11995  455241 −43 455241 −10587

continued on following page

Fields where H ≃ ℤ_{16}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
458193 −163 458193 −2811  876985 −43 876985 −20395
471409 −43 471409 −10963  880689 −163 880689 −5403
472417 −67 472417 −7051  904969 −67 904969 −13507
475513 −19 475513 −25027  937576 −163 234394 −1438
488296 −67 122074 −1822  958565 −235 191713 −20395
495016 −43 123754 −2878  962857 −67 962857 −14371
497209 −43 497209 −11563  1004665 −67 1004665 −14995
515785 −43 515785 −11995  1005985 −43 1005985 −23395
527953 −19 527953 −27787  1042385 −163 1042385 −6395
538085 −115 107617 −23395  1074985 −163 1074985 −6595
538489 −43 538489 −12523  1080364 −163 270091 −1657
541561 −67 541561 −8083  1086481 −43 1086481 −25267
550777 −163 550777 −3379  1120441 −67 1120441 −16723
563765 −235 112753 −11995  1141489 −163 1141489 −7003
564289 −43 564289 −13123  1172785 −163 1172785 −7195
572889 −43 572889 −13323  1274497 −163 1274497 −7819
593553 −67 593553 −8859  1282849 −67 1282849 −19147
617953 −43 617953 −14371  1377676 −163 344419 −2113
624616 −163 156154 −958 1444017 −163 1444017 −8859
636969 −67 636969 −9507  1530481 −67 1530481 −22843
644785 −43 644785 −14995  1549641 −163 1549641 −9507
654121 −67 654121 −9763  1567465 −67 1567465 −23395
659009 −163 659009 −4043  1661785 −163 1661785 −10195
668521 −43 668521 −15547  1680856 −163 420214 −2578
704765 −235 140953 −14995  1725681 −163 1725681 −10587
713777 −163 713777 −4379  1810441 −163 1810441 −11107
721601 −163 721601 −4427  1955185 −163 1955185 −11995
726481 −67 726481 −10843  2041249 −163 2041249 −12523
734521 −67 734521 −10963  2097484 −163 524371 −3217
751756 −163 187939 −1153  2139049 −163 2139049 −13123
771304 −67 192826 −2878  2664561 −163 2664561 −16347
774721 −67 774721 −11563  2725849 −163 2725849 −16723
793816 −67 198454 −2962  3844681 −163 3844681 −23587

Table 6.7: Fields where H ≃ ℤ_{8} × ℤ_{2}
f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
264 −33  2 −66  1155 −55  21 −1155
264  −1 66 −66  1160 −58  10 −145
280  −5 70 −14  1160 −10  58 −145
328 −82  2 −41  1160 −2 290 −145
328  −2 82 −41  1220 −5  61 −305
456 −57  2 −114  1221 −11 1221 −111
456  −1 114 −114  1240 −10 155  −62
520 −130  2 −65  1240 −5 310  −62
520  −2 130 −65  1320 −55  6 −330
580  −1 145 −145  1320 −15 110  −66
663  −3 221 −663  1320 −10 165  −66
680 −10 170 −17  1416 −6  59 −354
680  −5 170 −34  1428 −17  21 −357
712  −1 178 −178  1463 −19  77 −1463
728 −13 182 −14  1508 −13  29 −377
740  −1 185 −185  1540 −35  55  −77
744 −93  6 −62  1540 −5 385  −77
777 −259 21 −111  1551 −11 141 −1551
780 −15 39 −65  1596 −19 399  −21
780  −5 195 −39  1596 −7 399  −57
884 −13 221 −17  1608 −6  67 −402
884  −1 221 −221  1624 −58  7 −406
897 −39 69 −299  1624 −2 203 −406
915 −15 305 −183  1624 −1 406 −406
920 −5 230 −46 1640 −2 205 −410
924 −33 21 −77  1672 −1 418 −418
924 −21 33 −77  1705 −11 1705 −155
924  −3 231 −77  1716 −13  33 −429
984  −6 246 −41  1716 −11  39 −429
1032  −1 258 −258  1716 −1 429 −429
1045 −55 209 −95  1736 −14 217  −62
1096  −2 274 −137  1740 −15  87 −145
1140  −5 57 −285  1740 −3 435 −145
1140  −3 95 −285  1767 −3 589 −1767
1140  −1 285 −285  1785 −3 1785 −595

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
1820  −7 455  −65  2580  −1 645 −645
1848 −42 77  −66  2584 −19  34 −646
1848 −14 33 −462  2584 −19 646  −34
1848  −3 462 −154  2604 −93  21 −217
1860  −5 93 −465  2604 −21  93 −217
1860  −3 155 −465  2604  −1 651 −651
1860  −1 465 −465  2632 −14 329  −94
1864  −1 466 −466  2712  −6 339 −226
1924  −1 481 −481  2712  −6 678 −113
1932 −21 161  −69  2715  −3 905 −2715
1992  −1 498 −498  2760 −115  30 −138
2020  −1 505 −505  2760 −10  69 −690
2067  −3 689 −2067  2760 −10 345 −138
2072 −37 14 −518  2760  −3 230 −690
2072  −2 259 −518  2760  −2 345 −690
2072  −1 518 −518  2856 −102  21 −238
2120  −2 530 −265  2856 −51  42 −238
2136  −6 267 −178  2856 −51 714  −14
2244 −187  3 −561  2856  −6 357 −238
2244 −51 11 −561  2856  −3 714 −238
2244 −11 51 −561  2860 −13 715  −55
2244  −3 187 −561  2945 −155 589  −95
2248  −1 562 −562  2980  −5 149 −745
2316  −1 579 −579  2996  −7 107 −749
2328  −3 194 −582  3003 −39  77 −3003
2343 −11 213 −2343  3036 −253  33  −69
2436  −3 203 −609  3036 −33 253  −69
2436  −1 609 −609  3045 −15 3045 −203
2460 −123 15 −205  3135 −15 1045 −627
2460 −15 123 −205  3144  −6 131 −786
2460  −3 615 −205  3160 −10 395 −158
2504  −2 626 −313  3160  −5 158 −790
2568  −6 107 −642  3160  −5 790 −158
2580 −15 43 −645  3160  −2 395 −790
2580  −3 215 −645  3160  −1 790 −790

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
3172  −1 793 −793  3885 −15 3885 −259
3180  −3 795 −265  3916 −1 979 −979
3192  −7 798 −114  3944 −58 986  −17
3201 −11 3201 −291  3948 −21 329 −141
3220 −115  7 −805  3948 −7 987 −141
3220  −1 805 −805  3948 −1 987 −987
3245 −55 649 −295  4008 −6 167 −1002
3245 −11 3245 −295  4008 −6 334 −501
3255 −35 465 −651  4008 −2 1002 −501
3255 −15 1085 −651  4008 −1 1002 −1002
3288  −6 822 −137  4017 −39 309 −1339
3336  −6 139 −834  4017 −3 4017 −1339
3460  −1 865 −865  4020 −15  67 −1005
3477 −19 3477 −183  4020 −3 335 −1005
3608 −22 902 −41 4020 −1 1005 −1005
3612 −21 129 −301  4040 −10 202 −505
3640  −7 910 −130  4040 −2 1010 −505
3656  −2 914 −457  4044 −3 1011 −337
3752  −2 938 −469  4053 −7 4053 −579
3752  −1 938 −938  4081 −11 4081 −371
3756  −1 939 −939  4081 −7 4081 −583
3784 −22  43 −946  4161 −19 4161 −219
3795 −115  33 −3795  4260 −15  71 −1065
3795 −15 253 −3795  4260 −15 355 −213
3795  −3 1265 −3795  4260 −5 213 −1065
3828 −11  87 −957  4260 −5 1065 −213
3828  −1 957 −957  4260 −3 355 −1065
3836  −7 959 −137  4260 −1 1065 −1065
3864 −483  14 −138  4264 −13 1066  −82
3864 −42  69 −322  4296 −2 537 −1074
3864 −42 161 −138  4305 −35 4305 −123
3864  −6 322 −483  4305 −3 4305 −1435
3864  −3 966 −322  4340 −35 155 −217
3880 −10 970  −97  4340 −5 1085 −217
3885 −35 777 −555  4345 −55 869 −395

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
4380 −15 1095  −73  4972 −11 1243 −113
4380  −5 1095 −219  4984  −7 178 −1246
4424 −14 553 −158  5012  −7 179 −1253
4452  −3 371 −1113  5016 −627  6 −418
4452  −1 1113 −1113  5016 −627  22 −114
4485 −115 897 −195  5016 −627  38  −66
4485 −115 4485  −39  5016 −19 1254  −66
4488 −187  6 −1122  5016 −11 1254 −114
4488 −187 102  −66  5016  −3 1254 −418
4488 −51  22 −1122  5032 −37 1258  −34
4488 −51 374  −66  5064  −2 633 −1266
4488 −22 1122  −51  5160 −43  30 −1290
4488 −11 102 −1122  5160 −43 1290  −30
4488 −11 1122 −102  5160 −15  86 −1290
4488  −6 1122 −187  5160 −15 430 −258
4488  −3 374 −1122  5160 −10 645 −258
4515 −43 105 −4515  5160  −3 430 −1290
4515 −35 129 −4515  5180  −5 1295 −259
4515  −7 645 −4515  5187 −91  57 −5187
4551 −123 1517 −111  5187 −19 273 −5187
4584  −6 382 −573  5187  −7 741 −5187
4584  −2 1146 −573  5192  −2 649 −1298
4632  −6 386 −579  5208 −42  62 −651
4669  −7 4669 −667  5208  −2 1302 −651
4692 −51 391  −69  5215 −35 745 −1043
4712  −2 1178 −589  5236 −187 119  −77
4712  −1 1178 −1178  5244 −57 437  −69
4740  −5 237 −1185  5244 −19 1311  −69
4740  −3 395 −1185  5313 −483  33 −1771
4740  −1 1185 −1185  5313  −3 5313 −1771
4920 −15 1230  −82  5340 −267  15 −445
4935 −235 105 −987  5340 −15 267 −445
4935 −15 1645 −987  5340  −3 1335 −445
4953 −39 381 −1651  5380  −5 269 −1345
4956  −7 1239 −177  5412 −123  11 −1353

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
5412 −11 123 −1353  6195  −7 885 −6195
5412  −3 451 −1353  6195  −3 2065 −6195
5484  −3 1371 −457  6204 −33 517 −141
5592  −6 699 −466  6279 −91 897 −483
5605 −19 5605 −295  6312  −6 263 −1578
5640 −235  30 −282  6312  −1 1578 −1578
5640 −10 705 −282  6315  −3 2105 −6315
5668  −1 1417 −1417  6328  −7 1582 −226
5704 −46 713  −62  6344  −2 1586 −793
5736  −6 478 −717  6360  −2 1590 −795
5736  −2 1434 −717  6405 −427 105 −915
5740 −35 287 −205  6405 −35 1281 −915
5740  −7 1435 −205  6405  −7 6405 −915
5820 −15 1455  −97  6420  −1 1605 −1605
5820  −5 1455 −291  6440 −115  70 −322
5848 −43  34 −1462  6440 −115 322  −70
5852 −133 209  −77  6440 −115 1610  −14
5860  −1 1465 −1465  6440 −35 230 −322
5883  −3 1961 −5883  6440 −35 1610  −46
5896 −22  67 −1474  6567 −11 597 −6567
5943  −3 1981 −5943  6580 −35  47 −1645
5964 −21 497 −213  6580  −1 1645 −1645
5980 −115 299  −65  6612  −3 551 −1653
5992  −1 1498 −1498  6612  −1 1653 −1653
5995 −11 545 −5995  6636 −21 237 −553
6028 −11 1507 −137  6636  −1 1659 −1659
6040 −10 151 −1510  6692  −7 239 −1673
6040  −5 302 −1510  6744  −6 843 −562
6040  −2 755 −1510  6745 −355 1349  −95
6040  −1 1510 −1510  6765 −123 6765  −55
6060 −15 303 −505  6792  −6 283 −1698
6060  −3 1515 −505  6808 −37 1702  −46
6188 −91 1547  −17  6868  −1 1717 −1717
6195 −35 177 −6195  6888 −123 1722  −14
6195 −15 413 −6195  6924  −1 1731 −1731

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
6952  −1 1738 −1738  7896  −2 1974 −987
7035 −67 105 −7035  8008 −91 286 −154
7035 −35 201 −7035  8008 −11 182 −2002
7035  −7 1005 −7035  8008  −2 1001 −2002
7052 −43 1763  −41  8040 −67 2010  −30
7084 −253 161  −77  8052 −11 183 −2013
7212  −1 1803 −1803  8052  −1 2013 −2013
7293 −187 7293  −39  8060 −403 155  −65
7320 −10 366 −915  8060 −13 2015 −155
7320 −2 1830 −915 8120 −10 2030 −203
7332 −13 1833 −141  8216 −13 2054 −158
7368 −6 307 −1842 8265 −435 57 −2755
7384 −13 1846 −142  8265 −15 1653 −2755
7420 −35 371 −265 8265 −3 8265 −2755
7437 −67 7437 −111  8360 −22 1045 −190
7464 −6 622 −933 8463 −403 273 −651
7505 −19 7505 −395  8463 −91 1209 −651
7640 −10 955 −382  8556 −93 713  −69
7640  −5 1910 −382  8648 −46 1081  −94
7656 −58 957  −66  8652  −1 2163 −2163
7720 −10 1930 −193  8680 −35  62 −2170
7752  −6 1938 −323  8680 −10 217 −2170
7755 −235  33 −7755  8680  −7 310 −2170
7755 −15 517 −7755  8680  −2 1085 −2170
7755  −3 2585 −7755  8715 −15 581 −8715
7761 −39 597 −2587  8715  −7 1245 −8715
7780  −5 389 −1945  8715  −3 2905 −8715
7832 −22 178 −979  8760 −10 2190 −219
7832 −11 1958 −178  8844 −67 2211  −33
7832  −2 1958 −979  8855 −115 385 −1771
7896 −42  94 −987  8855 −35 1265 −1771
7896 −42 141 −658  8932 −11 203 −2233
7896 −42 329 −282  9015 −15 3005 −1803
7896  −7 1974 −282  9048 −58  78 −1131
7896  −3 1974 −658  9048 −58 2262  −39

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
9048  −2 2262 −1131  10668  −1 2667 −2667
9145 −155 1829 −295  10696 −14 1337  −382
9192  −6  766 −1149  10696  −7  382 −2674
9228  −1 2307 −2307  10716 −57  893  −141
9291  −3 3097 −9291  10716 −19 2679  −141
9345  −35 9345 −267  10804 −37 2701  −73
9372  −33  781 −213  10815 −15 3605 −2163
9384  −51  782 −138  10824 −123  902  −66
9384  −51 2346  −46  10824 −22 2706  −123
9412  −13  181 −2353  10856  −2 2714 −1357
9436  −7 2359 −337  10945 −55 2189  −995
9548  −11 2387 −217  10945 −11 10945  −995
9560  −10 1195 −478  11060 −35  395  −553
9560  −5 2390 −478  11060  −5 2765  −553
9579  −3 3193 −9579  11067 −51 3689  −651
9640  −2 1205 −2410  11067  −7 1581 −11067
9804  −43 2451  −57  11076 −13 2769  −213
9835  −7 1405 −9835  11240 −10 1405  −562
9944 −1243  22 −226  11284 −403  91  −217
9960  −10 1245 −498  11284 −91  403  −217
10005  −15 10005 −667  11308  −1 2827 −2827
10024  −7  358 −2506  11352 −43 2838  −66
10212  −37 2553  −69  11352 −11 2838  −258
10329  −11 10329 −939  11480 −35 2870  −82
10392  −6  866 −1299  11580 −15 2895  −193
10392  −2 2598 −1299  11640 −10 2910  −291
10412  −19 2603 −137  11784  −2 1473 −2946
10472 −187  154 −238  11816  −2 2954 −1477
10472 −187  238 −154  11868 −43 2967  −69
10472 −187 2618  −14  12056  −2 3014 −1507
10472  −22 1309 −238  12232  −2 1529 −3058
10472  −11 2618 −238  12245 −155 2449  −395
10488  −19 2622 −138  12255 −15 4085 −2451
10505  −11 10505 −955  12376 −91  238  −442
10549  −7 10549 −1507  12376 −91 3094  −34

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
12376  −7 3094 −442  14140 −35  707  −505
12460 −35  623 −445  14140  −7 3535  −505
12460  −7 3115 −445  14168 −11 3542  −322
12616  −2 1577 −3154  14168  −2 3542 −1771
12760 −58 3190  −55  14344  −2 1793 −3586
12831 −91 1833 −987  14413 −203  497 −2059
12857 −43 12857 −299  14413  −7 14413 −2059
13048  −7 3262 −466  14424  −6 1202 −1803
13080 −10  654 −1635  14484 −51 1207  −213
13080  −2 3270 −1635  14536 −46 1817  −158
13145 −55 2629 −1195  14601  −3 14601 −4867
13160 −235  70 −658  14728  −7  526 −3682
13160 −235  658  −70  14735 −35 2105 −2947
13160 −235 3290  −14  14952 −267 3738  −14
13160 −35  470 −658  14973 −483  713  −651
13209 −51 13209 −259  14973  −7 14973 −2139
13224 −58 1653 −114  15276 −67 3819  −57
13244 −43 3311  −77  15576 −22  354 −1947
13244 −11 3311 −301  15652 −91  559  −301
13272  −2 3318 −1659  15652 −13 3913  −301
13288 −22  151 −3322  15656  −2 3914 −1957
13288  −1 3322 −3322  15715  −7 2245 −15715
13335 −35 1905 −2667  15736  −7 3934  −562
13335 −15 4445 −2667  15771  −3 5257 −15771
13384 −14 1673 −478  15953 −43 15953  −371
13384  −7 3346 −478  15980 −235 3995  −17
13452 −19 3363 −177  15996 −43 3999  −93
13468 −13 3367 −259  16008 −58 2001  −138
13629  −7 13629 −1947  16188 −57 1349  −213
13748  −7  491 −3437  16188 −19 4047  −213
13848  −2 3462 −1731  16215 −235  345 −3243
13912 −37 3478  −94  16215 −115  705 −3243
13944 −42  581 −498  16284 −177 1357  −69
13944  −7 3486 −498  16401  −3 16401 −5467
14028  −1 3507 −3507  16467 −11 1497 −16467

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
16492 −133  589  −217  19005  −7 19005 −2715
16492 −19 4123  −217  19272 −22 2409  −438
16492  −1 4123 −4123  19272 −22 4818  −219
16536  −2 4134 −2067  19272 −11 4818  −438
16616  −2 4154 −2077  19299  −3 6433 −19299
16643 −187 1513  −979  19352 −82  118 −2419
16744 −91  598  −322  19448 −187  286  −442
16779 −51 5593  −987  19448 −11 4862  −442
16936 −58 4234  −73  19491 −267 6497  −219
17017 −91 17017  −187  19560 −163 4890  −30
17068  −1 4267 −4267  19608  −6 1634 −2451
17080 −10 4270  −427  19745 −55 3949 −1795
17112  −2 4278 −2139  19815 −15 6605 −3963
17121 −39 1317 −5707  20251  −7 2893 −20251
17121  −3 17121 −5707  20504 −22 2563  −466
17240 −10 2155  −862  20539 −19 1081 −20539
17240  −5 4310  −862  20636 −67 5159  −77
17304 −42  206 −2163  20811  −3 6937 −20811
17304  −2 4326 −2163  20945 −355 4189  −295
17484 −93 1457  −141  21027  −3 7009 −21027
17535 −15 5845 −3507  21028  −7  751 −5257
17608 −142 2201  −62  21045 −115 4209  −915
17841 −19 17841  −939  21336  −2 5334 −2667
17841  −3 17841 −5947  21432 −19 5358  −282
17963 −11 1633 −17963  21489 −19 21489 −1131
18145 −955 3629  −95  21516 −163 5379  −33
18249 −11 18249 −1659  21720  −2 5430 −2715
18456  −6 1538 −2307  21868  −1 5467 −5467
18460 −13 4615  −355  22040 −58  190 −2755
18696 −123 1558  −114  22040 −10 1102 −2755
18705 −43 18705  −435  22040  −2 5510 −2755
18907  −7 2701 −18907  22083 −51 7361 −1299
18921 −51 1113 −6307  22165 −403 22165  −55
18921  −3 18921 −6307  22184  −2 5546 −2773
19005 −35 3801 −2715  22204 −427  91  −793

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
22204  −7 5551  −793  27224  −82  166 −3403
22568 −403 5642  −14  27260 −235 1363 −145
22616  −2 5654 −2827  27404 −403 6851  −17
22701 −483  329 −3243  27993  −43 27993 −651
22701 −483 1081  −987  28045 −355 5609 −395
22701  −7 22701 −3243  28056  −42  334 −3507
22940  −37 5735  −155  28203  −51 9401 −1659
23124 −123 1927  −141  28424  −22 7106 −323
23144  −22  526 −2893  28756  −91 1027 −553
23144  −2 5786 −2893  28756  −13 7189 −553
23485 −427  385 −3355  28952  −11 7238 −658
23485 −427 23485  −55  29337  −11 29337 −2667
23485  −35 4697 −3355  29393  −91 29393 −323
23485  −7 23485 −3355  29427  −51 9809 −1731
23621 −1027 1817  −299  29428  −7 1051 −7357
23736  −43 5934  −138  29545 −1555 5909  −95
24072  −6 6018 −1003  29656  −22 7414 −337
24136  −14 3017  −862  29829 −163 29829 −183
24136  −7 6034  −862  29832  −6 7458 −1243
24211  −11 2201 −24211  29928  −58 3741 −258
24252  −43 6063  −141  30107 −187 2737 −1771
24564 −267 2047  −69  30261  −7 30261 −4323
24728  −22 3091  −562  30520  −10 7630 −763
24808  −7  886 −6202  30845 −155 6169 −995
24897  −43 24897  −579  31521  −19 31521 −1659
24940  −43 6235  −145  31704  −2 7926 −3963
25305  −35 25305  −723  31960 −235 7990  −34
25388  −11 6347  −577  32116  −37 8029 −217
26085 −235 5217  −555  32712  −58 4089 −282
26412  −93 2201  −213  32889  −19 32889 −1731
26691 −123 8897  −651  32945  −55 6589 −2995
26696 −142 3337  −94  32984  −2 8246 −4123
26840  −2 6710 −3355  33099 −187 3009 −1947
26961  −11 26961 −2451  33099  −51 11033 −1947
27020  −35 6755  −193  33276 −177 2773 −141

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
33352  −2 4169 −8338  39345 −43 39345  −915
33580 −115 8395  −73  39545 −55 7909 −3595
33901 −203 1169 −4843  39676 −13 9919  −763
33943 −91 4849 −2611  39711 −427 5673  −651
34136 −34  502 −4267  40216 −22 10054  −457
34216 −91 1222 −658  40467 −123 13489  −987
34408 −187 8602  −46  40584 −267 3382  −114
34892 −11 8723 −793  40641 −19 40641 −2139
34932 −123 2911 −213  40940 −115 2047  −445
35061 −403 2697 −1131  42028 −133 1501  −553
35096 −82  214 −4387  42028 −19 10507  −553
35096  −2 8774 −4387  42380 −163 10595  −65
35144 −46 4393 −382  42441 −43 42441  −987
35581 −91 2737 −5083  42945 −35 42945 −1227
35581 −7 35581 −5083 43240 −235 10810 −46
35960 −58 8990 −155  43593 −11 43593 −3963
36168  −6 9042 −1507  43665 −123 43665  −355
36636 −43 9159 −213  43736  −2 10934 −5467
36771 −51 12257 −2163  43833 −19 43833 −2307
36984 −67 9246 −138  44776 −58 11194  −193
36993 −627 1121 −1947  44872 −142 5609  −158
36993 −19 36993 −1947  44988 −163 11247  −69
37045 −155 7409 −1195  45353 −11 45353 −4123
37164 −163 9291 −57 45505 −19 45505 −2395
37605 −115 7521 −1635  45591 −91 6513 −3507
37788 −67 9447 −141  46145 −11 46145 −4195
37848 −19 9462 −498  46248 −123 3854  −282
38152  −2 4769 −9538  47357 −667 1633 −2059
38248  −7 1366 −9562  47576  −2 11894 −5947
38540 −235 1927 −205  47704 −67 11926  −178
38577 −11 38577 −3507  47705 −235 47705  −203
38793 −67 38793 −579  47804 −37 11951  −323
38913 −51 38913 −763  48059 −187 4369 −2827
39109 −259 1057 −5587  48328  −7 1726 −12082
39219 −51 13073 −2307  48433 −187 48433  −259

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
48488 −58 6061  −418  65417 −11 65417 −5947
48504 −43 12126  −282  65593 −67 65593 −979
49128 −267 12282  −46  66129 −67 66129 −987
49252  −7 1759 −12313  66185 −427 66185 −155
49749 −483 2369 −2163  66385 −187 66385 −355
49820 −235 2491  −265  66504 −163 16626 −102
50268 −177 4189  −213  67512 −58 16878 −291
50673 −19 50673 −2667  67804 −67 16951 −253
50808 −58 12702  −219  68019 −123 22673 −1659
51688 −91 12922  −142  68305 −19 68305 −3595
52948 −427 1891  −217  68385 −235 68385 −291
53537 −11 53537 −4867  69153 −267 69153 −259
53599 −403 1729 −4123  69864 −123 17466 −142
54417 −187 54417  −291  70252 −91 17563 −193
54520 −58 13630  −235  71020 −67 17755 −265
54808 −403 1054  −442  71071 −91 10153 −5467
54940 −67 13735  −205  72124 −13 18031 −1387
55131 −51 18377 −3243  73676 −163 18419 −113
55384 −43 13846  −322  74245 −155 14849 −2395
55448 −58 6931  −478  74936 −58 18734 −323
55645 −155 11129 −1795  75336 −43 18834 −438
55913 −187 3289 −5083  75576 −67 18894 −282
55913 −11 55913 −5083  76328 −58 9541 −658
56440 −10 14110 −1411  76845 −235 15369 −1635
57084 −67 14271 −213 77165 −115 15433 −3355
57577 −43 57577 −1339  78364 −13 19591 −1507
57939 −267 19313  −651  78568 −427 2806 −322
57964 −43 14491  −337  78568 −427 19642  −46
59619 −51 19873 −3507  79304 −46 9913 −862
61305 −67 61305 −915 79705 −19 79705 −4195
61457 −11 61457 −5587  79827 −123 26609 −1947
62445 −115 12489 −2715  79849 −187 79849 −427
63271 −403 2041 −4867  80661 −483 3841 −3507
63624 −22 15906 −723 80668 −67 20167 −301
64077 −403 4929 −2067  81073 −323 4769 −4267

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
81844  −37 20461 −553  117465 −123 117465 −955
82137 −627  2489 −4323  118465  −43 118465 −2755
83145 −115 83145 −723  120801  −67 120801 −1803
84760 −163 21190 −130  122056  −22 30514 −1387
84845 −355 16969 −1195  122476  −67 30619 −457
85656  −43 21414 −498  122845 −1555 24569 −395
86241 −267 86241 −323  124168  −22 31042 −1411
86296  −67 21574 −322  124684 −427 31171  −73
87745 −115 87745 −763  127445 −355 25489 −1795
88264 −22 22066 −1003 127605 −235 25521 −2715
88683 −123 29561 −2163  130045 −155 26009 −4195
88780 −115 22195 −193  130169 −403 130169 −323
90193  −19 90193 −4747  130429 −1027 10033 −1651
91180 −235 22795  −97  130449  −67 130449 −1947
91273  −91 91273 −1003  132184 −403 33046  −82
91848 −267  7654 −258  132963 −123 44321 −3243
92845 −155 18569 −2995  133464  −67 33366 −498
92939 −187  8449 −5467  134932 −427  4819 −553
93009  −43 93009 −2163  135340  −67 33835 −505
93849 −123 93849 −763  135752 −142 16969 −478
94540 −163 23635 −145  137137  −91 137137 −1507
95029 −163 95029 −583  138489  −67 138489 −2067
95116  −43 23779 −553  140056 −427 35014  −82
96664  −43 24166 −562  141340  −37 35335 −955
98332 −403  1891 −793  143065 −403 143065 −355
99201  −43 99201 −2307  143313  −67 143313 −2139
99992  −58 12499 −862  143787 −123 47929 −3507
105781 −1027  8137 −1339  144265  −43 144265 −3355
108031 −427 15433 −1771  144921  −67 144921 −2163
108488 −142 13561 −382  147352 −163 36838 −226
109545  −67 109545 −1635  148444  −37 37111 −1003
111241  −43 111241 −2587  150801  −43 150801 −3507
111389 −667  3841 −4843  151585 −427 151585 −355
112024  −67 28006 −418  152889 −123 152889 −1243
113176  −43 28294 −658  155176 −163 38794 −238

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
156364 −403 39091  −97  266505 −163 266505 −1635
164217  −67 164217 −2451  267801 −267 267801 −1003
165676 −427 41419  −97  276241  −67 276241 −4123
166969  −43 166969 −3883  282153 −163 282153 −1731
170045 −355 34009 −2395  284664  −58 71166 −1227
173283 −267 57761 −1947  284809 −427 284809 −667
177243 −123 59081 −4323  285576 −163 71394 −438
177288 −267 14774 −498  288184 −163 72046 −442
177289  −43 177289 −4123  288345 −235 288345 −1227
179305 −235 179305 −763  288376  −58 72094 −1243
183864 −163 45966 −282  290140 −163 72535 −445
184353 −163 184353 −1131  290785 −187 290785 −1555
185889 −43 185889 −4323 297845 −355 59569 −4195
190045 −955 38009 −995  307489 −403 307489 −763
190371 −267 63457 −2139 321784 −58 80446 −1387
192049 −187 192049 −1027  324696 −163 81174 −498
192507 −267 64169 −2163  329260 −163 82315 −505
196252 −163 49063 −301  329644 −427 82411 −193
197449 −67 197449 −2947 331585 −235 331585 −1411
204373 −1027 15721 −2587  333487 −427 47641 −5467
205323 −267 68441 −2307  342845 −955 68569 −1795
212524  −67 53131 −793  354161 −1411 20833 −4267
218139 −267 72713 −2451  360760  −58 90190 −1555
221560  −58 55390 −955  366424 −163 91606 −562
228245 −955 45649 −1195  371645 −1555 74329 −1195
228904 −403 57226 −142 374329 −67 374329 −5587
229449 −187 229449 −1227  376204 −163 94051 −577
230140  −37 57535 −1555  376737 −267 376737 −1411
234969  −67 234969 −3507  394297 −163 394297 −2419
238264  −58 59566 −1027  404209 −403 404209 −1003
242536 −427 60634 −142  407785 −427 407785 −955
244808 −142 30601 −862  429016 −163 107254 −658
251503 −427 35929 −4123  438529 −427 438529 −1027
251753 −1003 14809 −4267  442545 −163 442545 −2715
255245 −355 51049 −3595  449065 −163 449065 −2755

continued on following page

Fields where H ≃ ℤ_{8} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
450853 −1027 34681 −5707  704649 −163 704649 −4323
457445 −955 91489 −2395  801245 −955 160249 −4195
554689 −163 554689 −3403  828529 −163 828529 −5083
558245 −1555 111649 −1795  931445 −1555 186289 −2995
686645 −955 137329 −3595  1304645 −1555 260929 −4195

Table 6.8: Fields where H ≃ ℤ_{4} × ℤ_{4}
f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
420 −21  5 −105  2392 −46  13 −598
420  −5 105  −21  2480 −15 870  −58
520 −10  26  −65  2580 −43  15 −645
840 −42  5 −210  2632  −1 658 −658
952 −17  14 −238  2755 −19 145 −2755
952 −14  17 −238  2788 −17  41 −697
1015 −35 145 −203  2820 −235  15 −141
1020 −15 255  −17  2968 −14  53 −742
1131 −39  29 −1131  2980  −1 745 −745
1220  −1 305 −305  3080 −35 110 −154
1240 −155  2 −310  3080 −10 385 −154
1380 −115  15  −69  3108 −37 777  −21
1380 −15 115  −69  3180 −15 159 −265
1443 −39  37 −1443  3192 −19  42 −798
1508  −1 377 −377  3315 −15 221 −3315
1560 −15 130  −78  3335 −115 145 −667
1560  −6 130 −195  3355 −55  61 −3355
1677 −43 1677  −39  3432 −11  78 −858
1736 −14  62 −217  3480  −2 870 −435
1736  −2 434 −217  3640 −91 130  −70
1752 −219  2 −438  3660  −5 183 −915
1768 −34  13 −442  3720 −30  93 −310
1780  −1 445 −445  3720 −15 186 −310
1785 −35 1785  −51  3720  −6 465 −310
1820 −91  35  −65  3720  −6 930 −155
1820 −35  91  −65  3720  −3 930 −310
2035 −55  37 −2035  3828  −3 319 −957
2040 −51 170  −30  4004 −91 143  −77
2067 −39  53 −2067  4020 −67  15 −1005
2120 −10 106 −265  4060  −5 1015 −203
2280 −15  38 −570  4424  −2 1106 −553
2280 −15 190 −114  4440  −6 370 −555
2280 −10 285 −114  4845 −15 4845 −323
2280  −3 190 −570  4872  −6 1218 −203
2380 −35 595  −17  4920 −123 410  −30

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{4}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
5720 −22 130 −715  11305 −35 11305  −323
6020 −35 215 −301  11544  −2 2886 −1443
6148  −1 1537 −1537  12056 −22  274 −1507
6307  −7 901 −6307  12155 −187 1105  −715
6420 −15 107 −1605  12220 −235  611  −65
6420  −3 535 −1605  12792 −123 1066  −78
6540  −5 327 −1635  13340  −5 3335  −667
6612 −19  87 −1653  13515 −51 4505  −795
6916 −13 1729 −133  13580 −35 3395  −97
7735 −91 1105 −595  15045 −15 15045 −1003
7780  −1 1945 −1945  15080 −58 1885  −130
7820 −115 1955  −17  15260  −5 3815  −763
7995 −123 2665 −195  16008  −6 4002  −667
8295 −35 1185 −1659  16280 −10  814 −2035
8295 −15 2765 −1659  16280  −2 4070 −2035
8364 −123  51 −697  16555 −43  385 −16555
8364 −123 2091  −17  16728 −123 1394  −102
8364 −51 123 −697  16728 −123 4182  −34
8364  −3 2091 −697  16728 −51 4182  −82
8520 −15 2130 −142  17080 −427  610  −70
8520  −6 2130 −355  17353 −67 17353  −259
8701  −7 8701 −1243  17355 −267 5785  −195
8904  −6 1113 −742  17835 −123 5945  −435
8904  −3 2226 −742  17864 −22 4466  −203
9412  −1 2353 −2353  18145 −19 18145  −955
9435 −51 3145 −555  18645 −15 18645 −1243
9816  −6 818 −1227  18788 −427  671  −77
10220 −35 2555  −73  19516  −7 4879  −697
10556 −13 2639 −203  19929 −91 19929  −219
10680 −267 890  −30  22605 −15 22605 −1507
10695 −115 465 −2139  22920  −6 5730  −955
10695 −15 3565 −2139  23023 −91 3289 −1771
10860  −5 543 −2715  24380 −115 1219  −265
11096  −2 2774 −1387  25851  −3 8617 −25851
11256 −67 2814  −42  26312 −22 3289  −598

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{4}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
26312 −11 6578  −598  48705 −51 48705  −955
26481 −91 26481  −291  49105 −115 49105  −427
26572 −91 6643  −73  49395 −267 16465  −555
27363  −3 9121 −27363  50043  −3 16681 −50043
27379 −19 1441 −27379  50347 −11  4577 −50347
27384 −163 6846  −42  51843 −11  4713 −51843
27417 −19 27417 −1443  53265 −67 53265  −795
27768 −267 2314  −78  53320 −43 13330  −310
28443  −3 9481 −28443  53464 −163 13366  −82
29643 −123 9881  −723  54604 −187 13651  −73
29883  −3 9961 −29883  56067 −11  5097 −56067
30668 −187  451  −697  56392 −19 14098  −742
30668 −11 7667  −697  58696 −22 14674  −667
30705 −115 30705  −267  59563  −7  8509 −59563
32403  −3 10801 −32403  60088 −58 15022  −259
35105 −35 35105 −1003  61336 −187 15334  −82
35308 −91 8827  −97  64227  −3 21409 −64227
36507  −3 12169 −36507  67144 −22 16786  −763
37145 −115 37145  −323  68620 −235 17155  −73
37320  −6 9330 −1555  69843  −3 23281 −69843
38643 −11 3513 −38643  70755 −267 23585  −795
41080 −10 10270 −1027  70905 −163 70905  −435
41547 −11 3777 −41547  71323  −7 10189 −71323
42123  −3 14041 −42123  73444 −427  2623  −301
44620 −115 11155  −97  78729 −163 78729  −483
44689 −67 44689  −667  82041 −123 82041  −667
44733 −403 3441 −1443  83080 −67 20770  −310
45339 −51 15113 −2667  83923 −19  4417 −83923
45448 −19 11362  −598  87505 −43 87505 −2035
46483 −43 1081 −46483  87576 −267 21894  −82
46543 −427 6649  −763  88257 −403 88257  −219
46805 −115 9361 −2035  90376 −22 22594 −1027
47163  −3 15721 −47163  96145 −67 96145 −1435
48441 −67 48441  −723  100491 −123 33497 −2451
48545 −35 48545 −1387  102201 −163 102201  −627

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{4}

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f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
109347 −123 36449  −2667  181420 −235 45355 −193
115345 −115 115345  −1003  198856 −67 49714 −742
115404 −163 28851  −177  200001 −163 200001 −1227
117676 −403 29419  −73  202120 −163 50530 −310
122683 −19  6457 −122683  235209 −163 235209 −1443
123369 −123 123369  −1003  241345 −235 241345 −1027
137921 −427 137921  −323  259369 −187 259369 −1387
141505 −91 141505  −1555  291369 −403 291369 −723
142945 −115 142945  −1243  292105 −235 292105 −1243
144364 −187 36091  −193  325945 −235 325945 −1387
147651 −267 49217  −1659  354145 −235 354145 −1507
160264 −67 40066  −598  402369 −267 402369 −1507
173305 −115 173305  −1507  523929 −427 523929 −1227

Table 6.9: Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2}
f  d_{1} d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
552 −46  6  −69  1560  −5 390 −78
616 −14 22  −77  1596  −3 399 −133
660 −33  5 −165  1624  −7 58 −406
663 −51 13 −663  1640 −10 41 −410
780 −3 195  −65  1640  −5 410 −82
840 −70  6 −105  1640  −2 410 −205
840 −30 14 −105  1704 −142  6 −213
840 −10 42 −105  1716  −3 143 −429
840 −10 210  −21  1752 −73  6 −438
840 −6 35 −210  1752  −6 438 −73
840 −6 70 −105  1752  −2 219 −438
840 −5 42 −210  1752  −1 438 −438
840 −5 210  −42  1768 −13 34 −442
840 −2 210 −105  1848 −42 33 −154
840 −1 210 −210  1848 −33 14 −462
884 −13 17 −221  1848 −22 231 −42
952 −2 119 −238  1848  −7 462 −66
952 −1 238 −238  1848  −1 462 −462
1015 −35 29 −1015  1860 −15 31 −465
1092 −21 13 −273 1932 −7 483 −69
1140 −19 15 −285  2040 −15 170 −102
1140 −15 19 −285  2040 −10 255 −102
1320 −30 22 −165  2040  −6 510 −85
1320 −22 15 −330  2040  −5 510 −102
1320 −22 30 −165  2072  −7 74 −518
1320 −10 66 −165  2184 −78 14 −273
1320 −10 330  −33  2184 −13 546 −42
1320 −6 55 −330  2184  −6 182 −273
1320 −6 110 −165  2184  −2 546 −273
1320 −5 66 −330  2244  −1 561 −561
1320 −1 330 −330  2280 −19 30 −570
1428 −21 17 −357  2280 −10 57 −570
1560 −13 390  −30  2280 −10 570 −57
1560 −10 195  −78  2280  −2 285 −570
1560 −6 195 −130  2328  −6 97 −582

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2}

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f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
2408 −14  86 −301  3560  −2 890 −445
2436  −7  87 −609  3588 −13  69 −897
2580  −5 129 −645  3588  −3 299 −897
2760 −115  6 −690  3660  −1 915 −915
2760 −15  46 −690  3720 −10 930  −93
2760 −15 230 −138  3864 −483  6 −322
2760 −10 138 −345  3864  −7 966 −138
2760  −6 230 −345  4020  −5 201 −1005
2760  −2 690 −345  4060 −35 203 −145
2856 −102  14 −357  4060  −7 1015 −145
2856  −6 238 −357  4092 −11 1023  −93
2860 −11 715  −65  4180 −19  55 −1045
3080 −22  70 −385  4180 −11  95 −1045
3080 −10 154 −385  4180  −5 209 −1045
3080  −2 770 −385  4251  −3 1417 −4251
3108 −37  21 −777  4424 −14 158 −553
3108  −7 111 −777  4440 −37 1110  −30
3108  −3 259 −777  4452  −7 159 −1113
3192 −42 133 −114  4515 −15 301 −4515
3192  −7 114 −798  4515  −3 1505 −4515
3192  −6 133 −798  4524 −13  87 −1131
3192  −6 798 −133  4524  −1 1131 −1131
3192  −3 266 −798  4632  −6 1158 −193
3220 −35  23 −805  4872 −58 1218  −21
3220  −7 115 −805  4935 −35 705 −987
3220  −5 161 −805  4940 −19 1235  −65
3315 −195  17 −3315  5016 −22 1254  −57
3315 −51  65 −3315  5115 −15 341 −5115
3315 −51 1105 −195  5160 −10 129 −1290
3315  −3 1105 −3315  5160  −2 645 −1290
3432  −3 286 −858  5208  −7 186 −1302
3432  −2 429 −858  5304 −51  78 −442
3480 −58 435  −30  5304  −3 1326 −442
3480  −6 290 −435  5320 −10 1330 −133
3560 −10 178 −445  5412  −1 1353 −1353

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
5640 −15 470 −282  7395 −51 2465  −435
5720 −22 715 −130  7420 −7 1855  −265
5772 −37  39 −1443  7480 −22 1870  −85
5772 −13 111 −1443  7917 −91  609 −1131
5964  −7 1491 −213  7917 −7 7917 −1131
6052  −1 1513 −1513  8008 −91  22 −2002
6072 −11 1518 −138  8008 −7  286 −2002
6072  −6 1518 −253  8052 −3  671 −2013
6105 −555  33 −2035  8120 −58 1015  −70
6105  −3 6105 −2035  8140 −37  55 −2035
6123  −3 2041 −6123  8140 −5  407 −2035
6216 −37 1554  −42  8184 −22 2046  −93
6216  −6 1554 −259  8268 −13  159 −2067
6315 −15 421 −6315  8268 −1 2067 −2067
6360  −6 530 −795  8680 −70  217  −310
6380 −11 1595 −145  8715 −35  249 −8715
6440 −10 805 −322  8835 −19  465 −8835
6460  −5 1615 −323  9020 −11 2255  −205
6540  −1 1635 −1635  9048 −58 1131  −78
6580 −235  7 −1645  9640 −10  241 −2410
6580  −5 329 −1645  9835 −35  281 −9835
6708 −43  39 −1677  9867 −3 3289 −9867
6708 −13 129 −1677  9880 −13 2470  −190
6708  −3 559 −1677  9960 −15  830  −498
6771  −3 2257 −6771  10065 −15 2013 −3355
6820 −11 155 −1705  10065 −11 10065  −915
6820  −5 341 −1705  10065 −3 10065 −3355
7035  −3 2345 −7035  10101 −7 10101 −1443
7080 −10 1770 −177  10120 −10 2530  −253
7161 −11 7161 −651  10360 −37 2590  −70
7176 −78  69 −598  10420 −5  521 −2605
7176  −3 1794 −598  10635 −15  709 −10635
7224 −42 301 −258  10815 −35 1545 −2163
7315 −35 209 −7315 10857 −11 10857 −987
7315 −11 665 −7315  10860 −1 2715 −2715

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
10948  −7  391 −2737  15096 −37 3774  −102
10948  −1 2737 −2737  15132 −13 3783  −291
11020  −5  551 −2755  15405 −15 15405 −1027
11067 −51  217 −11067  15576  −6 1298 −1947
11067  −3 3689 −11067  15580 −19 3895  −205
11180 −43 2795  −65  15715 −35  449 −15715
11388 −13 2847  −219  16215 −15 5405 −3243
11544 −37 2886  −78  16555 −11 1505 −16555
11704 −22 2926  −133  16555  −7 2365 −16555
11704 −19 2926  −154  16653 −427 16653  −39
11704  −7 2926  −418  16744 −91 4186  −46
11715 −15  781 −11715  16744  −7 4186  −598
11832 −58 1479  −102  17985 −11 17985 −1635
12136 −37 3034  −82  18088 −19 4522  −238
12264  −7 3066  −438  18156 −267 4539  −17
12408 −11 3102  −282  18312  −6 4578  −763
12580 −37 3145  −85  18655 −91 2665 −1435
12920 −10 3230  −323  18753 −19 18753  −987
13160 −10 1645  −658  19065 −123 19065  −155
13192  −2 1649 −3298  19285 −35 3857 −2755
13272 −42  158 −1659  19285  −7 19285 −2755
13340 −115  667  −145  19788 −51 4947  −97
13420  −5  671 −3355  20060  −5 5015 −1003
13420  −1 3355 −3355  20140 −19 5035  −265
13640 −22 3410  −155  20332 −13  391 −5083
13640 −11 3410  −310  20332  −1 5083 −5083
14155 −19  745 −14155  20805 −15 20805 −1387
14245  −7 14245 −2035  21576 −58 5394  −93
14469 −91 1113 −2067  21835 −11 1985 −21835
14469  −7 14469 −2067  22220 −11 5555  −505
14547  −3 4849 −14547  22632 −123 5658  −46
14685 −267 14685  −55  22737 −11 22737 −2067
14763  −7 2109 −14763  22792 −22 5698  −259
14892 −51 3723  −73  23560 −190  589  −310
14973 −483  217 −2139  23560 −19 5890  −310

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
24472 −19 6118 −322  61341 −483  2921 −2667
24860  −5 6215 −1243  61617 −19 61617 −3243
27740  −5 6935 −1387  62049 −43 62049 −1443
28120 −37 7030 −190  63365 −115 12673 −2755
30140 −5 7535 −1507 63784 −67 15946 −238
30856 −58 7714 −133  63804 −13 15951 −1227
31240 −22 7810 −355  73491 −51 24497 −4323
32648 −11 8162 −742  82137 −19 82137 −4323
33820 −19 8455 −445  82360 −58 20590 −355
34584  −6 2882 −4323  84040 −22 21010 −955
36312 −267 9078  −34  86860 −43 21715 −505
37596 −13 9399 −723  93513 −427 93513 −219
38157 −483 1817 −1659  94785 −267 94785 −355
38665 −19 38665 −2035 94940 −235 4747 −505
38860 −67 9715 −145  94956 −123 23739 −193
39372 −51 9843 −193  96577 −19 96577 −5083
40120 −10 10030 −1003  102856 −43 25714 −598
40664 −2 10166 −5083 103873 −19 103873 −5467
40936 −43 10234 −238  107004 −37 26751 −723
41064 −58 10266 −177  118105 −115 118105 −1027
43068 −37 10767 −291  119833 −19 119833 −6307
43505 −35 43505 −1243  136345 −67 136345 −2035
45580 −43 11395 −265  136396 −43 34099 −793
46460 −115 2323 −505  136840 −22 34210 −1555
47724 −123 11931 −97 139449 −43 139449 −3243
49720 −10 12430 −1243  151656 −267 37914 −142
50008 −19 12502 −658  157685 −235 31537 −3355
50456 −2 12614 −6307 159505 −115 159505 −1387
51585 −19 51585 −2715  172780 −163 43195 −265
52745 −35 52745 −1507  178585 −187 178585 −955
52972 −19 13243 −697  181596 −37 45399 −1227
53599 −91 7657 −4123  183964 −37 45991 −1243
55480 −10 13870 −1387  186796 −67 46699 −697
58696 −58 14674 −253  217281 −67 217281 −3243
60268 −19 15067 −793  232696 −58 58174 −1003

continued on following page

Fields where H ≃ ℤ_{4} × ℤ_{2} × ℤ_{2}

continued from previous page

f  d_{1}  d_{2}  d_{3}  f  d_{1}  d_{2}  d_{3}
235705 −235 235705 −1003  384865 −403 384865 −955
268801 −403 268801 −667  415185 −267 415185 −1555
274209 −267 274209 −1027  422569 −67 422569 −6307
308721 −427 308721 −723  454444 −163 113611 −697
327352 −58 81838 −1411  483784 −163 120946 −742
336921 −163 336921 −2067  517036 −163 129259 −793
340561 −67 340561 −5083  626665 −403 626665 −1555

Table 6.10: Fields where H ≃ ℤ_{2} × ℤ_{2} × ℤ_{2} × ℤ_{2}

\begin{matrix}f & d_{1} & d_{2} & d_{3} & f & d_{1} & d_{2} & d_{3} \\ 1320 & −2 & 330 & −165 & 5208 & −6 & 217 & −1302 \\ 1848 & −21 & 22 & −462 & 5304 & −13 & 1326 & −102 \\ 1848 & −2 & 231 & −462 & 6105 & −11 & 6105 & −555 \\ 2652 & −13 & 663 & −51 & 6460 & −19 & 1615 & −85 \\ 2760 & −30 & 46 & −345 & 7315 & −19 & 385 & −7315 \\ 2856 & −2 & 714 & −357 & 7755 & −11 & 705 & −7755 \\ 3080 & −70 & 22 & −385 & 8835 & −15 & 589 & −8835 \\ 3795 & −11 & 345 & −3795 & 8932 & −7 & 319 & −2233 \\ 4420 & −13 & 1105 & −85 & 11305 & −19 & 11305 & −595 \\ 4488 & −2 & 561 & −1122 & 11715 & −11 & 1065 & −11715 \\ 5016 & −2 & 1254 & −627 & 14763 & −3 & 4921 & −14763 \\ 5115 & −11 & 465 & −5115 & 18204 & −37 & 4551 & −123 \end{matrix}

Bibliography

[1] S. Arno. The imaginary quadratic fields of class number 4. Acta Arithmetica, LX.4:321– 334, 1992.

[2] E. Brown. The power of 2 dividing the class-number of a binary quadratic discriminant. J. Number Theory, 5:413–419, 1973.

[3] E. Brown. Class numbers of complex quadratic fields. J. Number Theory, 6:185–191, 1974.

[4] E. Brown. Class numbers of real quadratic number fields. Trans. Amer. Math. Soc., 190:99–107, 1974.

[5] E. Brown and C.J. Parry. The imaginary bicyclic biquadratic fields with class number 1. Reine Angew. Math., 266:118–120, 1974.

[6] D.A. Buell. Small class numbers and extreme values of l-functions of quadratic fields. Math. Comp., 31:786–796, 1977.

[7] D.A. Buell, H.C. Williams, and K.S. Williams. On the imaginary bicyclic biquadratic fields with class number 2. Math. Comp., 31:1034–1042, 1977.

Ramona R Ranalli  Chapter 5. Fields with Class Numbers 4, 8 and 16  92

[8] Hoffstein. On the siegel-tatuzawa theorem. Acta Arithmetica, 38:167–174, 1980/81.

[9] T. Kubota. Über den bizyklischen biquadratischen zahlkörper. Nagoya Math. J., 10:65–85, 1956.

[10] T.M. McCall, C.J. Parry, and R.R. Ranalli. The 2-rank of the class group of imaginary bicyclic biquadratic fields. Can. J. Math., 49:283–300, 1997.

[11] H.M. Stark. A complete determination of the complex quadratic fields of class number one. Michigan Math. J., 14:1–27, 1967.

[12] H.M. Stark. On complex quadratic fields with class number two. Math. Comp., 29:289–302, 1975.

Vita

Ramona Renee Ranalli was born on January 22, 1966, in Mount Holly, New Jersey while her parents awaited visas in order to begin their missionary work. In 1967 the family moved to Costa Rica and, in 1968, to Guayaquil, Ecuador. Here they remained until 1978 when they moved to the capital city of Quito. In 1983, at the age of 17, Ramona graduated from the Alliance Academy and began attending Houghton College, near Buffalo, N.Y.. She graduated from Houghton in 1987 with a math major and minors in physics, secondary education, Bible and Spanish. That same year she headed south for warmer weather and began working towards her MA in math from Wake Forest University. Upon receiving this degree in 1989 she moved to Virginia Polytechnic Institute and State University to begin work on her doctorate, again in math. She worked at Virginia Tech as a full-time instructor for three and half of the last four years of her work towards her PhD, which may go a little way towards explaining her long sojourn there.