Fields of Definition of Some

Three Point Ramified Field Extensions

Gunter Malle * Mathematisches Institut und IWR Im Neuenheimer Feld 368, D – 69120 Heidelberg

1 Introduction

In this paper we collect some computational data on fields of definition of three point ramified field extensions of ℚ̄(t) of degree n ≤ 13. No general results on such fields of definition, except for upper bounds on their degree and restrictions on possible ramified primes, are known at present. A more precise description of these fields would probably lead to results in the inverse problem of Galois theory, i.e., the question whether every finite group occurs as geometric Galois group over the field ℚ(t). The data presented here already gave hints for the possible validity of a translation theorem. This could then be proved to hold in general in [4].

Without any further knowledge one would expect the Galois group of a field of definition to be the full symmetric group. Our lists contain a number of examples where the Galois group is strictly smaller, although no known result predicts this.

There have been some previous investigations in this direction, for example by Atkin and Swinnerton-Dyer [1], who were interested in fields of definition attached to factor groups of the modular group SL_{2}(ℤ), and by Thompson [8], who considered a particular example in the context of the Guralnick–Thompson conjecture on groups of genus zero.

2 Background

We are interested in the fields of definition of finite field extensions K of

* The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft ℂ(t) ramified over the three points {0, 1, ∞}. There is a natural bijection between these extensions and the finite unramified coverings of the Riemann sphere ℙ^{1}(ℂ) with the three points {0, 1, ∞} removed, which in turn are in bijection with the set of dessins d’enfant (see [3^{∗}] and [6^{∗}] in this volume). The latter are classified by conjugacy classes of subgroups Δ of finite index of the fundamental group

π_{1} = ⟨γ_{1}, γ_{2}, γ_{3} | γ_{1}γ_{2}γ_{3} = 1⟩

of ℙ^{1}(ℂ)\{0, 1, ∞} (cf. [6^{∗}]). Any such subgroup Δ of index n gives rise to a transitive permutation representation ϕ of π_{1} into S_{n}. The Galois group of the Galois closure of K/ℂ(t) is then isomorphic to the image of π_{1} under this homomorphism.

It is well known that the same picture applies to finite extensions K of ℚ̄(t) ramified only over {0, 1, ∞} (see for example [5], I, §4, Satz 3). Any such extension K/ℚ̄(t) is already defined over a number field k of finite degree over ℚ. It is these fields that we are interested in.

Let N denote the Galois closure of K/ℚ̄(t). The homomorphism ϕ : π_{1} → S_{n} maps the generators γ_{i} onto generators σ_{i} of the group G = ϕ(π_{1}) ≤ S_{n}, which is isomorphic to the Galois group Gal(N/ℚ̄(t)). Conversely, for any triple (σ_{1}, σ_{2}, σ_{3}) of elements generating G ≤ S_{n} with σ_{1}σ_{2}σ_{3} = 1, there exists a homomorphism from π_{1} to S_{n} mapping the γ_{i} to σ_{i}. Hence any such triple (σ_{1}, σ_{2}, σ_{3}) gives rise to a field extension K/ℚ̄(t). Triples which are conjugate in S_{n} yield the same extension. More precisely the field extensions of ℚ̄(t) of degree n unramified outside {0, 1, ∞} and having group G ≤ S_{n} are parametrized by sets Σ_{G}(C_{1}, C_{2}, C_{3}) :=

{σ_{1}, σ_{2}, σ_{3} | σ_{i} ∈ C_{i}, σ_{1}σ_{2}σ_{3} = 1, G = ⟨σ_{1}, σ_{2}, σ_{3}⟩} mod S_{n},

where the C_{i} run over the conjugacy classes of S_{n} meeting G. This is called the Hurwitz classification [5]. The field extension corresponding to σ = (σ_{1}, σ_{2}, σ_{3}) is denoted by K_{σ}. It is easily seen that if two extensions K/ℚ̄(t), K′/ℚ̄(t) are algebraically conjugate, then they are classified by the same class triple C = (C_{1}, C_{2}, C_{3}) of the same subgroup G of S_{n}. So we get:

\underline{Proposition A}: An upper bound for the degree of the field of definition k_{σ} of K_{σ}/ℚ̄(t) is given by the number n_{G}(C) := |Σ_{G}(C)|.

Beckmann [2] has shown the following:

\underline{Proposition B}: For the choice \{0, 1, ∞\} of ramification points, only prime divisors of the group order |G| can ramify in k_{σ}/ℚ.

The reality type of k_{σ} can be read off from the tuple σ, as was noted by Serre [7]:

\underline{Proposition C}: The action of complex conjugation on the fields of definition induces an action

ρ : [σ_{1}, σ_{2}, σ_{3}] ↦ [σ_{1}^{−1}, σ_{2}^{−1}, σ_{2}σ_{3}^{−1}σ_{2}^{−1}]

on the corresponding classes modulo conjugation [σ] of element triples.

In the case where the class vector C admits symmetries, a little bit more is known. In those cases, either there is a nontrivial action of geometric automorphisms on Σ_{G}(C_{1}, C_{2}, C_{3}), which forces k/ℚ to be imprimitive, or the class vector can be obtained by a Galois translation from another class triple [5]. In some additional cases, the translation by a non-Galois extension also leads to information on the field of definition (see [4]).

Apart from that, the precise nature of k is at present a mystery.

3 The calculations

The calculation divides up into several different parts. First, for given degree n the triples σ of elements modulo conjugation of S_{n} with product 1 must be determined. Then the groups generated by these triples must be identified. This step was done using a C–program written by B. Przywara.

Next, the corresponding field extensions have to be constructed. This is done as described for example in [5], using Gröbner basis techniques. The occurring systems of nonlinear equations were solved using a Pascal program written by the author. The construction of the field extension immediately yields a generating polynomial f(x) for the field of definition k_{σ}/ℚ. In general, this polynomial is very badly conditioned, i.e., the order generated by a root of f has large index in the maximal order. The program system KANT II was used to calculate the field discriminant of k_{σ} from f and to obtain a small generating equation. We are grateful for having obtained access to the Kant II system prior to its publication. This program also determines the number of real embeddings r_{1} of k_{σ}. Of course, this can also be deduced without the explicit knowledge of k_{σ} from the purely group theoretical data using Proposition C. Independently, the field discriminant was also computed using the computer algebra system PARI.

The Galois group of the polynomial f and hence of the Galois closure of k_{σ}/ℚ was determined from the decomposition types modulo small primes in the case that it was the symmetric group. For groups of degree at most seven, the MAPLE Galois group program was used, and for degree 8 a program by Helmut Geyer for the determination of the Galois groups of degree 8 polynomials gave the answer.

4 The tables

The tables contain the data for field extensions of degree at most thirteen. However, the following restrictions have been made:

- Only extensions of genus zero are considered, since only for those a good and sufficiently mechanical algorithm for the computation of fields of definition is available.

- Only primitive permutation groups G were treated, since non primitive groups correspond to extensions containing a proper subextension, and so can be built together from smaller primitive ones.

- If a structure constant n_{G} equals one, then certainly the extension is defined over ℚ, so these have as well been omitted in the tables.

In addition to the above, the tables were further restricted as follows:

- For degree n = 9, only class vectors C containing at least one class of involutions are retained.

- For 10 ≤ n ≤ 13, we have only included those class vectors C containing one class of involutions and one class of elements of order three. In other words, the corresponding Galois groups are factor groups of the modular group SL_{2}(ℤ) modulo (in general: noncongruence) subgroups. Such extensions had already been studied in [1], 2.3 and 4.2. Contrary to the expectation put forward by these authors, we find examples where primes not dividing n or the element orders in the three classes ramify in the field of definition.

Thus the fields of definition considered here correspond to genus zero dessins of degree at most 13.

The tables are organized as follows. For each degree n we give a separate table for each primitive permutation group G having a class triple C satisfying the above restrictions. The first column contains the description of the class vector in terms of triples of partitions of n. (Trivial parts of partitions are suppressed.) The second counts the number n_{G}(C) of classes of triples generating G modulo conjugation in S_{n}. In the third, we give the Galois group of the Galois closure of the field of definition k. Note that by the Hurwitz classification it is a subgroup of the symmetric group S_{n(C)}. The fourth column contains the field discriminant d(k). By the Proposition B of Beckmann, d(k) is only divisible by prime divisors of |G|. In the case that neither KANT II nor PARI were able to compute the field discriminant, a ∗ is printed instead. The fifth column shows the number r_{1} of real embeddings of k. In the last column we give comments (”geo”, ”tr”) if the structure of the Galois group is restricted by the action of geometric automorphisms [5] or due to translation arguments [4].

In the second set of tables, we display generating equations for the fields of definition occurring in the previous tables. These are ordered by their degree d and then by their discriminant. The polynomials are given by the descending sequence of their coefficients, the highest a_{d} = 1 being omitted. These tables only cover fields of degree at most 9, since for larger fields the coefficients become to unwieldy. For field degrees 3 and 4 we have included an additional column counting the number of occurrences of this particular field of definition in the first set of tables.

Remark: The fields of definition found here also give a large number of examples for number fields ramified over only a few very small primes. It is not clear how such examples could be effectively constructed otherwise.

Degree 5, G = S_{5}

\begin{matrix}C & n(C) & Gal & d(k) & r_{1} \\ 3 − 4 − 3 2 & 2 & S_{2} & 2^{3}.3 & 2 \end{matrix}

Degree 6, G = S_{6}

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 3 2 − 6  3  S_{3}  −2^{2}.3^{3} 1
3 − 3 2 − 6  2  S_{2}  −2^{2} 0
3^{2} − 3 2 − 3 2  3  S_{3}  −2^{2}.3^{3} 1 as 2^{2}−32−6
4 − 4 − 4 2  2  S_{2}  −2^{2}.5 0
4 − 4 2 − 3 2  4  S_{4}  −2^{6}.3^{3}.5 2
4 − 5 − 3 2  4  S_{4}  2^{6}.3^{3}.5^{2} 0
4 2 − 3 2 − 3 2  4  2 ≀ S_{2}  −2^{6}.3^{3}.5 2  geo
5 − 3 2 − 3 2  7 (2 ≀ S_{3})^{+} × 1 2^{6}.3^{4}.5^{4} 2 geo, as 2^{3}−5−32
G = A_{6}
C  n_{G}(C)  Gal d(k) r_{1}  remarks
2^{2} − 4 2 − 5  2  S_{2} −3.5 0
2^{2} − 5 − 5  2  S_{2} −3.5 0 as 2^{2}−42−5
3 − 4 2 − 5  2  S_{2} 2^{3}.5 2

6 Fields of definition Degree 7, G = S_{7}

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 6 − 6  4  S_{4}  2^{4}.3^{3}.7 0 as 2^{3}−42−6
2^{2} − 6 − 5 2  3  S_{3}  −2^{2}.5.7 1
2^{2} − 6 − 4 3  3  S_{3}  −2^{2}.3^{3}.7 1
2^{2} − 5 2 − 5 2  3  S_{3}  −2^{2}.5.7 1 as 2^{3}−42−52
2^{2} − 5 2 − 4 3  2  S_{2}  3.5.7 2
2^{2} − 4 3 − 4 3  2  S_{2}  2^{2}.7 2 as 2^{3}−42−43
2^{3} − 4 2 − 6  4  S_{4}  2^{4}.3^{3}.7 0
2^{3} − 4 2 − 5 2  3  S_{3}  −2^{2}.5.7 1
2^{3} − 4 2 − 4 3  2  S_{2}  2^{2}.7 2
2^{3} − 5 − 6  3  S_{3}  −3^{3}.5 1
2^{3} − 5 − 4 3  2  S_{2}  −2^{2}.5 0
2^{3} − 3 2 − 7  3  S_{3}  −2^{2}.3.7^{2} 1
3 − 6 − 5 2  2  S_{2}  2^{2}.3.5 2
3 − 6 − 4 3  2  S_{2}  2^{3} 2
3 − 5 2 − 4 3  2  S_{2}  2^{3}.3.5 2
3^{2} − 4 − 6  2  S_{2}  −2^{3}.3 0
3^{2} − 4 − 5 2  3  S_{3}  −2^{3}.5^{2} 1
3^{2} − 3 2 − 6  6  A_{6}  2^{6}.3^{6}.7^{2} 2
3^{2} − 3 2 − 5 2  4  S_{4}  −2^{2}.3^{3}.5^{2}.7^{2} 2
3^{2} − 3 2 − 4 3  4  S_{4}  −2^{6}.3^{3}.7^{2} 2
4 − 4 2 − 6  4  2 ≀ S_{2}  2^{6}.3.5^{2}.7 0
4 − 4 2 − 5 2  4  S_{4}  −2^{6}.5^{2}.7 2
4 − 4 2 − 4 3  4  S_{4}  −2^{6}.3^{3}.5.7 2
4 − 5 − 5 2  2  S_{2}  2^{3} 2
4 − 5 − 4 3  3  S_{3}  2^{2}.3^{3}.5^{2} 3
4 − 3 2 − 7  3  S_{3}  −2^{3}.3^{3}.7^{2} 1
4 − 3 2^{2} − 6  3  S_{3}  −2^{3}.3^{3}.5 1
4 − 3 2^{2} − 4 3  2  S_{2}  2^{2}.3.5 2
4 2 − 3 2 − 6  12  S_{12} −2^{19}.3^{12}.5^{2}.7^{5} 2
4 2 − 3 2 − 5 2  9  S_{9} −2^{14}.3^{8}.5^{5}.7^{5} 3
4 2 − 3 2 − 4 3  9  S_{9}  −2^{16}.3^{6}.5.7^{5} 3
5 − 3 2 − 6  6  S_{6}  2^{6}.3^{6}.5^{3}.7^{2} 2
5 − 3 2 − 5 2  7  S_{7}  −2^{6}.3^{7}.5^{3}.7^{2} 1
5 − 3 2 − 4 3  5  S_{5}  −2^{8}.3^{3}.5^{3}.7^{2} 3
3 2 − 3 2 − 7  6  2 ≀ S_{3}  −2^{6}.3^{3}.7^{4} 0  geo
+3  S_{3}  −2^{2}.3.7^{2} 1 as 2^{3}−32−7
3 2 − 3 2^{2} − 6  6  S_{6}  −2^{8}.3^{6}.5.7^{2} 4
3 2 − 3 2^{2} − 5 2  4  S_{4}  −2^{2}.3^{3}.5^{2}.7^{2} 2
3 2 − 3 2^{2} − 4 3  3  S_{3}  −2^{3}.5.7^{2} 1

Degree 7, G = A_{7}

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 4 2 − 7  2  S_{2}  3.7 2
2^{2} − 5 − 7  2  S_{2}  3.7 2
2^{2} − 3 2^{2} − 7  2  S_{2}  3.7 2
3 − 4 2 − 7  2  S_{2}  −2.7 0
3^{2} − 4 2 − 4 2  4  S_{4}  2^{4}.3^{3}.7 0 from 2^{3}−42−6
3^{2} − 4 2 − 5  5  S_{5}  2^{6}.3^{3}.5^{3}.7 1
3^{2} − 4 2 − 3 2^{2}  3  S_{3}  −2^{3}.3^{3}.7 1
3^{2} − 5 − 5  3  S_{3}  −3^{3}.5 1 from 2^{3}−5−6
3^{2} − 5 − 3 2^{2}  3  S_{3}  2^{2}.3^{3}.5^{2} 3
4 2 − 4 2 − 4 2  12  S_{3} ≀ 2 2^{22}.3^{9}.5^{3}.7^{3} 0  geo
4 2 − 4 2 − 5  8  2 ≀ S_{4} −2^{12}.3^{6}.5^{5}.7 2  geo
+3  S_{3}  −2^{2}.5.7 1 from 2^{3}−42−52
4 2 − 4 2 − 3 2^{2}  4  2 ≀ S_{2}  2^{6}.3.5^{2}.7 0  geo
+2  S_{2}  2^{2}.7 2 from 2^{3}−42−43
4 2 − 5 − 5  4  2 ≀ S_{2}  2^{6}.3^{3}.5.7 0  geo
4 2 − 5 − 3 2^{2}  4  S_{3} × 1 −2^{3}.3.5^{2}.7 1
4 2 − 3 2^{2} − 3 2^{2}  2  S_{2}  3.5.7 2  geo
5 − 5 − 3 2^{2}  2  S_{2}  −2^{2}.5 0  geo
+2  S_{2}  2^{2}.3 2 from 2^{3}−5−43

G = L_{2}(7)
C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 3^{2} − 7  2  S_{2}  −7 0
2^{2} − 4 2 − 7  2  S_{2}  −7 0
3^{2} − 3^{2} − 4 2  4  2 ≀ S_{2} −2^{6}.3^{2}.7 2  geo
3^{2} − 4 2 − 4 2  2  S_{2}  2^{2}.7 2  geo
4 2 − 4 2 − 4 2  4  2 ≀ S_{2} −2^{6}.7 2  geo

8 Fields of definition Degree 8, G = S_{8}

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 6 − 8  2  S_{2}  2^{2}.7 2
2^{2} − 3^{2} 2 − 8  2  S_{2}  −2^{2} 0
2^{2} − 8 − 5 2  5  S_{5}  2^{9}.3.5^{3}.7 1
2^{2} − 8 − 4 3  5  S_{5}  2^{9}.3^{4}.7 1
2^{3} − 4 2 − 8  6  S_{6}  2^{11}.3.5^{2}.7 2
2^{3} − 4 2^{2} − 7  3  S_{3}  −2^{3}.3.7^{2} 1
2^{3} − 4 2^{2} − 5 3  2  S_{2}  2^{3}.5 2
2^{3} − 4^{2} − 5 2  3  S_{3}  −2^{3}.5^{2} 1
2^{3} − 4^{2} − 4 3  2  S_{2}  2^{3} 2
2^{3} − 5 − 8  2  S_{2}  2^{3}.3.5 2
2^{3} − 3 2^{2} − 8  6  S_{6}  2^{9}.3^{4}.5^{2} 2
2^{3} − 6 − 6 2  3  S_{3}  −2^{3}.3^{3}.7 1
2^{3} − 6 − 7  5  S_{5}  2^{3}.3^{3}.7^{4} 1
2^{3} − 6 − 5 3  4  S_{4} −2^{3}.3^{3}.5^{3}.7 2
2^{3} − 3^{2} 2 − 6 2  2  S_{2}  2^{3}.3 2
2^{3} − 3^{2} 2 − 7  3  S_{3}  −2^{3}.5.7^{2} 1
2^{3} − 3^{2} 2 − 5 3  2  S_{2}  2^{3}.3 2
2^{3} − 6 2 − 5 2  7  S_{7} −2^{12}.3^{3}.5^{5}.7 1
2^{3} − 6 2 − 4 3  5  S_{5}  2^{10}.3^{4}.7 1
2^{3} − 7 − 5 2  9  S_{9} 2^{12}.3.5^{6}.7^{6} 1
2^{3} − 7 − 4 3  9  S_{9} 2^{14}.3^{6}.5.7^{6} 1
2^{3} − 5 2 − 5 3  5  S_{5}  2^{10}.3^{2}.5.7 1
2^{3} − 4 3 − 5 3  5  S_{5} −2^{10}.3.5^{3}.7 3
2^{4} − 6 − 4 3  2  S_{2}  −3 0
2^{4} − 5 2 − 5 2  3  S_{3}  −2^{3}.5^{2} 1 as 2^{3}−4^{2}−52
2^{4} − 4 3 − 4 3  2  S_{2}  2^{3} 2 as 2^{3}−4^{2}−43
3 − 8 − 5 2  2  S_{2}  −2^{2}.5 0
3 − 8 − 4 3  2  S_{2}  −2^{3}.3 0
3^{2} − 4 2^{2} − 6  5  S_{5}  −2^{6}.3^{4}.7^{2} 3
3^{2} − 4 2^{2} − 5 2  4  S_{4} −2^{5}.3^{3}.5^{2}.7^{2} 2
3^{2} − 4 2^{2} − 4 3  7  S_{7}  2^{12}.3^{7}.7^{2} 3
3^{2} − 3 2 − 8  6  S_{6} −2^{14}.3^{4}.7^{2} 0
3^{2} − 6 − 6  3  S_{3}  −2^{3}.3^{3}.7 1 as 2^{3}−6−62
3^{2} − 6 − 3^{2} 2  4  S_{4}  −2^{5}.3^{3}.7^{2} 2
3^{2} − 6 − 5 2  9  S_{9} 2^{16}.3^{6}.5^{6}.7^{3} 1
3^{2} − 6 − 4 3  6  S_{6}  2^{8}.3^{6}.7^{3} 2
3^{2} − 3^{2} 2 − 3^{2} 2  2  S_{2}  2^{3}.3 2 as 2^{3}−3^{2}2−62

C  n_{G}(C) Gal  d(k) r_{1}  remarks
3^{2} − 3^{2} 2 − 5 2  8  S_{8}  −2^{10}.3^{4}.5^{5}.7^{2} 2
3^{2} − 3^{2} 2 − 4 3  4  S_{4}  −2^{5}.3^{3}.7^{2} 2
3^{2} − 5 2 − 5 2  6 2 ≀ S_{3}  −2^{10}.5^{3}.7^{4} 0  geo
+7  S_{7}  −2^{12}.3^{3}.5^{5}.7 1 as 2^{3}−62−52
+1  1  1 1 as 2^{4}−6−52
+1  1  1 1 as 2^{4}−3^{2}2−52
3^{2} − 5 2 − 4 3  12 S_{6} ≀ 2  2^{20}.3^{11}.5^{9}.7^{5} 4
3^{2} − 4 3 − 4 3  6 (2 ≀ S_{3})^{+}  2^{12}.3^{2}.7^{4} 2  geo
+5  S_{5}  2^{10}.3^{4}.7 1 as 2^{3}−62−43
+2  S_{2}  −3 0 as 2^{4}−6−43
4 − 4 2 − 8  3  S_{3}  −2^{2}.5.7 1
4 − 4 2^{2} − 7  3  S_{3}  −2^{2}.5.7^{2} 1
4 − 4 2^{2} − 5 3  3  S_{3}  2^{2}.3^{3}.5^{2} 3
4 − 4^{2} − 5 2  3  S_{3}  −2^{3}.5^{2} 1
4 − 3 2^{2} − 8  3  S_{3}  −3^{3}.5 1
4 − 6 − 6 2  2  S_{2}  3.5.7 2
4 − 6 − 5 3  3  S_{3}  2^{3}.3^{3}.5^{2}.7 3
4 − 3^{2} 2 − 6 2  2  S_{2}  −3.5 0
4 − 3^{2} 2 − 7  3  S_{3}  −2^{3}.5.7^{2} 1
4 − 6 2 − 5 2  4 S_{3} × 1  −2^{3}.3.5^{2}.7 1
4 − 6 2 − 4 3  6  S_{6}  2^{9}.3^{7}.5^{2}.7 2
4 − 7 − 5 2  4  S_{4}  2^{3}.5^{2}.7^{2} 0
4 − 7 − 4 3  4  S_{4}  2^{5}.3^{3}.5.7^{2} 0
4 − 5 2 − 5 3  4  S_{4}  −2^{3}.3^{3}.5^{2}.7 2
4 − 4 3 − 5 3  4  S_{4}  −2^{5}.3^{3}.5^{2}.7 2
4 2 − 4 2^{2} − 6  9  S_{9}  −2^{12}.3^{10}.5.7^{5} 3
4 2 − 4 2^{2} − 3^{2} 2  4  S_{4}  −2^{6}.3^{3}.5 2
4 2 − 4 2^{2} − 5 2  13 S_{13} −2^{27}.3^{6}.5^{10}.7^{5} 3
4 2 − 4 2^{2} − 4 3 12 S_{12} −2^{27}.3^{6}.5.7^{5} 2
4 2 − 3 2 − 8  12 S_{12} −2^{35}.3^{10}.5^{3}.7^{5} 2
4 2 − 6 − 6  6 2 ≀ S_{3}  2^{6}.3^{6}.5^{3}.7^{4} 2  geo
4 2 − 6 − 3^{2} 2  9  S_{9}  −2^{12}.3^{6}.5^{3}.7^{5} 3
4 2 − 6 − 5 2  18 S_{18}  ∗ 2
4 2 − 6 − 4 3  18 S_{18}  ∗ 4
4 2 − 3^{2} 2 − 3^{2} 2  4 2 ≀ S_{2}  2^{6}.3^{2}.5 0  geo
4 2 − 3^{2} 2 − 5 2  11 S_{11}  2^{24}.3^{4}.5^{6}.7^{5} 3
4 2 − 3^{2} 2 − 4 3  11 S_{11} −2^{23}.3^{10}.5^{4}.7^{5} 5
4 2 − 5 2 − 5 2  32 ≤ 2 ≀ S_{16}  ∗ 2  geo

C  n_{G}(C) Gal  d(k) r_{1} remarks
4 2 − 5 2 − 4 3  28 S_{28}  ∗ 4
4 2 − 4 3 − 4 3  24 ≤ 2 ≀ S_{12}  ∗ 6  geo
4 2^{2} − 5 − 6  4  S_{4}  −2^{5}.3^{3}.5^{2}.7^{2} 2
4 2^{2} − 5 − 3^{2} 2  3  S_{3}  2^{2}.3^{3}.5^{2} 3
4 2^{2} − 5 − 5 2  6  S_{6}  2^{12}.3^{3}.5^{4}.7^{2} 2
4 2^{2} − 5 − 4 3  7  S_{7}  2^{10}.3^{3}.5^{4}.7^{2} 3
4 2^{2} − 3 2 − 7  9  S_{9}  −2^{16}.3^{8}.5.7^{6} 3
4 2^{2} − 3 2 − 5 3  5  S_{5}  2^{11}.3^{3}.5^{3} 1
4 2^{2} − 3 2^{2} − 6  6  S_{6}  2^{6}.3^{6}.5.7^{2} 2
4 2^{2} − 3 2^{2} − 3^{2} 2  4  S_{4}  −2^{6}.3^{3}.5 2
4 2^{2} − 3 2^{2} − 5 2  11 S_{11}  2^{23}.3^{8}.5^{9}.7^{2} 3
4 2^{2} − 3 2^{2} − 4 3  6  S_{6}  2^{12}.3^{3}.5.7^{2} 2
4^{2} − 3 2 − 6  4  S_{4}  −2^{6}.3^{3}.7^{2} 2
4^{2} − 3 2 − 3^{2} 2  2  S_{2}  2^{3}.3 2
4^{2} − 3 2 − 5 2  6 A_{6}  2^{14}.3^{4}.5^{4}.7^{2} 2
4^{2} − 3 2 − 4 3  6  S_{6}  −2^{14}.3^{4}.7^{2} 0
5 − 3 2 − 8  4  S_{4}  2^{11}.3^{3}.5.7^{2} 0
5 − 6 − 3^{2} 2  5  S_{5}  2^{6}.3^{4}.5^{3}.7^{2} 1
5 − 6 − 5 2  4  S_{4}  2^{5}.3^{3}.5.7^{2} 0
5 − 6 − 4 3  6  S_{6}  −2^{12}.3^{6}.5^{4}.7^{2} 0
5 − 3^{2} 2 − 5 2  5  S_{5}  2^{5}.3^{3}.5^{2}.7^{2} 1
5 − 3^{2} 2 − 4 3  6  S_{6}  −2^{12}.3^{7}.5^{3}.7^{2} 4
5 − 5 2 − 5 2  8 2 ≀ S_{4}  −2^{9}.3^{6}.5^{4}.7^{4} 2  geo
+3  S_{3}  −2^{3}.5^{2} 1 as 2^{4}−52−52
5 − 5 2 − 4 3  14 ≤ S_{14}  2^{33}.3^{11}.5^{9}.7^{4} 2
5 − 4 3 − 4 3  6 (2 ≀ S_{3})^{+}  2^{6}.3^{4}.5^{4}.7^{4} 2  geo
+1  1  1 1 as 2^{4}−52−43
3 2 − 3 2^{2} − 8  12 S_{12}  2^{33}.3^{6}.5^{3}.7^{2} 0
3 2 − 6 − 6 2  10 S_{10}  2^{17}.3^{9}.5.7^{5} 2
3 2 − 6 − 7  8  S_{8}  2^{15}.3^{7}.7^{6} 0
3 2 − 6 − 5 3  8  S_{8}  −2^{17}.3^{6}.5^{5}.7^{5} 2
3 2 − 3^{2} 2 − 6 2  6  S_{6}  2^{11}.3^{6}.5 2
3 2 − 3^{2} 2 − 7  9  S_{9}  −2^{15}.3^{7}.5^{2}.7^{6} 3
3 2 − 3^{2} 2 − 5 3  5  S_{5}  2^{11}.3^{4}.5^{2} 1
3 2 − 6 2 − 5 2  14 S_{14}  2^{32}.3^{13}.5^{9}.7^{5} 2
3 2 − 6 2 − 4 3  14 S_{14} −2^{35}.3^{14}.5^{2}.7^{5} 4
3 2 − 7 − 5 2  17 S_{17}  ∗ 3
3 2 − 7 − 4 3  17 S_{17}  ∗ 3

C  n_{G}(C) Gal  d(k) r_{1} remarks
3 2 − 5 2 − 5 3  15 S_{15}  ∗ 5
3 2 − 4 3 − 5 3  11 S_{11}  2^{29}.3^{4}.5^{7}.7^{5} 3
3 2^{2} − 6 − 6  10 2 ≀ S_{5}  2^{12}.3^{9}.5^{2}.7^{6} 2  geo
+2  S_{2}  −3 0 as 2^{4}−6−43
3 2^{2} − 6 − 3^{2} 2  6  S_{6}  −2^{6}.3^{6}.5^{3}.7^{2} 4
3 2^{2} − 6 − 5 2  18 S_{18}  ∗ 0
3 2^{2} − 6 − 4 3  18 S_{18}  ∗ 0
3 2^{2} − 3^{2} 2 − 3^{2} 2  2  S_{2}  −2^{2} 0  geo
3 2^{2} − 3^{2} 2 − 5 2  7  S_{7}  2^{11}.3^{6}.5^{2}.7^{2} 3
3 2^{2} − 3^{2} 2 − 4 3  9  S_{9}  −2^{20}.3^{6}.5^{3}.7^{2} 3
3 2^{2} − 5 2 − 5 2  22 ≤ 2 ≀ S_{11}  ∗ 4  geo
+1  1  1 1 as 2^{4}−52−43
3 2^{2} − 5 2 − 4 3  25 S_{25}  ∗ 5
3 2^{2} − 4 3 − 4 3  18 ≤ 2 ≀ S_{9}  ∗ 2  geo
+2  S_{2}  2^{3} 2 as 2^{4}−43−43

Degree 8, G = A_{8}

C  n_{G}(C) Gal  d(k) r_{1}  remarks
2^{2} − 6 2 − 7  2  S_{2}  −2^{3}.7 0
2^{2} − 6 2 − 5 3  3  S_{3}  −2^{3}.3^{3}.5 1
2^{2} − 7 − 7  3  S_{3}  −2^{3}.3.7^{2} 1  from 2^{3}−42^{2}−7
2^{2} − 7 − 5 3  4 2 ≀ S_{2} −2^{3}.3^{3}.5^{2}.7^{2} 2
2^{2} − 5 3 − 5 3  3 S_{2} × 1  2^{3}.5 2 2^{3}−42^{2}−53,2^{4}−42−53
3 − 6 2 − 7  2  S_{2}  3.7 2
3 − 6 2 − 5 3  2  S_{2}  3.5.7 2
3 − 7 − 5 3  2  S_{2}  5 2
3^{2} − 3^{2} − 5 3  2  S_{2}  3.7 2  geo
3^{2} − 4 2 − 6 2  5  S_{5}  2^{9}.3^{4}.7 1
3^{2} − 4 2 − 7  6  S_{6}  2^{12}.3^{3}.5.7^{4} 2
3^{2} − 4 2 − 5 3  8  S_{8} −2^{15}.3^{6}.5^{6}.7^{2} 2
3^{2} − 4^{2} − 3 2^{2}  2  S_{2}  −3 0
3^{2} − 5 − 6 2  5  S_{5}  2^{4}.3^{4}.5^{3}.7 1
3^{2} − 5 − 7  3  S_{3}  −2^{2}.3^{3}.5.7 1
3^{2} − 5 − 5 3  3  S_{3}  −2^{2}.3^{3}.5.7 1
3^{2} − 3 2^{2} − 6 2  6  S_{6}  2^{10}.3^{6}.7 2
3^{2} − 3 2^{2} − 7  11  S_{11} −2^{14}.3^{10}.5.7^{7} 1
3^{2} − 3 2^{2} − 5 3  7  S_{7}  −2^{10}.3^{5}.5^{5}.7 1

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
4 2 − 4 2 − 4^{2}  6  S_{6}  2^{11}.3.5^{2}.7 2 from 2^{3}−42−8
4 2 − 4 2 − 6 2  16 ≤ 2 ≀ S_{8}  ∗ 2  geo
4 2 − 4 2 − 7  18 ≤ 2 ≀ S_{9}  ∗ 0  geo
4 2 − 4 2 − 5 3  15 2 ≀ S_{7} × 1 2^{28}.3^{7}.5^{10}.7^{4} 2 geo, as 2^{4}−42−53
4 2 − 4^{2} − 5  3  S_{3}  −2^{3}.3.5^{2}.7 1
4 2 − 4^{2} − 3 2^{2}  6  S_{6}  2^{11}.3^{3}.5^{2}.7 2
4 2 − 5 − 6 2  7  S_{7}  −2^{13}.3^{7}.5^{4}.7^{2} 1
4 2 − 5 − 7  6  S_{6}  2^{13}.3^{3}.5^{3}.7^{4} 2
4 2 − 5 − 5 3  8  S_{4} ≀ 2  2^{16}.3^{5}.5^{5}.7^{2} 4
4 2 − 3 2^{2} − 6 2  15  S_{15}  2^{35}.3^{15}.5^{6}.7^{2} 3
4 2 − 3 2^{2} − 7  22  S_{22}  ∗ 2
4 2 − 3 2^{2} − 5 3  14  S_{7} ≀ 2  2^{35}.3^{10}.5^{8}.7^{2} 6
4^{2} − 5 − 5  2  S_{2}  2^{3}.3.5 2 from 2^{3}−5−8
4^{2} − 5 − 3 2^{2}  4  S_{4}  2^{6}.3^{3}.5^{2} 0
4^{2} − 3 2^{2} − 3 2^{2}  6  S_{6}  2^{9}.3^{4}.5^{2} 2 from 2^{3}−32^{2}−8
5 − 5 − 6 2  2  S_{2}  7 2  geo
5 − 5 − 5 3  3 S_{2} × 1  3.7 2 geo, as 2^{4}−5−53
5 − 3 2^{2} − 6 2  6  S_{6}  2^{10}.3^{6}.5^{3}.7 2
5 − 3 2^{2} − 7  9  S_{9}  2^{12}.3^{8}.5^{5}.7^{6} 1
5 − 3 2^{2} − 5 3  5  S_{5}  2^{8}.3^{3}.5^{2}.7 1
3 2^{2} − 3 2^{2} − 6 2 12  2 ≀ A_{6}  −2^{20}.3^{12}.5^{5}.7 2  geo
3 2^{2} − 3 2^{2} − 7  17 ≤ 2 ≀ S_{8} × 1  ∗ 4 geo, as 2^{4}−32^{2}−7
3 2^{2} − 3 2^{2} − 5 3  9 2 ≀ S_{4} × 1 −2^{16}.3^{2}.5^{4}.7 2 geo, as 2^{4}−32^{2}−53

G = Hol(Z_{2}^{3})
C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 4^{2} − 7  2  S_{2}  −7 0
2^{2} − 6 2 − 7  2  S_{2}  −7 0
2^{2} − 7 − 7  2  S_{2}  −7 0 as 2^{4}−42−7
2^{4} − 4 2 − 7  2  S_{2}  −7 0
3^{2} − 4 2 − 4^{2}  2  S_{2}  2^{2}.7 2
3^{2} − 4 2 − 6 2  4  2 ≀ S_{2} −2^{6}.3^{2}.7 2
3^{2} − 4 2 − 7  4  V_{4}  2^{6}.7^{2} 0
4 2 − 4 2 − 4^{2}  4  2 ≀ S_{2} −2^{6}.7 2  geo
4 2 − 4 2 − 6 2  2  S_{2}  2^{2}.7 2  geo
4 2 − 4 2 − 7  2  S_{2}  −7 0 as 2^{4}−42−7

G = PGL_{2}(7)

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{3} − 3^{2} − 8 & 2 & S_{2} & 2^{3} & 2 \end{matrix}

G = L_{2}(7)
C  n_{G}(C)  Gal d(k) r_{1}  remarks
3^{2} − 3^{2} − 4^{2}  2  S_{2} 2^{3} 2 from 2^{3}−3^{2}−8

G = 2^{3}: 7: 3
C  n_{G}(C)  Gal d(k) r_{1}  remarks
3^{2} − 3^{2} − 6 2  4  2 ≀ S_{2} 3^{3}.7 0  geo
3^{2} − 3^{2} − 7  4  V_{4} 3^{2}.7^{2} 0  geo

Degree 9, G = S_{9}

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{2} − 6 3 − 8  5  S_{5}  2^{9}.3^{6} 1
2^{2} − 6 3 − 7 2  4  S_{4}  −2^{2}.3^{5}.7^{2} 2
2^{2} − 6 3 − 5 4  3  S_{3}  2^{2}.3^{5}.5 3
2^{2} − 8 − 8  6  S_{6}  −2^{10}.3^{5}.7 0 as 2^{4}−42^{2}−8
2^{2} − 8 − 7 2  5  S_{5}  2^{9}.3^{6}.7^{2} 1
2^{2} − 8 − 5 4  5  S_{5}  2^{9}.3^{5}.5^{3} 1
2^{2} − 7 2 − 7 2  5  S_{5}  2^{4}.3^{5}.7^{3} 1 as 2^{4}−42^{2}−72
2^{2} − 7 2 − 5 4  4  S_{4} −2^{2}.3^{5}.5^{2}.7^{2} 2
2^{2} − 5 4 − 5 4  3  S_{3}  2^{2}.3^{5}.5 3 as 2^{4}−42^{2}−54
2^{3} − 4 2^{2} − 9  9  S_{9}  −2^{7}.3^{18}.5.7 3
2^{3} − 4^{2} − 6 3  3  S_{3}  −2^{3}.3^{4} 1
2^{3} − 4^{2} − 8  6  S_{6}  −2^{11}.3^{4}.5^{2} 0
2^{3} − 4^{2} − 7 2  5  S_{5}  2^{5}.3^{4}.5.7^{3} 1
2^{3} − 4^{2} − 5 4  4  S_{4}  −2^{5}.3^{4}.5^{2} 2
2^{3} − 3^{2} 2 − 9  10  S_{10}  −2^{9}.3^{18}.5^{3} 0
2^{3} − 6 − 9  3  S_{3}  −2^{3}.3^{3}.7 1
2^{3} − 3 2^{3} − 9  3  S_{3}  −3^{3}.5 1
2^{3} − 6 2 − 6 3  9  S_{9}  2^{14}.3^{12}.7 1
2^{3} − 6 2 − 8  14  S_{14} −2^{33}.3^{15}.5^{2}.7^{3} 0

C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{3} − 6 2 − 7 2  13  S_{13}  ∗ 1
2^{3} − 6 2 − 5 4  11  S_{11}  2^{16}.3^{12}.5^{8}.7 3
2^{3} − 6 3 − 7  7  S_{7}  −2^{10}.3^{4}.5.7^{4} 1
2^{3} − 6 3 − 5 2^{2}  4  S_{4}  −2^{2}.3^{4}.5^{3}.7 2
2^{3} − 6 3 − 4 3 2  6  S_{6}  2^{6}.3^{8}.7 2
2^{3} − 6 3 − 5 3  10  S_{10}  2^{10}.3^{11}.5^{7} 2
2^{3} − 7 − 8  10  S_{10}  2^{25}.3^{5}.7^{6} 2
2^{3} − 7 − 7 2  6  S_{5} × 1  2^{10}.3^{5}.7^{2} 1 tr 2^{3}−3^{3}−72
2^{3} − 7 − 5 4  7  S_{7}  −2^{12}.3^{4}.5^{5}.7^{4} 1
2^{3} − 8 − 5 2^{2}  6  S_{6}  2^{9}.3^{4}.5^{4}.7^{2} 2
2^{3} − 8 − 4 3 2  12  S_{12} −2^{30}.3^{11}.5^{2}.7^{2} 2
2^{3} − 8 − 5 3  16  S_{16}  ∗ 0
2^{3} − 9 − 5 2  10  S_{10}  2^{14}.3^{18}.5^{7}.7 2
2^{3} − 9 − 4 3  10  S_{10}  2^{14}.3^{18}.5^{3}.7 2
2^{3} − 5 2^{2} − 7 2  6  S_{6}  2^{4}.3^{4}.5^{3}.7^{4} 2
2^{3} − 5 2^{2} − 5 4  4  S_{3} × 1  −3^{4}.5.7 1 tr 2^{3}−3^{3}−54
2^{3} − 4 3 2 − 7 2  9  S_{8} × 1 −2^{4}.3^{9}.5.7^{6} 2
2^{3} − 4 3 2 − 5 4  7  S_{7}  2^{8}.3^{8}.5^{4}.7 3
2^{3} − 7 2 − 5 3  10  S_{10}  2^{12}.3^{9}.5^{8}.7^{6} 2
2^{3} − 5 3 − 5 4  8  A_{7} × 1  2^{12}.3^{8}.5^{2} 3
2^{4} − 4 2^{2} − 8  6  S_{6}  −2^{10}.3^{5}.7 0
2^{4} − 4 2^{2} − 7 2  5  S_{5}  2^{4}.3^{5}.7^{3} 1
2^{4} − 4 2^{2} − 5 4  3  S_{3}  2^{2}.3^{5}.5 3
2^{4} − 3^{2} 2 − 6 3  3  S_{3}  −3^{5} 1
2^{4} − 3^{2} 2 − 8  6  S_{6}  −2^{11}.3^{6}.5^{2} 0
2^{4} − 3^{2} 2 − 7 2  4  S_{4}  −2^{2}.3^{5}.7^{2} 2
2^{4} − 3^{2} 2 − 5 4  3  S_{3}  −3^{5} 1
2^{4} − 6 − 6 3  2  S_{2}  −7 0
2^{4} − 6 − 8  4  S_{4}  2^{8}.3^{3}.7 0
2^{4} − 6 − 5 4  2  S_{2}  −5.7 0
2^{4} − 6 3 − 5 2  5  S_{5}  2^{3}.3^{5}.5^{3}.7 1
2^{4} − 6 3 − 4 3  4  S_{4}  −2^{4}.3^{5}.7 2
2^{4} − 8 − 5 2  9  S_{9}  2^{22}.3^{5}.5^{6}.7 1
2^{4} − 8 − 4 3  9  S_{9}  2^{22}.3^{9}.5^{2}.7 1
2^{4} − 5 2 − 7 2  5  S_{5}  2^{5}.3^{5}.5^{3}.7^{2} 1
2^{4} − 5 2 − 5 4  5  S_{5}  2^{4}.3^{5}.5^{2}.7 1
2^{4} − 4 3 − 7 2  4  S_{4}  −2^{2}.3^{5}.7^{2} 2
2^{4} − 4 3 − 5 4  4  S_{4}  −2^{5}.3^{5}.5.7 2

G = A_{9}
C  n_{G}(C) Gal  d(k) r_{1}  remarks
2^{2} − 4^{2} − 9  3 S_{3}  −3^{5} 1
2^{2} − 6 2 − 9  6 S_{6}  2^{5}.3^{11}.7 2
2^{2} − 7 − 9  3 S_{3}  2^{3}.3^{5}.7 3
2^{2} − 9 − 5 2^{2}  3 S_{3}  3^{5}.5.7 3
2^{2} − 9 − 4 3 2  6 S_{6}  2^{4}.3^{11}.7 2
2^{2} − 9 − 5 3  6 S_{3} ≀ 2 −2^{3}.3^{11}.5^{4} 0
2^{4} − 4 2 − 9  3 S_{3}  −3^{4}.5.7 1
2^{4} − 4^{2} − 6 2  4 A_{4}  2^{6}.3^{4} 0
2^{4} − 4^{2} − 7  4 A_{4}  2^{6}.7^{2} 0
2^{4} − 4^{2} − 4 3 2  3 S_{3}  −2^{2}.3^{4} 1
2^{4} − 4^{2} − 5 3  5 A_{5}  2^{6}.3^{4}.5^{2} 1
2^{4} − 3 2^{2} − 9  6 S_{6}  2^{4}.3^{10}.5 2
2^{4} − 6 2 − 6 2  4 A_{4}  2^{6}.3^{4} 0  as 2^{4}−4^{2}−62
2^{4} − 6 2 − 7  6 S_{6}  2^{6}.3^{7}.7^{3} 2
2^{4} − 6 2 − 5 2^{2}  3 S_{3}  −3^{4}.5.7 1
2^{4} − 6 2 − 4 3 2  6 S_{6}  2^{7}.3^{8}.7 2
2^{4} − 6 2 − 5 3  8 S_{8}  2^{6}.3^{9}.5^{5}.7 0
2^{4} − 7 − 7  4 A_{4}  2^{6}.7^{2} 0  as 2^{4}−4^{2}−7
2^{4} − 7 − 4 3 2  4 S_{4}  2^{5}.3^{4}.7^{2} 0
2^{4} − 7 − 5 3  8 S_{8} 2^{6}.3^{7}.5^{6}.7^{5} 0
2^{4} − 5 2^{2} − 4 3 2  3 S_{3}  2^{2}.3^{4}.5 3
2^{4} − 5 2^{2} − 5 3  3 S_{3}  −3^{4}.5^{2}.7 1
2^{4} − 4 3 2 − 4 3 2  3 S_{3}  −2^{2}.3^{4} 1  as 2^{4}−4^{2}−432
2^{4} − 4 3 2 − 5 3  4 S_{4} −2^{5}.3^{4}.5.7 2
2^{4} − 5 3 − 5 3  7 A_{5} × 1^{2} 2^{6}.3^{4}.5^{2} 1  geo, as 2^{4}−4^{2}−53

G = Hol(Z_{3}^{2})

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{3} − 3^{3} − 8 & 2 & S_{2} & −2^{3} & 0 \\ 2^{3} − 6 2 − 8 & 2 & S_{2} & −2^{3} & 0 \end{matrix}

G = ΓL_{2}(8)
C  n_{G}(C)  Gal  d(k) r_{1}  remarks
2^{4} − 3^{2} − 9  2  S_{2}  −3 0
2^{4} − 6 2 − 6 2  8  2.S_{4} 2^{8}.3^{8}.7^{2} 0  geo

G = L_{2}(8)

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 7 − 7 & 2 & 1^{2} & 1 & 2 \end{matrix}

G = G_{144}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{3} − 4^{2} − 8 & 2 & S_{2} & −2^{3} & 0 \end{matrix}

Degree 10, G = S_{10}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{3} − 3^{3} − 10 & 5 & F_{20} & 2^{4}.5^{5} & 1 \end{matrix}

G = A_{10}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{3} − 8 2 & 3 & S_{3} & −2^{3}.5^{2} & 1 \\ 2^{4} − 3^{3} − 9 & 6 & S_{6} & −3^{8}.5^{2}.7 & 0 \\ 2^{4} − 3^{3} − 6 4 & 3 & S_{3} & −2^{2}.3.5^{2} & 1 \\ 2^{4} − 3^{3} − 7 3 & 3 & S_{3} & −5^{2}.7 & 1 \end{matrix}

Degree 11, G = S_{11}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{5} − 3^{3} − 10 & 6 & S_{6} & −5^{2}.7^{2}.11 & 0 \\ 2^{5} − 3^{3} − 9 2 & 3 & S_{3} & −3^{4}.11 & 1 \\ 2^{5} − 3^{3} − 8 3 & 2 & S_{2} & 2^{3}.11 & 2 \\ 2^{5} − 3^{3} − 7 4 & 3 & S_{3} & −2^{2}.11 & 1 \\ 2^{5} − 3^{3} − 6 5 & 2 & S_{2} & −11 & 0 \end{matrix}

G = A_{11}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{3} − 11 & 8 & S_{8} & −5^{2}.7^{3}.11^{6} & 2 \end{matrix}

G = L_{2}(11)

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{3} − 11 & 2 & S_{2} & −11 & 0 \end{matrix}

Degree 12, G = S_{12}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{5} − 3^{3} − 12 & 9 & S_{9} & 2^{10}.3^{9}.5^{2}.7^{4}.11 & 1 \\ 2^{5} − 3^{4} − 9 2 & 3 & S_{3} & −2^{2}.3^{4} & 1 \\ 2^{5} − 3^{4} − 8 3 & 2 & S_{2} & −2^{3} & 0 \\ 2^{5} − 3^{4} − 7 4 & 2 & S_{2} & −7 & 0 \\ 2^{5} − 3^{4} − 6 5 & 2 & S_{2} & 2^{2}.3 & 2 \\ 2^{5} − 3^{4} − 7 3 2 & 2 & S_{2} & 2^{2}.7 & 2 \end{matrix}

G = A_{12}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{4} − 11 & 3 & S_{3} & −2^{2}.11^{2} & 1 \\ 2^{4} − 3^{4} − 7 5 & 2 & S_{2} & 2^{2}.7 & 2 \end{matrix}

G = M_{12}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{4} − 10 2 & 2 & S_{2} & −2^{2}.5 & 0 \\ 2^{4} − 3^{4} − 11 & 2 & S_{2} & −11 & 0 \\ 2^{6} − 3^{3} − 11 & 2 & S_{2} & −11 & 0 \end{matrix}

G = PGL_{2}(11)

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{5} − 3^{4} − 10 & 2 & S_{2} & 5 & 2 \end{matrix}

Degree 13, G = S_{13}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{5} − 3^{4} − 12 & 20 & S_{20} & ∗ & 0 \\ 2^{5} − 3^{4} − 11 2 & 10 & S_{10} & ∗ & 0 \\ 2^{5} − 3^{4} − 10 3 & 10 & S_{10} & ∗ & 0 \\ 2^{5} − 3^{4} − 9 4 & 9 & A_{9} & 2^{6}.3^{12}.13^{6} & 1 \\ 2^{5} − 3^{4} − 8 5 & 7 & D_{7} & −2^{9}.13^{6} & 1 \\ 2^{5} − 3^{4} − 7 6 & 8 & S_{8} & −2^{4}.3^{5}.7^{3}.13^{6} & 2 \end{matrix}

G = A_{13}

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{4} − 13 & 10 & S_{10} & −2^{2}.5^{2}.7^{2}.13^{8} & 0 \\ 2^{6} − 3^{3} − 13 & 5 & S_{5} & 7.11.13^{3} & 1 \\ 2^{6} − 3^{4} − 8 4 & 4 & S_{4} & 2^{5}.3^{3}.13 & 0 \\ 2^{6} − 3^{4} − 9 3 & 4 & S_{4} & 2^{2}.3^{4}.13 & 0 \\ 2^{6} − 3^{4} − 10 2 & 6 & S_{6} & −3^{3}.5^{5}.13 & 0 \\ 2^{6} − 3^{4} − 11 & 6 & S_{6} & −2^{4}.3^{4}.7.11^{4} & 0 \\ 2^{6} − 3^{4} − 6 4 3 & 2 & S_{2} & 13 & 2 \\ 2^{6} − 3^{4} − 8 3 2 & 3 & S_{3} & −2^{3}.13 & 1 \\ 2^{6} − 3^{4} − 7 4 2 & 3 & S_{3} & −2^{2}.7.13 & 1 \\ 2^{6} − 3^{4} − 6 5 2 & 2 & S_{2} & −3.13 & 0 \\ 2^{6} − 3^{4} − 7 5 & 4 & S_{4} & 2^{2}.3^{3}.5.7.13 & 0 \end{matrix}

G = L_{3}(3)

\begin{matrix}C & n_{G}(C) & Gal & d(k) & r_{1} \\ 2^{4} − 3^{4} − 13 & 4 & 2 ≀ S_{2} & 13^{3} & 0 \end{matrix}

Gunter Malle 19

field degree 3

d(k)  a_{2}, a_{1}, a_{0}  d(k)  a_{2}, a_{1}, a_{0}
−2^{2}.3^{3}  0, 0, 2  2  −3^{4}.5.7  0, −12, 19  3
−2^{2}.3^{4}  0, −3, 4  3  3^{5}.5.7  0, −27, −51  1
−2^{3}.3^{4}  0, −3, −10  1  −5^{2}.7  1, 2, 3  1
−3^{5}  0, 0, 3  3  −2^{3}.3.5^{2}.7  −1, 2, −38  3
−3^{3}.5  0, 3, −1  4  2^{3}.3^{3}.5^{2}.7  0, −45, 110  1
−2^{3}.3^{3}.5  3, 6, −2  2  −3^{4}.5^{2}.7  0, 15, 40  1
2^{2}.3^{4}.5  0, −12, −14  1  −2^{2}.3.7^{2}  −1, 5, 1  2
2^{2}.3^{5}.5  0, −18, 12  3  −2^{3}.3.7^{2}  1, −2, 6  2
−2^{3}.5^{2}  1, 2, −2  6  −2^{3}.3^{3}.7^{2}  0, −21, 70  1
−2^{2}.3.5^{2}  −1, −3, −3  1  −2^{2}.5.7^{2}  −1, 5, −13  1
2^{2}.3^{3}.5^{2}  0, −15, 20  4  −2^{3}.5.7^{2}  1, 5, −15  3
−2^{2}.3^{3}.7  0, −6, 12  1  −2^{2}.11  −1, 1, 1  1
−2^{3}.3^{3}.7  0, −6, −16  4  −3^{4}.11  0, 6, −1  1
2^{3}.3^{5}.7  0, −27, 30  1  −2^{2}.11^{2}  1, 4, −2  1
−2^{2}.5.7  0, 2, 2  5  −2^{3}.13  0, −1, 2  1
−2^{2}.3^{3}.5.7  0, −3, 12  2  −2^{2}.7.13  0, 4, 2  1

field degree 4

d(k)  a_{3}, a_{2}, a_{1}, a_{0}  d(k)  a_{3}, a_{2}, a_{1}, a_{0}
2^{6}.3^{4}  2, 6, 6, 3 2 −2^{3}.3^{3}.5^{3}.7  −1, 6, 9, −9 1
−2^{6}.3^{3}.5  2, 6, 0, −6^{(a)} 3 −2^{2}.3^{4}.5^{3}.7  0, 0, 10, −15 1
−2^{6}.3^{3}.5 2, −3, −4, −2^{(b)} 1  2^{6}.7^{2}  2, 2, 0, 2^{(e)} 2
2^{6}.3^{2}.5  −2, −4, 8, 10 1  2^{6}.7^{2}  −2, 1, 0, 14^{(f)} 1
2^{6}.3^{3}.5^{2}  2, 3, 4, 20^{(c)} 1  3^{2}.7^{2}  −1, −1, −2, 4 1
2^{6}.3^{3}.5^{2} 0, 6, −16, 12^{(d)} 1  −2^{5}.3^{3}.7^{2}  0, −3, −6, −6 2
−2^{5}.3^{4}.5^{2} 0, −9, −24, −18 1  −2^{6}.3^{3}.7^{2}  0, −6, −8, −24 2
−2^{6}.7  2, 1, 2, 1 2  2^{5}.3^{4}.7^{2}  0, 9, 34, 30 1
−2^{6}.3^{2}.7  2, −1, −8, −8 2  −2^{2}.3^{5}.7^{2} 0, −6, 14, −12^{(g)} 1
3^{3}.7  −1, 3, −1, 1 1  −2^{2}.3^{5}.7^{2}  −1, 0, 16, −4 2
2^{4}.3^{3}.7  2, 0, 0, 3 3  2^{5}.3^{3}.5.7^{2}  0, 15, 6, 18 2
2^{8}.3^{3}.7  0, 6, 4, 3 1  2^{11}.3^{3}.5.7^{2}  0, 36, 176, 942 1
−2^{4}.3^{5}.7  0, 3, 14, −3 1  2^{3}.5^{2}.7^{2}  1, −3, 0, 8 1
−2^{6}.3^{3}.5.7  2, 3, 14, 7 1 −2^{2}.3^{3}.5^{2}.7^{2} 1, −9, −29, −14 2
2^{6}.3^{3}.5.7 −8, 30, −32, 16 1 −2^{3}.3^{3}.5^{2}.7^{2}  0, 18, 0, −24 1
−2^{5}.3^{4}.5.7 0, −9, −16, −6 1 −2^{5}.3^{3}.5^{2}.7^{2}  2, −9, 60, 60 2
−2^{5}.3^{5}.5.7  −2, 0, 30, −33 1 −2^{2}.3^{5}.5^{2}.7^{2} −1, 9, −35, −70 1
−2^{6}.5^{2}.7  2, −2, −6, −5 1  2^{5}.3^{3}.13  0, −3, 2, 6 1
2^{6}.3.5^{2}.7  0, 2, 0, 21 2  2^{2}.3^{4}.13  1, 3, −1, 2 1
−2^{3}.3^{3}.5^{2}.7  1, −3, −6, 6 1 2^{2}.3^{3}.5.7.13  1, 9, −5, 4 1
−2^{5}.3^{3}.5^{2}.7 −2, 0, −20, −20 1  13^{3}  −1, 2, 4, 3 1

(a) = S_{4}, ^{(b)} = 2 ≀ S_{2}, ^{(c)} = 4 − 5 − 32, ^{(d)} = 4^{2} − 5 − 32^{2}, ^{(e)} = A_{4}, ^{(f)} = V_{4},

(g) = 2^{2} − 63 − 72.

20 Fields of definition field degree 5

d(k)  a_{4}, …, a_{0}  d(k)  a_{4}, …, a_{0}
2^{9}.3^{6}  −1, −2, 6, −6, 6  2^{4}.3^{4}.5^{3}.7  −3, 9, −15, 21, −21
2^{6}.3^{4}.5^{2}  0, −2, 4, −6, 4  2^{3}.3^{5}.5^{3}.7  −1, 8, −4, 7, 37
2^{11}.3^{4}.5^{2}  1, −8, 0, 18, −30  −2^{6}.3^{4}.7^{2}  −1, −5, 7, −7, 7
2^{11}.3^{3}.5^{3} −1, −6, 30, −45, 45  2^{10}.3^{5}.7^{2}  −1, −8, −20, 2, −2
2^{9}.3^{5}.5^{3} 1, 14, −10, 5, −35  2^{9}.3^{6}.7^{2}  −2, 16, −8, 37, −14
2^{4}.5^{5}  0, 0, 0, 0, −2 2^{5}.3^{3}.5^{2}.7^{2}  1, 5, −5, −2, 2
2^{9}.3^{4}.7  −1, 4, −8, 17, −5 −2^{8}.3^{3}.5^{3}.7^{2}  −1, 0, 36, −72, 24
2^{10}.3^{4}.7  1, −8, −4, 14, 14 2^{6}.3^{4}.5^{3}.7^{2}  −1, −2, 22, −49, 49
2^{10}.3^{2}.5.7  2, 6, 4, 3, −2 2^{5}.3^{5}.5^{3}.7^{2}  2, 11, −12, −96, −96
2^{8}.3^{3}.5^{2}.7 0, −4, −8, 12, −16  2^{4}.3^{5}.7^{3}  −1, −10, 24, −15, 3
2^{4}.3^{5}.5^{2}.7  −1, −7, 5, 7, 7  2^{5}.3^{4}.5.7^{3}  1, −3, −3, 6, 18
2^{9}.3.5^{3}.7 1, −6, 10, 25, −55  2^{3}.3^{3}.7^{4}  1, −1, 4, −8, 5
−2^{10}.3.5^{3}.7 −2, −6, 12, −9, 10  7.11.13^{3}  1, 1, 5, 2, 1
2^{6}.3^{3}.5^{3}.7  1, 2, −2, −20, 20

field degree 6

d(k)  a_{5}, …, a_{0}  d(k)  a_{5}, …, a_{0}
2^{11}.3^{6}.5  0, −3, −8, −3, 0, 5  2^{6}.3^{6}.5.7^{2} 3, −3, −18, −21, −15, −5
2^{4}.3^{10}.5  0, 3, −4, 9, −6, 1 2^{6}.3^{6}.5^{3}.7^{2}  0, −9, 4, 42, −60, 20
2^{9}.3^{4}.5^{2} 2, −1, −6, −2, 4, −1 −2^{6}.3^{6}.5^{3}.7^{2}  0, −3, 6, −12, −24, −8
−2^{11}.3^{4}.5^{2}  2, 3, 0, −2, 0, 2 −2^{12}.3^{7}.5^{3}.7^{2}  0, −36, −64, 24, 192, 64
−2^{11}.3^{6}.5^{2}  0, 0, −8, 9, 0, 18 2^{12}.3^{3}.5^{4}.7^{2}  −2, −5, 20, 25, −62, 19
2^{6}.3^{4}.5^{4}  0, 3, 10, 3, 0, 1 2^{9}.3^{4}.5^{4}.7^{2}  −2, 1, −6, 26, 20, −1
−2^{3}.3^{11}.5^{4} 0, −9, −18, 54, 36, 96 2^{14}.3^{4}.5^{4}.7^{2} 0, 0, −120, −150, 0, 2000
−2^{10}.3^{5}.7  2, −2, −4, 2, 4, 4 −2^{12}.3^{6}.5^{4}.7^{2} 0, 18, −140, 333, −360, 166
2^{10}.3^{6}.7  0, 3, 4, 3, 0, −1  2^{8}.3^{6}.7^{3}  0, 9, 4, 15, 18, −10
2^{6}.3^{8}.7  0, 6, −2, 3, −6, 2  2^{6}.3^{7}.7^{3}  −3, 3, −4, 12, −6, −6
2^{7}.3^{8}.7  0, 0, −6, −3, 0, −2  2^{12}.3^{2}.7^{4}  0, −10, 0, 31, 0, −36
2^{4}.3^{11}.7 −3, 6, −16, 15, −21, 16  −2^{6}.3^{3}.7^{4}  2, 8, 18, 24, 18, 6
2^{5}.3^{11}.7 3, −3, −17, −12, 30, 46  2^{12}.3^{3}.5.7^{4}  2, −9, 4, 4, 0, 28
2^{11}.3.5^{2}.7  2, 1, 4, 5, 6, 3  −2^{10}.5^{3}.7^{4}  0, −13, 0, 40, 0, 20
2^{11}.3^{3}.5^{2}.7  0, 6, 0, 6, 12, 2 2^{13}.3^{3}.5^{3}.7^{4}  2, 11, 36, −54, 60, −6
2^{9}.3^{7}.5^{2}.7 0, −3, 2, 12, −24, −38 2^{4}.3^{4}.5^{3}.7^{4}  0, −2, 2, −12, 8, −12
−3^{8}.5^{2}.7  −3, 6, −3, 3, −3, 2 2^{6}.3^{6}.5^{3}.7^{4}  −3, 18, 14, 63, 147, 56
2^{10}.3^{6}.5^{3}.7  0, −3, 8, 0, 0, 20 2^{6}.3^{4}.5^{4}.7^{4}  0, −3, 0, 3, 0, −36
−2^{14}.3^{4}.7^{2}  0, 6, 16, 18, 0, 4  −5^{2}.7^{2}.11  3, 5, 5, 0, −1, 2
2^{6}.3^{6}.7^{2}  0, 3, −4, 0, 0, −4 −2^{4}.3^{4}.7.11^{4}  3, 9, 11, 30, 18, 46
2^{12}.3^{3}.5.7^{2}  0, −2, 0, −3, −4, 2  −3^{3}.5^{5}.13  2, 0, 0, 5, 0, 5
−2^{8}.3^{6}.5.7^{2}  0, −9, −4, 15, 12, 5

field degree 7

\begin{matrix}d(k) & a_{6}, …, a_{0} \\ 2^{12}.3^{8}.5^{2} & 0, 0, 0, −3, 12, −2, −12 \\ 2^{8}.3^{8}.5^{4}.7 & 1, 3, −19, 11, −45, 35, −7 \\ −2^{12}.3^{3}.5^{5}.7 & 2, 2, 10, 10, 4, 28, −12 \\ −2^{10}.3^{5}.5^{5}.7 & −3, 10, −30, 40, −48, 84, −60 \\ 2^{12}.3^{7}.7^{2} & −1, −3, −1, −11, 3, −11, −1 \\ 2^{11}.3^{6}.5^{2}.7^{2} & −3, 3, 1, −12, 24, −12, −12 \\ −2^{6}.3^{7}.5^{3}.7^{2} & 1, −6, −2, 16, 0, −8, 10 \\ 2^{10}.3^{3}.5^{4}.7^{2} & 0, −6, 4, 8, −4, −14, −4 \\ −2^{13}.3^{7}.5^{4}.7^{2} & −1, 6, −26, 26, 30, −140, 140 \\ −2^{10}.3^{4}.5.7^{4} & −2, 3, 7, −22, 36, −30, 10 \\ −2^{12}.3^{4}.5^{5}.7^{4} & −2, 2, 10, −15, −138, −204, 954 \\ −2^{9}.13^{6} & −2, −2, −4, 3, −2, −6, −4 \end{matrix}

field degree 8

\begin{matrix}d(k) & a_{7}, …, a_{0} \\ −2^{16}.3^{2}.5^{4}.7 & 4, 6, 4, −13, −28, 10, 24, −3 \\ −2^{12}.3^{6}.5^{5}.7 & 0, 8, 0, 18, 0, −4, 0, −35 \\ 2^{6}.3^{9}.5^{5}.7 & 0, 9, 5, 30, 0, 65, −15, 45 \\ 2^{8}.3^{8}.7^{2} & 0, 0, 0, −3, 0, 12, −18, 9 \\ −2^{10}.3^{4}.5^{5}.7^{2} & −1, 0, −3, 0, 2, 8, −10, 4 \\ 2^{16}.3^{5}.5^{5}.7^{2} & −2, −9, 38, −46, −42, 181, −242, 111 \\ −2^{15}.3^{6}.5^{6}.7^{2} & 0, −10, −20, 60, 200, 370, 300, −325 \\ −2^{9}.3^{6}.5^{4}.7^{4} & 2, −9, −10, 34, 12, 0, 90, −30 \\ −2^{17}.3^{6}.5^{5}.7^{5} & −2, −20, 104, −496, 1364, −328, −1792, −1828 \\ 2^{6}.3^{7}.5^{6}.7^{5} & −3, 23, −36, 75, −291, 53, 522, 256 \\ 2^{15}.3^{7}.7^{6} & 0, 3, −10, −9, −24, −34, 108, 138 \\ −2^{4}.3^{9}.5.7^{6} & −2, 10, −5, 1, 58, −68, 91, −32 \\ −5^{2}.7^{3}.11^{6} & 0, 2, −3, 10, −14, 14, −8, 1 \\ −2^{4}.3^{5}.7^{3}.13^{6} & −1, −10, 1, 55, 0, −120, −6, 96 \end{matrix}

\begin{matrix}d(k) & a_{8}, …, a_{0} \\ 2^{14}.3^{12}.7 & −3, 12, −22, 42, −48, 56, −36, 24, −4 \\ −2^{7}.3^{18}.5.7 & 0, 0, −9, 9, 0, −6, 9, 0, −2 \\ 2^{22}.3^{9}.5^{2}.7 & −2, 8, −8, 20, −8, 68, −32, 38, −4 \\ 2^{22}.3^{5}.5^{6}.7 & −1, −12, 12, 116, −116, −440, 360, 540, −540 \\ −2^{20}.3^{6}.5^{3}.7^{2} & −1, −4, −16, −16, −16, 16, 24, 36, 12 \\ 2^{16}.3^{6}.5^{6}.7^{3} & 3, 14, 32, 76, 120, 140, 160, 100, 100 \\ −2^{12}.3^{10}.5.7^{5} & 3, −6, −10, 69, 135, −88, −312, −108, 76 \\ −2^{12}.3^{6}.5^{3}.7^{5} & 1, −4, −16, −17, 27, 30, −30, −60, 60 \\ −2^{14}.3^{8}.5^{5}.7^{5} & 3, −15, −85, −135, 87, 251, −1215, −2670, −1070 \\ −2^{16}.3^{6}.5.7^{5} & 3, 2, −2, 6, 38, 54, 30, −9, −3 \\ 2^{14}.3^{6}.5.7^{6} & 4, 7, 4, −17, −56, −23, 4, −32, −36 \\ −2^{16}.3^{8}.5.7^{6} & 1, −19, 5, 99, −249, 15, 1167, −768, 132 \\ −2^{15}.3^{7}.5^{2}.7^{6} & −2, −2, −20, 46, 82, −20, −200, −200, 300 \\ 2^{12}.3^{8}.5^{5}.7^{6} & 2, −16, −32, 36, 60, 420, 1800, 3375, 2250 \\ 2^{12}.3.5^{6}.7^{6} & 2, 3, 9, −10, 69, 168, −8, −64, 80 \\ 2^{10}.3^{9}.5^{2}.7^{4}.11 & 3, −12, −42, 30, 216, 234, 36, −9, −9 \\ 2^{6}.3^{12}.13^{6} & 3, −6, −30, 0, 90, 86, −30, −9, 87 \end{matrix}

References

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[2] S. Beckmann, Ramified primes in the field of moduli of branched coverings of curves, J. Algebra 125 (1989), 236–255.

[3^{∗}] J. Birch, Noncongruence subgroups, covers and drawings, this volume.

[4] G. Malle, Genus zero translates of three point ramified Galois extensions, Manuscripta Math. 71 (1991), 97–111.

[5] B.H. Matzat, Konstruktive Galoistheorie, Springer Lecture Notes 1284, Berlin - Heidelberg - New York 1987.

[6^{∗}] L. Schneps, Dessins d’enfant on a Riemann sphere, this volume.

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