Explicit Galois realization of transitive groups of degree up to 15

JÜRGEN KLÜNERS AND GUNTER MALLE

Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany

We describe methods for the construction of polynomials with certain types of Galois groups. As an application we deduce that all transitive groups G up to degree 15 occur as Galois groups of regular extensions of ℚ(t), and in each case compute a polynomial f ∈ ℚ[x] with Gal(f) = G.

1. Introduction

Until now, the inverse problem of Galois theory, i.e., the question whether every finite group occurs as the Galois group of a field extension of ℚ, has not been solved. Even less is known in the direction of explicit results. Complete results for permutation groups of small degree were previously only published for degrees up to eleven (see Eichenlaub (1996), Malle and Matzat (1999)). More then ten years ago, Malle (1987) completed the explicit realization of primitive non-solvable permutation groups of degree d ≤ 15 as Galois groups over ℚ(t). The purpose of this paper is to extend this result to cover all transitive permutation groups up to this degree. Thus we give polynomials for the 477 transitive groups of degree between 12 and 15. Note that there exist 1954 distinct transitive groups of degree 16. In fact we verify the following stronger result:

Theorem 1.1. The transitive groups of degree d ≤ 15 have regular Galois realizations over ℚ(t).

The methods presented here would in principle allow to explicitly construct such regular field extensions in all cases.

We encounter two types of problems. First, as mentioned above, not all the groups in the range were even theoretically known to occur as Galois groups over ℚ. Secondly, there arises the practical problem how to come from theoretical existence results to explicit polynomials. We recall some of the approaches to the first problem and give algorithmic solutions for the second. The algorithms are not restricted to the ground field ℚ but work, for example, for any Hilbertian field.

An important tool in the constructions is a Galois group program which for a polynomial f ∈ ℤ[x] gives a permutation representation of Gal(f) on the set of complex (approximations to the) roots of f, as provided by the computer algebra system Kant (Daberkow et al., 1997). We are grateful to the Kant-team for obtaining access to this facility prior to its official release.

Conversely, the polynomials constructed here can be (and have been) used as comprehensive test input for Galois group programs.

The polynomials given here (and many more) will be made available shortly via a database in Kant (see Klüners and Malle (2000)). Meanwhile, they can be fetched from our respective homepages http://www.iwr.uni-heidelberg.de/˜ Juergen.Klueners/ http://www.mathematik.uni-kassel.de/˜ malle/

2. General methods

As noted in the introduction, it is not known whether every finite group is a Galois group over ℚ. In particular, there does not (yet) exist a general method which would allow, starting from some finite group G, to construct an extension N/ℚ with Galois group G. Nevertheless, there exist approaches allowing to handle certain subclasses of groups. In this section we recall and describe some of these methods for the (explicit) construction of field extensions with given Galois group. Very roughly, these divide into the rigidity method on the one hand, which first constructs regular extensions over some rational function field and then invokes Hilbert’s irreducibility theorem, and into applications of class field theory on the other. But we will stress another subdivision, which relates to the structure of groups to be considered. First, in order to get started, one needs to realize simple groups as Galois groups. As a second step, field extensions with composite groups can be obtained by solving embedding problems. Let’s first recall the approaches used to tackle simple groups.

2.1. Construction of extensions with simple Galois group

The most successful method so far for realizing finite simple groups as Galois groups is via rigidity. For cyclic groups of prime order, Kummer theory provides an alternative method. We first review a very special case of rigidity and then explain how in some cases a descent to subgroups is possible; for details and proofs we refer the reader to Malle and Matzat (1999), Chap. 1.

2.1.1. The rigidity method

Let G be a finite group, and assume that the center Z(G) has a complement in G. A triple C := (C_{1}, C_{2}, C_{3}) of conjugacy classes of G is called rigid, if the set

{(σ_{1}, σ_{2}, σ_{3}) | σ_{i} ∈ C_{i}, σ_{1}σ_{2}σ_{3} = 1, ⟨σ_{1}, σ_{2}⟩ = G}

is non-empty and a single orbit under G-conjugation. To any conjugacy class C we attach the cyclotomic field extension of ℚ generated by the values of the irreducible characters of G on C. Let ℚ_{C} be the composite of these extensions for the three classes in C. The class vector C is called rationally rigid if ℚ_{C} = ℚ.

Assume that C is rigid. Then the Rigidity Theorem states that there exists a regular Galois extension N/ℚ_{C}(t) with group G, i.e., an extension such that ℚ_{C} is algebraically closed in N. Moreover, N/ℚ_{C}(t) is ramified in precisely three points, and the ramification is described in terms of C. This allows to explicitly construct a generating polynomial for N/ℚ_{C}(t) by solving a system of non-linear equations which can be derived solely from the knowledge of C (see Malle (1987), Malle and Matzat (1999) for worked out examples).

By Hilbert’s irreducibility theorem, for any such extension there exist infinitely many specializations of t such that the specialized extension of ℚ_{C} is Galois with group G. In particular, a rationally rigid class vector leads to infinitely many Galois realizations of G over ℚ which can be constructed explicitly.

It has turned out that this criterion is particularly well-suited for an application to almost simple groups, i.e., to groups lying between a non-abelian simple group and its automorphism group (see Malle and Matzat (1999)). A variant, which takes into account the action of the cyclotomic character, can also be applied to cyclic groups (which have rigid, but not rationally rigid class vectors) and gives regular Galois realizations over ℚ(t) for all cyclic groups (see Malle and Matzat (1999), Chap. 3.4).

2.1.2. Descent to subgroups

The Galois extensions N/ℚ(t) constructed by the rigidity method lend themselves to a descent trick which under favorable circumstances allows the realization of subgroups as well. So assume that N/ℚ(t) is a regular Galois extension with group G ramified in three prime divisors of degree 1 of ℚ(t). Let H ≤ G be a subgroup and K := N^{H} its fixed field, so N/K is Galois with group H. If it can be shown that K is again a rational function field K = ℚ(u), then this yields a regular Galois realization of H over ℚ(t), which by the Hilbert irreducibility theorem has infinitely many specializations to Galois extensions of ℚ with group H. This descent procedure always works if H is of index 2 in G. Indeed, in this case H is normal, the Galois extension K/ℚ(t) has Galois group Z_{2} and is ramified in precisely two prime divisors. The Hurwitz relative genus formula then shows that K has genus 0, and moreover any of the ramified prime divisors has degree 1, so that K is a rational function field.

2.2. Construction of extensions with composite Galois group

We now describe some methods to obtain new Galois extensions from given ones. The ground field k is assumed to be Hilbertian, that is, for every irreducible polynomial f(t, X) ∈ k(t)[X] there exist infinitely many t_{0} ∈ k with f(t_{0}, X) irreducible.

2.2.1. Factor groups

Some of the constructions to be presented first yield a Galois extension N/k whose Galois group G = Gal(N/k) has the desired group G′ as a factor. By the main theorem of Galois theory, if H denotes the kernel of the canonical epimorphism G → G′, then G′ = G/H occurs as the Galois group Gal(N^{H}/k) of the fixed field of H, so theoretically the problem is solved. For the explicit construction using resolvents see Section 3.3.

2.2.2. Direct products

Let G_{1}, G_{2} be two groups which are known to occur as Galois groups over k. If at least for one of the two, say G_{2}, we have a regular realization over k(t), then also the direct product G_{1} × G_{2} occurs over k. Indeed, by the Hilbert irreducibility theorem, for any given G_{1}-extension N/k there exist infinitely many specializations of the regular G_{2}-extension linearly disjoint from N/k. The composite of any such specialization with N/k then gives a Galois extension with group G_{1} × G_{2}.

2.2.3. Wreath products

Let H, G be two finite groups such that G is realized as the Galois group of N/k and H is realized as a regular extension of k(t). We want to construct a field extension N′/k with Galois group G′ = H ≀ G the wreath product of H with G (with respect to some faithful transitive permutation representation of G). Let f(x) ∈ k[x] be a polynomial with group G such that the induced permutation representation of G on the roots of f is the desired one. Let g(t, x) ∈ k(t)[x] be a generating polynomial for a regular H-extension. Denote by α_{1}, …, α_{m} the ramification points of its splitting field in some algebraic closure of k. Let β be a primitive element of a stem field of N/k, i.e., of an intermediate field k ≤ M ≤ N whose Galois closure over k equals N. Furthermore denote by β_{1} = β, …, β_{n} the Galois conjugates of β. Since k is Hilbertian, it is infinite and there exists

γ ∈ k \ \{\frac{α_{i_{1}} − α_{i_{2}}}{β_{j_{1}} − β_{j_{2}}} | 1 ≤ i_{1}, i_{2} ≤ m, 1 ≤ j_{1} ≠ j_{2} ≤ n\} .

Then the splitting fields of the polynomials g(t − γβ_{j}, x), 1 ≤ j ≤ n, are pairwise linearly disjoint over k(t) since they are ramified in disjoint sets. Thus the splitting field of h(t, x) := ∏_{j=1}^{n} g(t − γβ_{j}, x) has as Galois group the wreath product H ≀ G. Since k is Hilbertian, there exist infinitely many specializations of t to t_{0} ∈ k such that the splitting field of h(t_{0}, x) has the same Galois group.

2.2.4. Split extensions with abelian kernel

Let G′ be a finite group with an abelian normal subgroup H having a complement G in G′. Then by Suzuki (1982), Th. 10.10 the group G′ is a factor group of the regular wreath product H ≀_{r}G. In particular, if G is realized as the Galois group of an extension N/k then G′ can be realized by combining the methods for wreath products and factor groups. One class of groups for which this method is particularly suited are the semi-abelian groups. They are defined inductively as follows. All finite abelian groups are semi-abelian, and a finite group G is called semi-abelian if it can be obtained as a quotient of a semi-direct product with abelian kernel and strictly smaller semi-abelian complement.

2.2.5. Subdirect products

Let K_{i}/k, i = 1, 2, be two Galois extensions inside some algebraic closure of k with groups G_{i}. Let G denote the Galois group of the composite of K_{1} and K_{2} over k and ϵ_{i} : G_{i} → H the epimorphisms onto H := Gal((K_{1} ∩ K_{2})/k) induced by restriction to K_{1} ∩ K_{2}. Then G is the subdirect product G_{1} ×_{H} G_{2}, i.e., the subgroup {(g_{1}, g_{2}) | g_{i} ∈ G_{i}, ϵ_{1}(g_{1}) = ϵ_{2}(g_{2})} of the direct product.

Thus, conversely, we can realize the subdirect product G_{1} ×_{H} G_{2} defined with respect to epimorphisms ϵ_{i} provided we have Galois extensions K_{i}/k with groups Gal(K_{i}/k) = G_{i} whose intersection has group H and such that the restriction maps Gal(K_{i}/k) → Gal(K_{1} ∩ K_{2}/k) coincide with ϵ_{i}, i = 1, 2.

3. Algorithms

Having reviewed the theoretical solutions, we now turn to the algorithmic and practical point of view. The setup will always be as follows. A Galois extension N/k (with group G) is given by the minimal polynomial f ∈ k[x] of a primitive element α of a stem field M of N/k. Thus G = Gal(N/k) is realized as a transitive permutation group on the set of roots of f, with point stabilizer equal to the fix group of M. To construct a Galois extension N′/k with group G′ from N/k will then mean to give an algorithm how to compute the minimal polynomial of a primitive element of a stem field of N′/k from f.

We first describe how the realization of the Galois group as permutation group on the roots can be obtained in practice. Then we address the different cases described in the previous section. Finally, we remark about how to reduce the size of the coefficients of the constructed polynomials.

3.1. The computation of Galois groups

Let f ∈ ℤ[x] be a monic irreducible polynomial. In this section we describe how to identify the Galois group G of f as a permutation group on (complex approximations of) the roots of f. For our purposes it is very important not only to know the abstract group G, but also its action on the roots of f. This has been implemented in KASH (Daberkow et al., 1997) for polynomials of degree up to 15. The algorithm is based on Stauduhar’s method and described in Geißler and Klüners (2000). The result of this algorithm is the name of the group and an ordering of the roots such that the action of the Galois group on this root ordering is equivalent to the computed transitive permutation group as classified in GAP (Schönert et al., 1997).

In the special situation that K/ℚ is an abelian number field, Acciaro and Klüners (1999) present an algorithm to compute the automorphism group. It is feasible to compute the automorphism group for abelian fields with degree bigger than 100. Klüners (1997) has extended this algorithm to normal number fields K/ℚ and to relative extensions L/K, where L/K is abelian.

3.2. The computation of H-invariant G-relative polynomials

Definition 3.1. Let H ≤ G be permutation groups on {x_{1}, …, x_{n}}. We call F ∈ ℤ[x_{1}, …, x_{n}] an H-invariant G-relative polynomial if

1 F^{σ} = F for all σ ∈ H, 2 F^{σ} ≠ F for all σ ∈ G \ H.

In this case

R_{G,H,F} := ∏_{σ∈G//H}(X − F^{σ}) ∈ ℤ[x_{1}, …, x_{n}, X]

is called a resolvent, where G//H denotes a full system of representatives of (right cosets of) G/H.

It is well known that H-invariant G-relative polynomials always exist:

Lemma 3.2. For H ≤ 𝔖_{n} let

F (x_{1}, …, x_{n}) := ∑_{σ∈H}(x_{1}^{1}x_{2}^{2} ⋯ x_{n}^{n})^{σ}.

Then Stab_{𝔖_{n}}(F) = H.

In practice it is not very efficient to use this polynomial. Our aim is to find an invariant of small degree. We use a very simple approach to compute H-invariant G-relative polynomials:

1 Set d := 1.

2 Compute all homogeneous invariants of H of degree d.

3 Check if these invariants are G-relative.

4 If there are H-invariant G-relative polynomials, return one with the smallest num-

ber of monomials. 5 Set d := d + 1 and go to 2.

For step 2 we use the algorithms implemented in Magma (Kemper and Steel, 1999). This part is the most expensive step of our algorithm. In the sequel we give some improvements which are useful in our situation.

Remark 3.3. Let H ≤ U ≤ G be permutation groups acting on {x_{1}, …, x_{n}} and F ∈ ℤ[x_{1}, …, x_{n}] be an H-invariant U-relative polynomial. Define Ũ := Stab_{G}(F). Then the following holds:

1 Ũ ∩ U = H.

2 Let F̃ ∈ ℤ[x_{1}, …, x_{n}] be an H-invariant Ũ-relative polynomial. If F and F̃ are

homogeneous of different degrees then F + F̃ is H-invariant G-relative. 3 If U is the only group with H ≨ U ≨ G, then F is H-invariant G-relative.

Proof. The first part follows directly from the definitions of U and Ũ. In the second clearly F +F̃ is H-invariant. If g ∈ G with (F +F̃)^{g} = F +F̃ then (F^{g}−F)+(F̃^{g}−F̃) = 0. Since F and F̃ have different degrees this implies F^{g} − F = 0 and F̃^{g} − F̃ = 0. This forces g ∈ U ∩ Ũ = H, proving that F + F̃ is H-invariant G-relative.

In the third part we get Ũ = H, therefore F is H-invariant G-relative. □

Using 2 we hope to find a smaller H-invariant G-relative polynomial. Apart from this remark we only consider homogeneous polynomials F.

We remarked that the most expensive part of this algorithm is the computation of H-invariants. In our examples we consider groups H of size up to 10^{10} acting on up to 84 points. In these cases it is impossible to compute the invariants of H of degree 3 or higher. The following theorem gives a criterion which under suitable conditions allows us to reduce to the computation of the invariants of a smaller group.

Theorem 3.4. Let H ≤ U ≤ G be permutation groups acting on {x_{1}, …, x_{n}}. Assume that U = Stab_{G}(Δ) := {g ∈ G | Δ^{g} = Δ} is the stabilizer of a block Δ ⊆ {x_{1}, …, x_{n}}. Define H̄ := H|_{Δ} and Ū := U|_{Δ}. If [Ū : H̄] = [U : H], then an H̄-invariant Ū-relative polynomial F is H-invariant U-relative.

Proof. From H ≤ U we get Δ^{H} = Δ. Since F only involves indeterminates x_{i} ∈ Δ we see that F σ̄ = F^{σ} for any σ ∈ U with image σ̄ ∈ Ū, and in particular F is H-invariant. Now let σ ∈ U \ H, so σ lies in a non-trivial H̄-coset. But then by assumption its image σ̄ ∈ Ū lies in a non-trivial H-coset, so σ̄ ∉ H̄. Since F is H̄-invariant Ū-relative this implies F^{σ} = F σ̄ ≠ F, and F is H-invariant U-relative. □

3.3. The computation of fixed fields

Let f ∈ ℤ[x] be a monic irreducible polynomial with zeros Ω := {α_{1}, …, α_{n}} and G be the Galois group of f acting on Ω. Let G̃ be another permutation group with point stabilizer H̃. Assume given an epimorphism ϕ : G → G̃. Our aim is the computation of a polynomial g with Galois group G̃. We denote by H the preimage of H̃ under ϕ. Then G acts on the right cosets G/H as G̃. Now we can compute g as the minimal polynomial of a primitive element of the fixed field Fix(H).

In the following we give a procedure to compute a fixed field.

Suppose that we have computed H, F, and R_{G,H,F}, where F is an H-invariant Grelative polynomial. Since G is acting on Ω it is easy to see that F (α_{1}, …, α_{n}) ∈ Fix(H) and R_{G,H,F} is the corresponding characteristic polynomial. If R_{G,H,F} is irreducible we have Gal(R_{G,H,F}) = G̃. Otherwise R_{G,H,F} is the power of an irreducible polynomial which is easy to check. In this case we have to apply a Tschirnhausen transformation to the roots of f and repeat the procedure. Girstmair (1983) has proven that one can compute a finite set of elements, which can be used for Tschirnhausen transformations. At least one of these transformations yields an irreducible R_{G,H,F}.

Algorithm 3.5. (ComputePolynomialFactorGroup)

\underline{Input}: A monic polynomial f ∈ ℤ[x], G := Gal(f), a group G̃, the roots \{α_{1}, …, α_{n}\} of f such that G acts on {α_{1}, …, α_{n}}.

\underline{Output}: A polynomial g ∈ ℤ[x] with Galois group G̃ if and only if there is an epimorphism from G → G̃, otherwise an indication of failure.

\underline{Step 1}: Try to find an epimorphism ϕ : G → G̃. If none exists then return ”Failure”.

\underline{Step 2}: Compute the preimage H of a point stabilizer of G̃ under ϕ.

\underline{Step 3}: Compute an H-invariant G-relative polynomial F ∈ ℤ[x_{1}, …, x_{n}].

\underline{Step 4}: Compute R_{G,H,F} (α_{1}, …, α_{n}) ∈ ℤ[x].

\underline{Step 5}: If gcd(R_{G,H,F} (α_{1}, …, α_{n}), R′_{G,H,F} (α_{1}, …, α_{n})) = 1 then return R_{G,H,F} (α_{1}, …, α_{n}).

\underline{Step 6}: Apply a Tschirnhausen transformation to f, update the roots \{α_{1}, …, α_{n}\} and go to step 4.

We remark that it is not necessary that the polynomial f is irreducible, i.e., that the group G acts transitively on the roots of f. The sorting of the roots can be computed with the Galois function of KASH. In the following sections we give special constructions which give the sorting of the roots.

3.4. The computation of polynomials for direct and subdirect products

Let G = G_{1} × G_{2} be a permutation group which is a direct product. Suppose we know two irreducible polynomials f_{1}, f_{2} ∈ ℤ[x] such that G_{1} = Gal(f_{1}) and G_{2} = Gal(f_{2}). We denote with N_{1} and N_{2} the splitting fields of f_{1} and f_{2}, respectively. Supposing N_{1} ∩ N_{2} = ℚ we know that Gal(N_{1}N_{2}/ℚ) ≅ G. In this case it remains to compute a polynomial f such that Gal(f) = G.

Let α = α_{1}, …, α_{n} and β = β_{1}, …, β_{m} be the roots of f_{1} and f_{2}, respectively. We want to compute a minimal polynomial of a primitive element of the extension ℚ(α, β)/ℚ. There is a well known algorithm for this task which computes a polynomial of degree deg(f_{1}) deg(f_{2}) using resultants. An irreducible factor g of this polynomial yields the wanted minimal polynomial. After computing the polynomial g ∈ ℤ[x] we have to compute its Galois group including the action on the roots. After this we can apply the function ComputePolynomialFactorGroup to compute f. The above approach has two disadvantages. It may happen that it is impractical to compute the Galois group of g in its action on the roots. Even if it is possible to compute this Galois group, this approach tends to yield polynomials with large coefficients. The following algorithm avoids this.

Algorithm 3.6. (Computation of a polynomial for a direct product.) \underline{Input}: Polynomials f_{1}, f_{2} ∈ ℤ[x] of degree n, m with linearly disjoint splitting fields,

G ≅ Gal(f_{1}) × Gal(f_{2}).

\underline{Output}: A polynomial f ∈ ℤ[x] with Gal(f) = G.

\underline{Step 1}: Compute the Galois groups G_{1} and G_{2} of f_{1} and f_{2}, respectively, including their action on the roots.

\underline{Step 2}: Set f̃:= f_{1}f_{2}.

\underline{Step 3}: Set G̃ to the direct product of G_{1} and G_{2} acting on \{α_{1}, …, α_{n}, β_{1}, …, β_{m}\} such that G|_{{α_{1},...,α_{n}}} = G_{1} and G|_{{β_{1},...,β_{m}}} = G_{2}.

\underline{Step 4}: Set f :=ComputePolynomialFactorGroup(f̃, G̃, G, \{α_{1}, …, α_{n}, β_{1}, …, β_{m}\}).

\underline{Step 5}: Return f.

The algorithm for computing polynomials for subdirect products is similar to algorithm 3.6. Recall the definition of a subdirect product.

Definition 3.7. Let G_{1}, G_{2}, H be groups and μ_{i} epimorphisms from G_{i} to H (i = 1, 2). The we define:

G_{1} ×_{H} G_{2} := {(g_{1}, g_{2}) | g_{1} ∈ G_{1}, g_{2} ∈ G_{2}, μ_{1}(g_{1}) = μ_{2}(g_{2})}.

Remark 3.8. Note that the notation G_{1} ×_{H} G_{2} may be ambiguous if there exist several epimorphisms μ_{i} : G_{i} → H for i = 1 or i = 2.

1 For example there exist two non-isomorphic subdirect products of the form D_{4} ×_{Z_{2}} S_{3}, the reason being that D_{4} has three different factor groups isomorphic to Z_{2} (these are the groups denoted 12T 12 and 12T 13 in our tables).

2 Consider subdirect products of the Frobenius group F_{20} with itself of the form F_{20} ×_{Z_{4}} F_{20}. There is only one factor group of F_{20} isomorphic to Z_{4}. Since the automorphism group of Z_{4} has two elements, there are two epimorphisms μ_{1} and μ_{2} from F_{20} to Z_{4}. Define G_{1} := {(g_{1}, g_{2} | g_{1}, g_{2} ∈ F_{20}, μ_{1}(g_{1}) = μ_{1}(g_{2})} and G_{2} := {(g_{1}, g_{2} | g_{1}, g_{2} ∈ F_{20}, μ_{1}(g_{1}) = μ_{2}(g_{2})}. These two subdirect products are not isomorphic. (Only one of them, denoted 10T 10, has a faithful permutation representation of degree at most 15.)

Suppose given two polynomials f_{1}, f_{2} ∈ ℤ[x] such that Gal(f_{1}) = G_{1}, Gal(f_{2}) = G_{2}, and the Galois group of the intersection of the corresponding splitting fields is isomorphic to H. Then we know that the splitting field of f_{1}f_{2} has Galois group isomorphic to G_{1}×_{H} G_{2} for suitable epimorphisms μ_{i} : G_{i} → H. From a computational point of view we are given G_{1} and G_{2} only up to automorphisms. In the following algorithm we fix the groups G_{1} and G_{2} and the epimorphism μ_{2}. We want to find an epimorphism μ_{1} such that μ_{1} and μ_{2} fit together.

Lemma 3.9. Let G, H be finite groups and μ : G → H an epimorphism with kernel N. Then all epimorphisms ν : G → H with kernel N are given by:

{τ ∘ μ | τ ∈ Aut(H)}.

Proof. It is clear that τ ∘ μ is an epimorphism with kernel N. Furthermore we have τ_{1} ∘ μ ≠ τ_{2} ∘ μ for τ_{1} ≠ τ_{2} ∈ Aut(H). Let μ_{1}, μ_{2} be two epimorphisms with kernel N. Then μ_{1} ∘ μ_{2}^{−1} is an automorphism of H. □

Algorithm 3.10. (Computation of a polynomial for a subdirect product.) \underline{Input}: Monic polynomials f_{1}, f_{2} ∈ ℤ[x] of degree n, m, G ≅ Gal(f_{1}) ×_{H} Gal(f_{2}).

\underline{Output}: A polynomial f ∈ ℤ[x] with Gal(f) = G.

\underline{Step 1}: Compute G_{1} := Gal(f_{1}) and G_{2} := Gal(f_{2}) including the action on the roots {α_{1}, …, α_{n}} and {β_{1}, …, β_{m}}, respectively.

\underline{Step 2}: Compute all normal subgroups N_{1} ⊴ G_{1} such that G_{1}/N_{1} ≅ H. For each N_{1} compute the fixed field using algorithm 3.5.

\underline{Step 3}: Compute all normal subgroups N_{2} ⊴ G_{2} such that G_{2}/N_{2} ≅ H. For each N_{2} compute the fixed field using algorithm 3.5.

\underline{Step 4}: Choose N_{1} ⊴ G_{1} and N_{2} ⊴ G_{2} such that the corresponding fixed fields are equal.

\underline{Step 5}: Compute epimorphisms μ_{i} : G_{i} → H with kernel N_{i} (i = 1, 2).

\underline{Step 6}: Compute A := Aut(H).

\underline{Step 7}: For all τ ∈ A do

1 μ̃_{1} := τ ∘ μ_{1}.
2 Set G̃ to the subdirect product of G_{1} and G_{2} acting on {α_{1}, …, α_{n},
β_{1}, …, β_{m}} corresponding to the epimorphisms μ̃_{1}, μ_{2}.
3 Set f :=ComputePolynomialFactorGroup(f_{1}f_{2}, G̃, G, {α_{1}, …, α_{n},
β_{1}, …, β_{m}}).

4 If f ∈ ℤ[x] then return f.

Proof. After having identified the correct kernels N_{1} and N_{2} our epimorphisms μ_{1}, μ_{2} have to satisfy μ_{1}(N_{1}) = μ_{2}(N_{2}) = 1. Applying Lemma 3.9 we compute all epimorphisms from G_{1} to H with kernel N_{1}. We know that one of these epimorphisms fits to our representations of the given Galois groups. □

We have to modify the function ComputePolynomialFactorGroup in such a way that this function can detect that the result is not in ℤ[x] which happens when the ordering of the roots was not correct. In Step 3 we identify the normal subgroups which are the kernels of the corresponding epimorphisms. In this step we can check which subdirect product we get (compare Remark 3.8).

3.5. Wreath products

Let G, H be finite groups such that we can realize G over ℚ and H regularly over ℚ(t). Recall from 2.2.3 that then the wreath product H ≀ G (with respect to any faithful transitive permutation representation of G) also occurs as Galois group over ℚ. Without loss (see Section 3.3) we may assume that the realization of G is already given by a polynomial f such that the induced permutation representation on the roots of f is the desired one. Let g(t, x) ∈ ℤ(t)[x] be a generating polynomial for a regular H-extension.

For a number field K/ℚ we denote by Norm the norm function extended to the polynomial ring K[x].

Algorithm 3.11. (WreathProduct) \underline{Input}: A monic polynomial g ∈ ℤ[x] with Gal(g) = G, h(t, x) ∈ ℤ[t][x] with Gal(h) =

H regular over ℚ(t).

\underline{Output}: A monic polynomial f ∈ ℤ[x] such that Gal(f) = H ≀ G.

\underline{Step 1}: Let K = ℚ(α), where α is a zero of g.

\underline{Step 2}: Choose a random algebraic integer γ of K.

\underline{Step 3}: Compute f := Norm(h(γ, x)) ∈ ℤ[x].

\underline{Step 4}: If gcd(f, f′) ≠ 1 then go to Step 2.

\underline{Step 5}: If f is reducible then go to Step 2.

\underline{Step 6}: If Gal(f) = H ≀ G then return f, otherwise go to Step 2.

The time critical step of the above algorithm is Step 6. This can be improved by using the fact that the Galois group is a subgroup of H ≀ G which is very useful for the Galois group computation. We remark that the right γ in Step 2 abound (see Section 2.2.3).

3.6. Split extensions with abelian kernel

Suppose we know that G is a factor group of A≀H, where A is an abelian group. When we are able to realize H over ℚ we can compute a polynomial g ∈ ℤ[x] for the wreath product. It may happen that we are not able to compute the Galois group including the ordering of the roots with general methods. We are looking for a special method in this case. In the following we denote with O_{K} the ring of algebraic integers of K. We only give an algorithm when A is a cyclic group, but it is not difficult to extend it to abelian groups.

Algorithm 3.12. (SplitExt)

\underline{Input}: A group G which is a factor group of A ≀ H (A cyclic), a monic polynomial g ∈ ℤ[x] with Gal(g) = H, K = ℚ(α), where α is a root of g, a polynomial h ∈ O_{K}[x] such that Gal(Norm(h)) ≅ A ≀ H.

\underline{Output}: A polynomial f ∈ ℤ[x] with Gal(f) = G.

\underline{Step 1}: Define L := K(β), where β is a root of h.

\underline{Step 2}: Compute the automorphisms of the abelian extension L/K.

\underline{Step 3}: Let h̃ := Norm(h) and α_{1}, …, α_{n} be the roots of g.

\underline{Step 4}: Let σ ∈ Aut(L/K) be a generator of order m.

\underline{Step 5}: Compute Gal(g) including the action on \{α_{1}, …, α_{n}\}.

\underline{Step 6}: Compute the images h^{(i)} of h under the mapping α ↦ α_{i} (1 ≤ i ≤ n).

\underline{Step 7}: Compute a root γ_{i,1} of h^{(i)} (1 ≤ i ≤ n).

\underline{Step 8}: Set γ_{i,j} := γ_{i,1}^{σ^{j−1}} (1 ≤ i ≤ n, 2 ≤ j ≤ m).

\underline{Step 9}: Let H = ⟨τ̃_{1}, …, τ̃_{r}⟩. Denote by τ_{1}, …, τ_{r} the permutations which operate simultaneously on {γ_{1,j}, …, γ_{n,j}} (1 ≤ j ≤ m) in the same way as τ̃_{1}, …, τ̃_{r}

on {α_{1}, …, α_{n}}. Define σ_{i} = (γ_{i,1}, …, γ_{i,m}) (1 ≤ i ≤ n).

\underline{Step 10}: Set G̃ := ⟨σ_{1}, …, σ_{n}, τ_{1}, …, τ_{r}⟩.

\underline{Step 11}: Set f :=ComputeFactorGroup(h̃, G̃, G).

\underline{Step 12}: Return f.

We remark that the automorphisms in Step 2 can be computed using an algorithm described in Klüners (1997).

3.7. Finding ”nicer” polynomials

Let f ∈ ℤ[x] be an irreducible monic polynomial of degree n and K be the stem field of f. We want to find an irreducible polynomial g ∈ ℤ[x] such that the stem fields of f and g are isomorphic and g is simpler or nicer in some sense. For instance that could mean that the coefficients or the discriminant of g are small. It is very difficult to solve this problem. A description of a possible approach to this problem can be found in Cohen (1993), section 4.4.2. The algorithm presented there works as follows. Let β ∈ O_{K} and m_{β} be the characteristic polynomial. We denote with β_{1}, …, β_{n} ∈ ℂ the zeros of m_{β}. The idea of the algorithm is to find a β ∈ O_{K} such that the T_{2}-norm

T_{2}(β) := ∑_{i=1}^{n} |β_{i}|^{2}

is small. If such an element is primitive one hopes that the corresponding polynomial m_{β} has small coefficients.

The first step of this algorithm is to compute the maximal order O_{K} of K. Then we can associate a lattice to O_{K}. Elements with small norm in the lattice are associated to elements of O_{K} with small T_{2}-norm. By computing an LLL-basis (Lenstra et al., 1982) of the lattice we get a basis of the lattice consisting of elements of small norms. This algorithm is implemented in PARI (polred) and in KASH (OrderShort).

There are three main problems of the polred and the OrderShort function. The first one is the computation of the maximal order. Here the main obstacle is the factorization of the polynomial discriminant. The second problem is the computation of the LLL reduced basis of the corresponding lattice. When the index of the equation order in the maximal order is very large we have to work with a high real precision in the lattice reduction. The third problem occurs when the given field has many subfields. In this case it is very probable that short vectors in the lattice correspond to elements which are not primitive. In the following section we give some ideas on how to overcome the second and third problem.

We remark that Fieker (1997) gives a relative version of this algorithm, where it is possible to find a nicer polynomial for an extension K/L. When a number field K has non-trivial subfields, very often the following procedure is useful.

1 Compute a subfield L of K = ℚ(α).

2 Compute the minimal polynomial of α over L.

3 Try to find a nicer polynomial for the relative extension K/L.

4 Compute a primitive element from the new representation.

We remark that subfield computation can be done with algorithms described in Klüners and Pohst (1997) and Klüners (1998).

3.8. The computation of maximal orders

One important task in algorithmic number theory is the computation of the maximal order of a number field. Well known algorithms for this are described for instance in Pohst and Zassenhaus (1989) and Cohen (1993). All these algorithms have in common that the first step is the factorization of the polynomial discriminant or some related large number. In our present applications it is often impractical to factorize this number. But since we have constructed our polynomials in a special way we are in a better situation. For instance in many cases we know the field discriminant in advance from theory. The purpose of this section is to describe how to use special information to compute the maximal order of a number field. The main idea of all approaches consists in avoiding the factorization of large integers.

Let K = ℚ(α) be a number field, where α is an algebraic integer. We first describe an algorithm which works if the field extension K/ℚ is not primitive, i.e., has proper intermediate fields. Using the subfield algorithm described in Klüners and Pohst (1997) and Klüners (1998) we can compute a subfield ℚ(β) of K including the embedding

β := \frac{1}{d} ∑_{i=0}^{n−1} b_{i}α^{i}, (d ∈ ℕ, b_{i} ∈ ℤ)

into K. We can now proceed as follows.

1 Compute all subfields of K.

2 Compute the maximal orders of all proper subfields.

3 Set A := {1, α, …, α^{n−1}}.

4 For each subfield L do:

Move each basis element of O_{L} to a representation in K and add it to the set A. 5 Compute a ℤ-basis of the multiplicative closure of A. 6 Set O to the order generated by this ℤ-basis. 7 Compute the maximal order O_{K} starting from O.

The idea of the above algorithm is to find integral elements which are not contained in ℤ[α]. The computation of a ℤ-basis can be done using the Hermite normal form algorithm.

In practice the discriminant of the order O is much smaller than the discriminant of ℤ[α]. For primitive extensions K/ℚ we have two other possibilities to find integral elements. When the extension is normal we can compute the automorphisms which in general produces elements not contained in ℤ[α]. The other possibility is to factorize the minimal polynomial of α over K. When the factorization is not trivial (a linear factor and a factor of degree n − 1) the coefficients of the factors are in general not contained in ℤ[α]. This approach is very useful for dihedral groups, but it cannot work when the group is at least 2-transitive.

Another approach is based on a theorem of Buchmann and Lenstra (1994).

Theorem 3.13. There are polynomial time algorithms that given an algebraic number field K and one of (a), (b), determine the other:

(a) the ring of algebraic integers of K,

(b) the largest square free divisor of the discriminant of K.

Since our fields are constructed in a special way, we are very often in the situation that we know (b) of the above theorem. This result is not only theoretical. Buchmann and Lenstra (1994) also provide an algorithm which works well in practice. This has been implemented in KASH (Daberkow et al., 1997) by the first author.

4. Determination of polynomials with groups of degree ≤ 15

In the preceding sections we have presented some general methods applicable to all groups with a certain structure. When trying to apply these to a particular group G, one has to investigate the structure of G to see whether it falls into one of the above categories. This was done for the transitive groups of degree at most 15 by using the computer algebra system GAP (Schönert et al., 1997). The results are collected in Section 5. It turned out that for the overwhelming majority of these groups at least one of the general methods can be used to construct a polynomial. Here, we do not consider the method of a subdirect product as a general method, since it depends on first finding fields with the desired properties.

In this section we consider the few remaining groups of degree between 12 and 15 and show how they can be realized explicitly as Galois groups. Finally, we give the proof of Theorem 1.1.

We denote by dT m the m-th transitive group of degree d. This is the same group one gets with TransitiveGroup(d, m) in GAP or Magma. This notation was introduced by Butler and McKay; we refer to Conway et al. (1998) for references and the most recent account of the classification of transitive groups.

4.1. The groups 12T57, 12T91, 12T93, 12T104

The permutation groups 12T91 and 12T93 are isomorphic as abstract groups. Moreover, all four groups contain a normal subgroup N isomorphic to 12T57. In the groups 12T91, 12T93, and 12T104 there exists an elementary abelian normal subgroup such that N is a supplement. Similar to the concept of semi-abelian groups we can prove that such a group is a factor group of a suitable wreath product.

It thus remains to construct a polynomial for the group 12T 57. We know that 12T 57 ≅ SL_{2}(3) ×_{𝔄_{4}} [4^{2}]3 ≅ 8T 12 ×_{𝔄_{4}} 12T 31. One can prove that all subdirect products of this type are isomorphic. Therefore it is sufficient to find an 𝔄_{4}-extension which is embeddable into both an SL_{2}(3)- and a [4^{2}]3-extension. Since these are Frattini embedding problems the solutions are regular provided the 𝔄_{4} extension is regular.

Let L/k be a normal 𝔄_{4} extension and K/k be the unique subfield of degree 3. Then K/k is cyclic and L/K has Galois group Z_{2} × Z_{2}. The theorem of Kochendörffer (see Malle and Matzat (1999), Th. IV.8.2) states in our case that

1 L/k is embeddable into an SL_{2}(3) extension, if and only if L/K is embeddable into

a Q_{8} extension. 2 L/k is embeddable into a [4^{2}]3 extension, if and only if L/K is embeddable into a

Z_{4} × Z_{4} extension.

Ledet (1996), Theorem 5.3, proved the following:

Theorem 4.1. Let k be a Hilbertian field of characteristic ≠ 2, and let K/k be a cyclic extension of degree 3 with Galois group ⟨σ⟩. Let a ∈ K\k be an element with the following properties:

1 aσ(a) ∉ K^{2},

2 a = 1 + r^{2} + r^{2}σ(r)^{2} for some r ∈ K \ k.

Then K(\sqrt{ab}, \sqrt{bc})/k is a normal 𝔄_{4} extension, where b := σ(a) and c := σ(b). Furthermore K(\sqrt{ab}, \sqrt{bc})/k is embeddable into a SL_{2}(3) extension.

This theorem gives us an approach how to construct an SL_{2}(3) extension. It is well known that a cyclic extension of degree 2 is embeddable into a Z_{4} extension if and only if the generator of the quadratic extension is the sum of two squares. Suppose that a, b are sums of two squares, then ab is the sum of two squares, too. We want to choose r in the above theorem in such a way that 1 + r^{2} is a square. In this case a and therefore b, c are sums of two squares. When we can prove that ab is not a square, we know that K(\sqrt{ab}, \sqrt{bc})/k is embeddable into an SL_{2}(3) and into a [4^{2}]3 extension and we are done. We have to solve two problems:

1 Find r ∈ K \ k such that 1 + r^{2} is a square.

2 Check, that ab is not a square in K.

A parametrization of all k-points on the genus 0 curve

s^{2} = 1 + r^{2}

is given by

r = \frac{2λ}{1 − λ^{2}} (λ ≠ ±1).

It remains to find an example such that ab is not a square in K. Let f(t, x) := x^{3} + 3tx^{2} − 9x − 3t ∈ ℚ(t)[x]. Since f is irreducible and disc(f) = 2^{2} · 3^{4} · (t^{2} + 3)^{2} is a square, we get that Gal(f) = Z_{3}. Define K := ℚ(t)(α), where α is a root of f. One can check that β := \frac{1}{2}(−3t − 6 + (3t − 1)α + α^{2}) is another root of f, hence β = σ(α) for a suitable generator σ of Gal(f). Now we compute:

r := \frac{2α}{1 − α^{2}} = \frac{1}{4}(−3t − α), σ(r) = \frac{2β}{1 − β^{2}} = \frac{1}{8}(−3t + 6 + (−3t + 1)α − α^{2}).

Therefore we can compute a := 1 + r^{2} + r^{2}σ(r)^{2} =

\frac{1}{512}(81t^{4} −135t^{3} +423t^{2} −9t+512+(81t^{4} +18t^{2} +216t−27)α+(27t^{3} −9t^{2} −3t+41)α^{2}).

Now we have to check that aσ(a) is not a square in K or equivalently that K(\sqrt{a})/ℚ(t) is not cyclic. The latter one can be easily checked by specializations and we get that Gal(K(\sqrt{a})/ℚ(t)) = Z_{2} ≀ Z_{3}. Therefore we have proved that K(\sqrt{aσ(a)})/ℚ(t) is an 𝔄_{4}(6) extension which can be embedded into 12T31 and 8T12.

4.2. The group 12T124

This group is the (unique) subdirect product of Z_{4} and 𝔖_{5} with kernel Z_{2}. In particular 12T124 is not solvable. In order to construct a 12T124-extension we need 𝔖_{5} and Z_{4}- extensions with a common subfield of degree 2. The following polynomials f_{i} ∈ ℚ(t_{i})[x], i ∈ {1, 2}, generate regular Galois extensions of ℚ(t_{i}) with Galois group 𝔖_{5} and Z_{4}, respectively.

f_{1}(x, t) := x^{5} − 5x^{4} + 256 t_{1}, f_{2}(x, t) := x^{4} + t_{2} x^{3} − 6x^{2} − t_{2} x + 1 .

Their quadratic subfields are generated by the square roots of the polynomial discriminants and equal ℚ(\sqrt{5t_{1}(t_{1} − 1)}), ℚ(\sqrt{t_{2}^{2} + 16}) respectively. If we specialize

t_{1} := 4(5t^{2} − 6t + 5)/(4t^{2} − 5), t_{2} := −3(4t^{2} − 15t + 5)/(4t^{2} − 5),

then the Galois groups are conserved, but the polynomial discriminants of f_{1}, f_{2} coincide up to squares. This means that the splitting fields of f_{1} and f_{2} contain the same quadratic subfield. It follows that the splitting field of f_{1}f_{2} has Galois group isomorphic to 12T124 over ℚ(t). Furthermore this extension is regular since 𝔖′_{5} = 𝔄_{5} is simple.

4.3. The groups 12T143, 12T192

The group 12T143 is solvable but not semi-abelian. The normal subgroup Z_{2}^{4} has a supplement which is isomorphic to 12T93.

The group 12T192 is semi-abelian, but not a split extension with abelian kernel. The normal subgroup Z_{2}^{5} has a supplement isomorphic to 12T98.

4.4. The groups 12T256, 12T287, 12T297, 12T298, 14T59, 14T60, 15T94, 15T95, 15T97, 15T98, 15T99, 15T100

The groups G mentioned in the title all occur as subgroups of index 2 in suitable wreath products G̃ or in groups with a rationally rigid class triple. We can realize them as Galois groups by first constructing a regular extension with group G̃ and then using the descent trick explained in Section 2.1.2. This works whenever only two divisors of degree 1 ramify in the fixed field of G. We explain this by means of the following table.

G̃  C̃  G  C
12T 270 = 2 ≀ 𝔖_{5}  2^{6} − 8 − 10  12T 256  4^{2} − 10 − 10
12T 293 = 2 ≀ 𝔖_{6}  2^{2} − 10 − 12  12T 287  5^{2} − 12 − 12
12T 299 = 𝔖_{6} ≀ 2  2 − 10.2 − 12  12T 297  10.2 − 10.2 − 6^{2}
12T 299 = 𝔖_{6} ≀ 2  2 − 10.2 − 12  12T 298  5^{2} − 12 − 12
14T 61 = 𝔖_{7} ≀ 2  2 − 12.2 − 14  14T 59  12.2 − 12.2 − 7^{2}
14T 61 = 𝔖_{7} ≀ 2  2 − 12.2 − 14  14T 60  6^{2} − 14 − 14
15T 97  2^{6} − 10 − 12.3  15T 94  2^{6} − 2^{6} − 5^{2} − 6^{2}.3
15T 97  2^{6} − 10 − 12.3  15T 95  5^{2} − 12.3 − 12.3
15T 101 = 𝔖_{5} ≀ 3 2 − 2 − 4^{2} − 15 − 15 15T 98 4^{2} − 4^{2} − 15 − 15 − 15 − 15
15T 102 = 𝔖_{5} ≀ 𝔖_{3} 2 − 2^{5} − 8.2 − 15 15T 99 8.2 − 8.2 − 15 − 15
15T 102 = 𝔖_{5} ≀ 𝔖_{3} 2 − 2^{5} − 8.2 − 15 15T 100  2^{5} − 2^{5} − 4^{2} − 15 − 15

It contains a list of groups G̃ with class vector C̃. A generating polynomial for a stem field of a G̃-Galois extension N/ℚ(t) with ramification type C̃ can be obtained by the wreath product construction described above. Any of the missing groups G is a subgroup of index at most 2 in one of the groups G̃ and we indicate these subgroups in the third column. The fixed field of G is a rational function field, and in the fourth column we give the corresponding class vector C for the Galois extension N/N^{G}.

4.5. The groups 14T33, 14T42

The group 14T33 is the non-split extension of the elementary abelian group of order 8 with the simple group L_{3}(2) (the split extension already has a faithful permutation representation of degree 8), and 14T42 is the direct product of 14T33 with the group of order 2. Let us denote by 3A, 4A respectively the conjugacy classes of L_{3}(2) containing elements of permutation types 3^{2}, 4.2 in the degree 7 permutation representation. There exists a regular Galois extension of ℚ(t) with group L_{3}(2) and class vector C = (3A, 4A, 4A), with generating polynomial given by

f := (y^{3} − 41 y^{2} + 5 y − 1) (y^{3} − 20 y^{2} + 19 y − 162) (y − 2) − (y − 1)^{3} y t ,

ramified in ∞ and ±1372\sqrt{7} (see Malle and Matzat (1985)). The behaviour at the finite ramification points can be seen by calculating the resultant of f and t^{2} − 13176688 as

(y^{2} − 4 y − 3)^{4} (y^{2} − 32 y + 4)^{2} (y^{2} − 46 y + 81) .

It turns out that 14T42 has a generating system with three generators of permutation types 3^{4}, 8.2^{2}, 8.2^{2} in the degree 14 representation. In the degree 7 permutation representation of the factor group L_{3}(2), these map to elements in the class vector C. The above equation for the ramification thus shows that we obtain a Galois extension of ℚ(t) with group 14T42 by extracting a square root from y^{2} − 4 y − 3 in the stem field ℚ(y)/ℚ(t) of the L_{3}(2) extension. A generating polynomial is given by

x^{14}−1764 x^{12}−27 (4 t−14063) x^{10}−378 (2 t−7203) x^{8}−(x^{2}+3) (x^{2}+6)^{3} (t^{2}−13176688).

It can be checked that the corresponding Galois extension of ℚ(t) with group 14T42 is regular. The group 14T33 is now obtained either via its fixed field of genus zero, or alternatively as a factor group in the way described in the previous section. We get an explicit polynomial over ℚ(u) using the substitution t := 1372(6u^{2} − 4u + 3)/(2u^{2} − 1).

4.6. Primitive non-solvable groups

Polynomials for the primitive non-solvable groups of permutation degree at most 15 were known previously, their determination was completed in Malle (1987). References for these polynomials can either be found in loc. cit., or in the Appendix of the forthcoming book of Malle and Matzat (1999).

More precisely, this concerns the alternating and symmetric groups and the almost simple groups given in the following table:

G  C
12T 218 ≅ P GL_{2}(11) 2^{5} − 4^{3} − 11
12T 295 ≅ M_{12}  4^{2} − 4^{2} − 10.2
13T 7 ≅ L_{3}(3)  2^{4} − 8.4 − 8.4
14T 30 ≅ L_{2}(13)  2^{6} − 6^{2} − 6^{2}
14T 39 ≅ P GL_{2}(13) 2^{7} − 4^{3} − 6^{2}

All of these groups were realized by the rigidity method described in Section 2.1.

4.7. Proof of Theorem 1.1

Let G be a transitive subgroup of 𝔖_{d} for d ≤ 15. If d ≤ 11 then the assertion of Theorem 1.1 follows for G by the tables in Malle and Matzat (1999), Appendix, except for the 7 groups for which no polynomial is given there. But all seven groups are semidirect products with abelian kernel and complement already known to occur regularly. Thus G itself occurs regularly by Section 2.2.4. So now assume that d ≥ 12. The arguments given in Section 2 show that G occurs regularly over ℚ(t) unless G is one of the groups treated in this section. But for all of these we have given explicit constructions of regular extensions with group G. This completes the proof of Theorem 1.1.

5. The groups of degree 12 ≤ d ≤ 15

In the following tables we give the information needed to realize the transitive groups of degree 12 ≤ d ≤ 15 as Galois groups. For each group G we give a line with five columns.

The first column gives the number of G, the second one the size of G. A * in the second column indicates that G acts primitively. In the third column we give the information if G is a wreath product, a direct product, or a subdirect product. In the fourth column we give the information whether G is a factor group of a non-trivial wreath product. This information was computed by checking whether G is a split extension with abelian kernel or an abelian normal subgroup of G has a proper supplement. In the case that there is no entry this means that no non trivial normal abelian subgroup has a supplement. In the last column we give groups such that G is a factor group. Here we give groups acting on less than d points and/or a list of the numbers of transitive groups acting on d points. In the degree 12 case we do not list all groups with that property. If a group is a factor group of a group G then all groups which are bigger than G having G as a factor group are not listed. For the group 14T5 the following information is given in the table:

No Size  Group  Factor  Factor
5 42  7T3 x 2T1  Z_{7} ≀ 6T 1 [ 14, 18, 44 ]

This means that this group has 42 elements and is a direct product of 7T3 and 2T1 (7.3 × Z_{2}). Furthermore this group can be realized as a factor group of Z_{7} ≀ 6T 1 (Z_{7} ≀ Z_{6}), 14T14, 14T18, and 14T44.

5.1. Transitive Groups of degree 12

No Size  Group  Factor  Factor
1 12  4T1 x 3T1  Z_{4} ≀ 3T 1 [ 19, 29, 73 ]
2 12  4T2 x 3T1  Z_{3} ≀ 4T 2 [ 14, 18, 25 ]
3 12  3T2 x 2T1  Z_{3} ≀ 4T 2 6T3 [ 10, 11, 12, 13, 16, 18, 21 ]

4 12 Z_{2} ≀ 3T 1 4T4 [ 6, 20, 31, 32, 122, 132 ]

5 12  3T 2 ×_{2T 1} 4T 1  Z_{3} ≀ 4T 1 [ 19, 27, 72 ]
6 24  4T4 x 2T1  Z_{2} ≀ 3T 1 6T6 [ 7, 25, 29, 43, 55, 56, 58, 60, 91,

176 ]

7 24 4T4 x 2T1 Z_{2} ≀ 3T 1 6T6 [ 6, 25, 29, 43, 55, 56, 58, 60, 91,

176 ]

8 24 Z_{2} ≀ 3T 2 4T5 [ 9, 21, 27, 44, 45, 62, 66, 157,

175, 177 ]

9 24 Z_{2} ≀ 3T 2 4T5 [ 8, 21, 27, 44, 45, 62, 66, 157,

175, 177 ]

10 24  3T2 x 4T2  Z_{3} ≀ 8T 3 [ 28, 37, 48 ]
11 24  3T2 x 4T1  Z_{3} ≀ 8T 2 [ 39, 53, 119 ]
12 24  4T 3 ×_{2T 1} 3T 2  Z_{4} ≀ 6T 2 [ 38, 54, 118 ]
13 24  4T 3 ×_{2T 1} 3T 2  Z_{2} ≀ 6T 2 [ 15, 38, 42, 49, 116 ]
14 24  4T3 x 3T1  Z_{2} ≀ 6T 1 [ 42, 51, 121 ]
15 24  4T 3 ×_{2T 1} 3T 2  Z_{2} ≀ 6T 2 [ 13, 38, 42, 49, 116 ]
16 36  3T2 x 3T2  Z_{3} ≀ 4T 2 6T9 [ 37, 38, 39, 70, 71, 83, 161, 268,

283 ]

17 36 Z_{3} ≀ 4T 1 6T10 [ 40, 72, 73, 160 ]

18 36  6T1 x 3T2  Z_{3} ≀ 4T 2 [ 42, 70, 158 ]
19 36  12T5 x 3T1  Z_{3} ≀ 4T 1 [ 131, 159 ]

20 36  4T4 x 3T1  Z_{2} ≀ 9T 2 [ 85, 194 ]
21 48  4T5 x 2T1  Z_{2} ≀ 3T 2 6T11 [ 22, 23, 24, 48, 49, 53, 54, 83, 95,

100, 108, 112, 213 ]

22 48 4T5 x 2T1 Z_{2} ≀ 3T 2 6T11 [ 21, 23, 24, 48, 49, 53, 54, 83, 95,

100, 108, 112, 213 ]

23 48 4T5 x 2T1 Z_{2} ≀ 3T 2 6T11 [ 21, 22, 24, 48, 49, 53, 54, 83, 95,

100, 108, 112, 213 ]

24 48 4T5 x 2T1 Z_{2} ≀ 3T 2 6T11 [ 21, 22, 23, 48, 49, 53, 54, 83, 95,

100, 108, 112, 213 ]

25 48  4T4 x 4T2  Z_{2} ≀ 6T 1 [ 26, 51, 87, 89, 90, 143 ]
26 48  4T4 x 4T2  Z_{2} ≀ 6T 1 [ 25, 51, 87, 89, 90, 143 ]
27 48  4T 5 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 5 [ 30, 98, 102 ]
28 48  4T3 x 3T2  Z_{2} ≀ 6T 3 [ 81, 86, 156 ]
29 48  4T4 x 4T1  Z_{4} ≀ 4T 4 [ 94, 99 ]
30 48  4T 5 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 5 [ 27, 98, 102 ]

31 48 Z_{4} ≀ 3T 1 [ 55, 57 ] 32 48 4T 4 ×_{3T 1} 4T 4 Z_{2} ≀ 6T 4 8T32 [ 56, 144, 271 ] 33 60 5T4 [ 75, 230 ] 34 72 Z_{3} ≀ 4T 3 6T13 [ 35, 36, 77, 79, 116, 118, 121,

172, 200 ]

35 72 6T 2 ≀ 2T 1 Z_{3} ≀ 4T 3 6T13 [ 34, 36, 77, 79, 116, 118, 121,

172, 200 ]

36 72 Z_{3} ≀ 4T 3 6T13 [ 34, 35, 77, 79, 116, 118, 121,

172, 200 ]

37 72  6T3 x 3T2  Z_{3} ≀ 8T 3 [ 81, 117, 195 ]
38 72  6T 9 ×_{4T 2} 4T 3  Z_{3} ≀ 8T 4 [ 169, 196 ]
39 72  6T 9 ×_{2T 1} 4T 1  Z_{3} ≀ 8T 2 [ 170, 197 ]
40 72  6T10 x 2T1  Z_{3} ≀ 8T 2 [ 41, 82, 119, 171, 198 ]
41 72  6T10 x 2T1  Z_{3} ≀ 8T 2 [ 40, 82, 119, 171, 198 ]
42 72  6T 1 ≀ 2T 1  Z_{2} ≀ 6T 5 [ 167, 208 ]
43 72  4T4 x 3T2  Z_{2} ≀ 9T 4 [ 206, 234 ]
44 72  4T 5 ×_{2T 1} 3T 2  Z_{2} ≀ 9T 5 [ 127, 233 ]
45 72  4T5 x 3T1  Z_{2} ≀ 9T 4 [ 205, 231 ]

46 72 Z_{3} ≀ 8T 1 9T15 [ 173 ] 47 72 Z_{3} ≀ 8T 5 9T14 [ 174 ]

48 96  4T5 x 4T2  Z_{2} ≀ 6T 3 [ 86, 136, 138, 139 ]
49 96  4T 5 ×_{2T 1} 4T 3  Z_{2} ≀ 6T 2 [ 50, 52, 135, 145, 147 ]
50 96  4T 5 ×_{2T 1} 4T 3  Z_{2} ≀ 6T 2 [ 49, 52, 135, 145, 147 ]
51 96  4T4 x 4T3  Z_{2} ≀ 6T 1 [ 134, 141 ]
52 96  4T 5 ×_{2T 1} 4T 3  Z_{2} ≀ 6T 2 [ 49, 50, 135, 145, 147 ]
53 96  4T5 x 4T1  Z_{4} ≀ 4T 5 [ 150, 153 ]
54 96  4T 5 ×_{2T 1} 4T 3  Z_{2} ≀ 12T 12 [ 151, 152 ]
55 96  12T31 x 2T1  Z_{4} ≀ 6T 1 [ 94 ]
56 96  12T32 x 2T1  Z_{2} ≀ 6T 4 [ 90, 187, 189 ]

57 96 12T 31 ×_{4T 4} 8T 12 58 96 Z_{2} ≀ 6T 1 8T33 [ 59, 87, 99, 280 ] 59 96 Z_{2} ≀ 6T 1 8T33 [ 58, 87, 99, 280 ] 60 96 Z_{4} ≀ 6T 1 [ 61, 89, 104 ] 61 96 Z_{4} ≀ 6T 1 [ 60, 89, 104 ] 62 96 Z_{4} ≀ 3T 2 [ 63, 64, 65, 95, 98 ] 63 96 Z_{4} ≀ 3T 2 [ 62, 64, 65, 95, 98 ] 64 96 Z_{4} ≀ 3T 2 [ 62, 63, 65, 95, 98 ] 65 96 Z_{4} ≀ 3T 2 [ 62, 63, 64, 95, 98 ] 66 96 4T 5 ×_{3T 2} 4T 5 Z_{2} ≀ 6T 2 8T34 [ 67, 68, 69, 100, 102, 184, 192,

281, 282 ]

67 96 4T 5 ×_{3T 2} 4T 5 Z_{2} ≀ 6T 2 8T34 [ 66, 68, 69, 100, 102, 184, 192,

281, 282 ]

68 96 4T 5 ×_{3T 2} 4T 5 Z_{2} ≀ 6T 2 8T34 [ 66, 67, 69, 100, 102, 184, 192,

281, 282 ]

69 96 4T 5 ×_{3T 2} 4T 5 Z_{2} ≀ 6T 2 8T34 [ 66, 67, 68, 100, 102, 184, 192,

281, 282 ]

70 108  6T5 x 3T2  Z_{3} ≀ 4T 2 [ 130 ]
71 108  6T 9 ×_{2T 1} 3T 2  Z_{3} ≀ 4T 2 [ 130 ]
72 108  6T 10 ×_{2T 1} 3T 2  Z_{3} ≀ 4T 1 [ 131 ]
73 108  6T10 x 3T1  Z_{3} ≀ 4T 1 [ 131 ]

74 120 5T5 [ 123, 124, 256, 257 ]

75 120  5T4 x 2T1  Z_{2} ≀ 5T 4 10T11 [ 76, 255 ]
76 120  5T4 x 2T1  Z_{2} ≀ 5T 4 10T11 [ 75, 255 ]
77 144  6T13 x 2T1  Z_{3} ≀ 8T 9 [ 78, 125, 156, 210, 235 ]
78 144  6T13 x 2T1  Z_{3} ≀ 8T 9 [ 77, 125, 156, 210, 235 ]
79 144  6T 13 ×_{2T 1} 4T 1  Z_{3} ≀ 8T 10 [ 80, 209, 211, 237 ]
80 144  6T 13 ×_{2T 1} 4T 1  Z_{3} ≀ 8T 10 [ 79, 209, 211, 237 ]
81 144  6T 9 ×_{2T 1} 4T 3  Z_{2} ≀ 6T 9 [ 217, 240 ]
82 144  6T 10 ×_{2T 1} 4T 3  Z_{3} ≀ 8T 10 [ 209, 241 ]
83 144  4T5 x 3T2  Z_{2} ≀ 9T 8 [ 239, 258 ]

84 144 Z_{3} ≀ 8T 8 9T19 [ 212 ]

85 144  4T4 x 4T4  Z_{2} ≀ 9T 2 [ 164 ]
86 192  4T5 x 4T3  Z_{2} ≀ 6T 3 [ 185, 186 ]
87 192  8T33 x 2T1  Z_{2} ≀ 6T 1 [ 88, 134 ]
88 192  8T33 x 2T1  Z_{2} ≀ 6T 1 [ 87, 134 ]
89 192  12T60 x 2T1  Z_{4} ≀ 12T 2 [ 92, 141 ]
90 192  4T 2 ≀ 3T 1  Z_{2} ≀ 6T 6 [ 222 ]

91 192 Z_{2}^{4} ≀_{r} 12T 57 [ 93 ] 92 192 12T60 x 2T1 Z_{2} ≀ 12T 60 [ 89, 141 ] 93 192 Z_{2}^{4} ≀_{r} 12T 57 [ 91 ]

94 192  4T 1 ≀ 3T 1  Z_{4} ≀ 3T 1
95 192  12T65 x 2T1  Z_{2} ≀ 12T 65 [ 96, 97, 150, 151 ]
96 192  12T64 x 2T1  Z_{2} ≀ 12T 64 [ 95, 97, 150, 151 ]
97 192  12T65 x 2T1  Z_{2} ≀ 12T 65 [ 95, 96, 150, 151 ]
98 192  12T 65 ×_{2T 1} 4T 1  Z_{4} ≀ 12T 5
99 192  8T 33 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 1 [ 105 ]
100 192  8T34 x 2T1  Z_{2} ≀ 6T 2 [ 101, 103, 106, 135, 139, 221, 223,

225, 226 ]

101 192 8T34 x 2T1 Z_{2} ≀ 6T 2 [ 100, 103, 106, 135, 139, 221, 223,

225, 226 ]

102 192  8T 34 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 5 [ 107 ]

103 192  8T34 x 2T1  Z_{2} ≀ 6T 2 [ 100, 101, 106, 135, 139, 221, 223,

225, 226 ] 104 192 Z_{2}^{4} ≀_{r} 12T 57

105 192  8T 33 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 1 [ 99 ]
106 192  8T34 x 2T1  Z_{2} ≀ 6T 2 [ 100, 101, 103, 135, 139, 221, 223,

225, 226 ]

107 192 8T 34 ×_{2T 1} 4T 1 Z_{2} ≀ 12T 5 [ 102 ] 108 192 Z_{2} ≀ 6T 3 8T41 [ 109, 110, 111, 136, 145, 152,

153, 289 ]

109 192 Z_{2} ≀ 6T 3 8T41 [ 108, 110, 111, 136, 145, 152,

153, 289 ]

110 192 Z_{2} ≀ 6T 3 8T41 [ 108, 109, 111, 136, 145, 152,

153, 289 ]

111 192 Z_{2} ≀ 6T 3 8T41 [ 108, 109, 110, 136, 145, 152,

153, 289 ]

112 192 Z_{4} ≀ 6T 3 [ 113, 114, 115, 138, 147 ] 113 192 Z_{4} ≀ 6T 3 [ 112, 114, 115, 138, 147 ] 114 192 Z_{4} ≀ 6T 3 [ 112, 113, 115, 138, 147 ] 115 192 Z_{4} ≀ 6T 3 [ 112, 113, 114, 138, 147 ]

116 216  6T 13 ×_{2T 1} 3T 2  Z_{3} ≀ 4T 3 [ 120, 167, 169 ]
117 216  6T9 x 3T2  Z_{3} ≀ 8T 3 [ 168 ]
118 216  6T 13 ×_{2T 1} 3T 2  Z_{3} ≀ 8T 4 [ 169 ]
119 216  6T10 x 3T2  Z_{3} ≀ 8T 2 [ 170 ]
120 216  6T 13 ×_{2T 1} 3T 2  Z_{3} ≀ 4T 3 [ 116, 167, 169 ]
121 216  6T13 x 3T1  Z_{3} ≀ 4T 3 [ 167 ]

122 216 Z_{3} ≀ 8T 12 9T23 [ 232 ] 123 240 5T5 x 2T1 Z_{2} ≀ 5T 5 10T22 [ 270 ] 124 240 5T 5 ×_{2T 1} 4T 1

125 288  6T 3 ≀ 2T 1  Z_{3} ≀ 8T 18 [ 248, 260 ]
126 288  6T 4 ≀ 2T 1  Z_{2} ≀ 6T 5 8T42 [ 128, 129, 158, 159, 205, 206 ]
127 288  4T 5 ×_{2T 1} 4T 5  Z_{2} ≀ 9T 5 [ 204 ]

128 288 Z_{2} ≀ 6T 5 8T42 [ 126, 129, 158, 159, 205, 206 ] 129 288 Z_{2} ≀ 6T 5 8T42 [ 126, 128, 158, 159, 205, 206 ]

130 324  3T 1 ≀ 4T 2  Z_{3} ≀ 4T 2
131 324  3T 1 ≀ 4T 1  Z_{3} ≀ 4T 1

132 324 Z_{3} ≀ 4T 4 9T25 [ 133, 194, 284 ] 133 324 Z_{3} ≀ 4T 4 9T25 [ 132, 194, 284 ]

134 384  2T 1 ≀ 6T 1  Z_{2} ≀ 6T 1 [ 142 ]
135 384  2T 1 ≀ 6T 2  Z_{2} ≀ 6T 2 [ 148 ]
136 384  8T41 x 2T1  Z_{2} ≀ 6T 3 [ 137, 186 ]
137 384  8T41 x 2T1  Z_{2} ≀ 6T 3 [ 136, 186 ]
138 384  12T112 x 2T1  Z_{2} ≀ 12T 112 [ 140, 185 ]
139 384  4T 2 ≀ 3T 2  Z_{2} ≀ 6T 11 [ 250 ]
140 384  12T112 x 2T1  Z_{2} ≀ 12T 112 [ 138, 185 ]

141 384 12T 51 ×_{6T 6} 12T 61 Z_{4} ≀ 6T 1 142 384 Z_{2} ≀ 6T 1 [ 134 ] 143 384 Z_{2}^{4} ≀_{r} 12T 93 144 384 Z_{2} ≀ 12T 4

145 384  8T 41 ×_{4T 2} 4T 3  Z_{2} ≀ 12T 13 [ 146 ]
146 384  8T 41 ×_{4T 2} 4T 3  Z_{2} ≀ 12T 13 [ 145 ]

147 384 12T 112 ×_{4T 2} 4T 3 Z_{4} ≀ 12T 13 [ 149 ] 148 384 Z_{2} ≀ 6T 2 [ 135 ] 149 384 12T 112 ×_{4T 2} 4T 3 Z_{4} ≀ 12T 13 [ 147 ]

150 384  4T 1 ≀ 3T 2  Z_{4} ≀ 3T 2
151 384  12T 65 ×_{2T 1} 4T 3  Z_{4} ≀ 6T 2
152 384  8T 41 ×_{4T 2} 4T 3  Z_{2} ≀ 12T 12 [ 154 ]
153 384  8T 41 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 11 [ 155 ]
154 384  8T 41 ×_{4T 2} 4T 3  Z_{2} ≀ 12T 12 [ 152 ]
155 384  8T 41 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 11 [ 153 ]
156 432  6T13 x 3T2  Z_{3} ≀ 8T 9 [ 217 ]

157 432 Z_{3} ≀ 8T 23 9T26 [ 259 ]

158 576  8T42 x 2T1  Z_{2} ≀ 6T 5 [ 208 ]
159 576  8T 42 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 19

160 576 Z_{2} ≀ 6T 10 8T46 [ 162, 198 ] 161 576 Z_{2} ≀ 6T 9 8T45 [ 163, 165, 195, 196, 197, 239 ] 162 576 Z_{2} ≀ 6T 10 8T46 [ 160, 198 ] 163 576 Z_{2} ≀ 6T 9 8T45 [ 161, 165, 195, 196, 197, 239 ] 164 576 12T 85 ×_{3T 1} 4T 4 Z_{2} ≀ 9T 2 165 576 Z_{2} ≀ 6T 9 8T45 [ 161, 163, 195, 196, 197, 239 ] 166 576 Z_{2} ≀ 9T 1

167 648  6T 5 ≀ 2T 1  Z_{3} ≀ 4T 3
168 648  6T 9 ×_{2T 1} 6T 9  Z_{3} ≀ 8T 3
169 648  6T 13 ×_{4T 2} 6T 9  Z_{3} ≀ 8T 4
170 648  6T 9 ×_{2T 1} 6T 10  Z_{3} ≀ 8T 2
171 648  6T 10 ×_{2T 1} 6T 10  Z_{3} ≀ 8T 2
172 648  6T 13 ×_{4T 3} 6T 13  Z_{3} ≀ 8T 4
173 648  9T 15 ×_{8T 1} 9T 15  Z_{3} ≀ 8T 1
174 648  9T 14 ×_{8T 5} 9T 14  Z_{3} ≀ 8T 5

175 648 Z_{3} ≀ 4T 5 9T29 [ 231, 290 ] 176 648 Z_{3} ≀ 6T 6 9T28 [ 234, 292 ] 177 648 Z_{3} ≀ 6T 8 9T30 [ 178, 233, 291 ] 178 648 Z_{3} ≀ 6T 8 9T30 [ 177, 233, 291 ] 179 660* 11T5 180 720 6T15 x 2T1 Z_{2} ≀ 6T 15 [ 286 ] 181 720 10T31 182 720 10T30 183 720 6T16 [ 219, 285, 287 ] 184 768 Z_{2} ≀ 12T 8 [ 190, 191 ] 185 768 12T 86 ×_{6T 11} 12T 114 Z_{4} ≀ 6T 3 186 768 Z_{2} ≀ 6T 3 [ 193 ] 187 768 Z_{2} ≀ 6T 4 [ 188 ] 188 768 2T 1 ≀ 6T 4 Z_{2} ≀ 6T 4 [ 187 ] 189 768 Z_{4} ≀ 6T 4 190 768 Z_{2} ≀ 12T 8 [ 184, 191 ] 191 768 Z_{2} ≀ 12T 8 [ 184, 190 ] 192 768 Z_{2}^{5} ≀_{r} 12T 98

193 768  2T 1 ≀ 6T 3  Z_{2} ≀ 6T 3 [ 186 ]
194 972  3T 1 ≀ 4T 4  Z_{3} ≀ 4T 4
195 1152  8T45 x 2T1  Z_{2} ≀ 6T 9 [ 240 ]
196 1152  8T 45 ×_{4T 2} 4T 3  Z_{2} ≀ 12T 38
197 1152  8T 45 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 39
198 1152  8T46 x 2T1  Z_{2} ≀ 6T 10 [ 199, 241 ]
199 1152  8T46 x 2T1  Z_{2} ≀ 6T 10 [ 198, 241 ]
200 1152  6T 8 ≀ 2T 1  Z_{2} ≀ 6T 13 8T47 [ 201, 202, 203, 235, 237 ]

201 1152 Z_{2} ≀ 6T 13 8T47 [ 200, 202, 203, 235, 237 ] 202 1152 Z_{2} ≀ 6T 13 8T47 [ 200, 201, 203, 235, 237 ] 203 1152 6T 7 ≀ 2T 1 Z_{2} ≀ 6T 13 8T47 [ 200, 201, 202, 235, 237 ] 204 1152 12T 127 ×_{3T 2} 4T 5 Z_{2} ≀ 9T 5

205 1152  8T 42 ×_{3T 2} 4T 5  Z_{2} ≀ 9T 4
206 1152  8T 42 ×_{3T 1} 4T 4  Z_{2} ≀ 9T 4

207 1152 Z_{2} ≀ 9T 3

208 1152  6T 6 ≀ 2T 1  Z_{2} ≀ 6T 5
209 1296  6T 13 ×_{2T 1} 6T 10  Z_{3} ≀ 8T 10
210 1296  6T 13 ×_{4T 2} 6T 13  Z_{3} ≀ 8T 9
211 1296  6T 13 ×_{4T 2} 6T 13  Z_{3} ≀ 8T 10
212 1296  9T 19 ×_{8T 8} 9T 19  Z_{3} ≀ 8T 8

213 1296 Z_{3} ≀ 6T 11 9T31 [ 258, 294 ] 214 1296 Z_{3} ≀ 8T 11 215 1296 Z_{3} ≀ 8T 7 216 1296 Z_{3} ≀ 8T 6 217 1296 6T 13 ×_{2T 1} 6T 9 Z_{3} ≀ 8T 9 218 1320* 219 1440 6T16 x 2T1 Z_{2} ≀ 6T 16 [ 293 ] 220 1440 10T35 221 1536 Z_{4} ≀ 6T 7 222 1536 4T 3 ≀ 3T 1 Z_{2} ≀ 6T 6 223 1536 Z_{2} ≀ 6T 8 [ 224 ] 224 1536 2T 1 ≀ 6T 8 Z_{2} ≀ 6T 8 [ 223 ] 225 1536 Z_{4} ≀ 6T 8 226 1536 Z_{2} ≀ 6T 7 [ 227 ] 227 1536 2T 1 ≀ 6T 7 Z_{2} ≀ 6T 7 [ 226 ] 228 1728 Z_{2} ≀ 9T 6 229 1728 Z_{2} ≀ 9T 7 230 1920 Z_{2} ≀ 15T 5 231 1944 3T 1 ≀ 4T 5 Z_{3} ≀ 4T 5 232 1944 Z_{3} ≀ 8T 12

233 1944  9T 30 ×_{2T 1} 3T 2  Z_{3} ≀ 8T 14
234 1944  9T 28 ×_{2T 1} 3T 2  Z_{3} ≀ 8T 13
235 2304  8T47 x 2T1  Z_{2} ≀ 6T 13 [ 236, 260 ]
236 2304  8T47 x 2T1  Z_{2} ≀ 6T 13 [ 235, 260 ]
237 2304  8T 47 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 79 [ 238 ]
238 2304  8T 47 ×_{2T 1} 4T 1  Z_{2} ≀ 12T 79 [ 237 ]
239 2304  8T 45 ×_{3T 2} 4T 5  Z_{2} ≀ 9T 8
240 2304  2T 1 ≀ 6T 9  Z_{2} ≀ 6T 9

241 2304  2T 1 ≀ 6T 10  Z_{2} ≀ 6T 10

242 2592 Z_{3} ≀ 8T 22 243 2592 Z_{3} ≀ 8T 15 244 2592 Z_{3} ≀ 8T 16 245 2592 Z_{3} ≀ 8T 19 [ 246, 247 ] 246 2592 Z_{3} ≀ 8T 19 [ 245, 247 ] 247 2592 Z_{3} ≀ 8T 19 [ 245, 246 ]

248 2592  6T 9 ≀ 2T 1  Z_{3} ≀ 8T 18
249 2592  6T 10 ≀ 2T 1  Z_{3} ≀ 8T 17
250 3072  4T 3 ≀ 3T 2  Z_{2} ≀ 6T 11

251 3456 Z_{2} ≀ 9T 11 [ 253 ] 252 3456 Z_{2} ≀ 9T 12 253 3456 Z_{2} ≀ 9T 11 [ 251 ] 254 3456 Z_{2} ≀ 9T 10 255 3840 2T 1 ≀ 6T 12 Z_{2} ≀ 6T 12 256 3840 Z_{2}^{5} ≀_{r} 12T 124 257 3840 Z_{2} ≀ 15T 10 258 3888 9T 31 ×_{2T 1} 3T 2 Z_{3} ≀ 8T 24 259 3888 Z_{3} ≀ 8T 23

260 4608  6T 11 ≀ 2T 1  Z_{2} ≀ 6T 13
261 5184  3T 2 ≀ 4T 2  Z_{3} ≀ 8T 29 [ 267 ]

262 5184 Z_{3} ≀ 8T 27 [ 264 ] 263 5184 Z_{3} ≀ 8T 30

264 5184  3T 2 ≀ 4T 1  Z_{3} ≀ 8T 27 [ 262 ]
265 5184  4T 4 ≀ 3T 1  Z_{2} ≀ 9T 17

266 5184 Z_{3} ≀ 8T 26 267 5184 Z_{3} ≀ 8T 29 [ 261 ] 268 6912 Z_{2} ≀ 9T 18

269 7200  6T 12 ≀ 2T 1  10T40
270 7680  2T 1 ≀ 6T 14  Z_{2} ≀ 6T 14

271 7776 Z_{3} ≀ 8T 32 272 7920* 11T6 273 10368 Z_{2} ≀ 9T 22 274 10368 6T 13 ≀ 2T 1 Z_{2} ≀ 8T 35 275 10368 4T 4 ≀ 3T 2 Z_{2} ≀ 9T 20 276 10368 Z_{2} ≀ 9T 21 277 11520 Z_{2} ≀ 15T 20 278 14400 10T42 279 14400 10T41 280 15552 3T 2 ≀ 4T 4 Z_{3} ≀ 8T 38 281 15552 Z_{3} ≀ 8T 39 282 15552 Z_{3} ≀ 8T 40 283 20736 Z_{2} ≀ 9T 24 284 20736 Z_{2} ≀ 9T 25 285 23040 Z_{2} ≀ 15T 28 286 23040 2T 1 ≀ 6T 15 Z_{2} ≀ 6T 15 287 23040 288 28800 6T 14 ≀ 2T 1 10T43 289 31104 3T 2 ≀ 4T 5 Z_{3} ≀ 8T 44 290 41472 Z_{2} ≀ 9T 29 291 41472 Z_{2} ≀ 9T 30 292 41472 4T 5 ≀ 3T 1 Z_{2} ≀ 9T 28 293 46080 2T 1 ≀ 6T 16 Z_{2} ≀ 6T 16 294 82944 4T 5 ≀ 3T 2 Z_{2} ≀ 9T 31 295 95040* 296 259200 6T 15 ≀ 2T 1 297 518400 298 518400 299 1036800 6T 16 ≀ 2T 1 300 12!/2* 301 12!*

5.2. Transitive Groups of degree 14

No Size  Group  Factor  Factor
1 14  7T1 x 2T1  Z_{7} ≀ 2T 1 [ 8, 9, 29 ]

2 14 Z_{7} ≀ 2T 1 7T2 [ 3, 8, 13, 27, 28, 38 ] 3 28 7T2 x 2T1 Z_{7} ≀ 4T 2 [ 13, 38 ] 4 42 Z_{7} ≀ 6T 1 7T4 [ 7, 14, 24, 40, 41, 48 ] 5 42 7T3 x 2T1 Z_{7} ≀ 6T 1 [ 14, 18, 44 ] 6 56 Z_{2} ≀ 7T 1 8T25 [ 9, 21, 29 ]

7 84  7T4 x 2T1  Z_{7} ≀ 12T 2 [ 24, 48 ]
8 98  7T 1 ≀ 2T 1  Z_{7} ≀ 2T 1
9 112  8T25 x 2T1  Z_{2} ≀ 7T 1 [ 29 ]

10 168 7T5 [ 17, 19, 33, 34, 42, 43, 50, 51 ] 11 168 Z_{2} ≀ 7T 3 8T36 [ 18, 35, 44 ] 12 196 Z_{7} ≀ 4T 1

13 196  7T2 x 7T2  Z_{7} ≀ 4T 2
14 294  7T 4 ×_{3T 1} 7T 3  Z_{7} ≀ 6T 1

15 294 Z_{7} ≀ 3T 2 16 336 8T43

17 336  7T5 x 2T1  Z_{2} ≀ 7T 5 [ 19, 42, 43, 51 ]
18 336  8T36 x 2T1  Z_{2} ≀ 7T 3 [ 44 ]
19 336  7T5 x 2T1  Z_{2} ≀ 7T 5 [ 17, 42, 43, 51 ]
20 392  7T 2 ≀ 2T 1  Z_{7} ≀ 4T 3
21 448  8T 25 ×_{7T 1} 8T 25  Z_{2} ≀ 7T 1 [ 29 ]

22 588 Z_{7} ≀ 12T 5 23 588 Z_{7} ≀ 12T 1 24 588 7T 4 ×_{3T 1} 7T 4 Z_{7} ≀ 12T 2 25 588 Z_{7} ≀ 6T 3 26 882 7T 3 ≀ 2T 1 Z_{7} ≀ 6T 5 27 896 Z_{2} ≀ 7T 2 [ 28, 38 ] 28 896 Z_{2} ≀ 7T 2 [ 27, 38 ] 29 896 2T 1 ≀ 7T 1 Z_{2} ≀ 7T 1 30 1092* 31 1176 Z_{7} ≀ 12T 13 32 1176 Z_{7} ≀ 12T 14 33 1344 [ 42 ] 34 1344 Z_{2} ≀ 7T 5 8T48 [ 43, 50, 51 ] 35 1344 8T 36 ×_{7T 3} 8T 36 Z_{2} ≀ 7T 3 [ 44 ] 36 1764 Z_{7} ≀ 12T 19 37 1764 Z_{7} ≀ 12T 18 38 1792 2T 1 ≀ 7T 2 Z_{2} ≀ 7T 2 39 2184* 40 2688 Z_{2} ≀ 7T 4 [ 41, 48 ] 41 2688 Z_{2} ≀ 7T 4 [ 40, 48 ]

42 2688  14T33 x 2T1  Z_{2} ≀ 14T 33
43 2688  8T48 x 2T1  Z_{2} ≀ 7T 5 [ 51 ]
44 2688  2T 1 ≀ 7T 3  Z_{2} ≀ 7T 3
45 3528  7T 4 ≀ 2T 1  Z_{7} ≀ 12T 42

46 5040 7T7 [ 49, 54, 55, 57 ]

47 5040  7T6 x 2T1  Z_{2} ≀ 7T 6 [ 56 ]
48 5376  2T 1 ≀ 7T 4  Z_{2} ≀ 7T 4

49 10080 7T7 x 2T1 Z_{2} ≀ 7T 7 [ 57 ] 50 10752 Z_{2} ≀ 7T 5 [ 51 ] 51 21504 2T 1 ≀ 7T 5 Z_{2} ≀ 7T 5 52 56448 7T 5 ≀ 2T 1 53 161280 Z_{2} ≀ 7T 6 [ 56 ] 54 322560 Z_{2} ≀ 7T 7 [ 55, 57 ] 55 322560 Z_{2} ≀ 7T 7 [ 54, 57 ] 56 322560 2T 1 ≀ 7T 6 Z_{2} ≀ 7T 6 57 645120 2T 1 ≀ 7T 7 Z_{2} ≀ 7T 7 58 12700800 7T 6 ≀ 2T 1 59 25401600 60 25401600 61 50803200 7T 7 ≀ 2T 1 62 14!/2* 63 14!*

5.3. Transitive Groups of degree 15

No Size  Group  Factor  Factor
1 15  5T1 x 3T1  Z_{5} ≀ 3T 1 [ 25, 36 ]
2 30  5T 2 ×_{2T 1} 3T 2  Z_{5} ≀ 6T 2 [ 31, 45 ]
3 30  5T2 x 3T1  Z_{5} ≀ 6T 1 [ 30, 46 ]
4 30  3T2 x 5T1  Z_{5} ≀ 3T 2 [ 32, 44 ]

5 60 5T4 [ 15, 16, 23, 53, 61, 69, 76, 88, 90 ]

6 60  5T 3 ×_{2T 1} 3T 2  Z_{5} ≀ 12T 5 [ 37, 54 ]
7 60  5T2 x 3T2  Z_{5} ≀ 6T 3 [ 40, 55 ]
8 60  5T3 x 3T1  Z_{5} ≀ 12T 1 [ 38, 56 ]

9 75 Z_{5} ≀ 3T 1 [ 25 ] 10 120 5T5 [ 21, 22, 24, 29, 62, 63, 70, 77, 78,

83, 89, 91, 93 ]

11 120 5T3 x 3T2 Z_{5} ≀ 12T 11 [ 49, 64 ] 12 150 Z_{5} ≀ 6T 1 [ 30 ] 13 150 Z_{5} ≀ 3T 2 [ 14, 31, 32 ] 14 150 Z_{5} ≀ 3T 2 [ 13, 31, 32 ]

15 180  5T4 x 3T1  Z_{3} ≀ 5T 4 [ 16, 69 ]
16 180  5T4 x 3T1  Z_{3} ≀ 5T 4 [ 15, 69 ]

17 300 Z_{5} ≀ 12T 5 [ 37 ] 18 300 Z_{5} ≀ 6T 3 [ 40 ] 19 300 Z_{5} ≀ 12T 1 [ 38 ] 20 360* 6T15

21 360  5T 5 ×_{2T 1} 3T 2  Z_{3} ≀ 10T 12 [ 22, 77 ]
22 360  5T 5 ×_{2T 1} 3T 2  Z_{3} ≀ 10T 12 [ 21, 77 ]
23 360  5T4 x 3T2  Z_{3} ≀ 10T 11 [ 76 ]
24 360  5T5 x 3T1  Z_{3} ≀ 5T 5 [ 78 ]
25 375  5T 1 ≀ 3T 1  Z_{5} ≀ 3T 1

26 405 Z_{3} ≀ 5T 1 [ 36 ] 27 600 Z_{5} ≀ 12T 11 [ 49 ] 28 720* 6T16

29 720  5T5 x 3T2  Z_{3} ≀ 10T 22 [ 83 ]
30 750  15T 12 ×_{2T 1} 5T 2  Z_{5} ≀ 6T 1
31 750  15T 14 ×_{2T 1} 5T 2  Z_{5} ≀ 6T 2
32 750  5T 1 ≀ 3T 2  Z_{5} ≀ 3T 2

33 810 Z_{3} ≀ 10T 1 [ 44 ] 34 810 Z_{3} ≀ 5T 2 [ 35, 45, 46 ] 35 810 Z_{3} ≀ 5T 2 [ 34, 45, 46 ]

36 1215  3T 1 ≀ 5T 1  Z_{3} ≀ 5T 1
37 1500  15T 17 ×_{4T 1} 5T 3  Z_{5} ≀ 12T 5
38 1500  15T 19 ×_{4T 1} 5T 3  Z_{5} ≀ 12T 1

39 1500 Z_{5} ≀ 4T 4 40 1500 15T 18 ×_{2T 1} 5T 2 Z_{5} ≀ 6T 3 41 1620 Z_{3} ≀ 5T 3 [ 42, 54, 56 ] 42 1620 Z_{3} ≀ 5T 3 [ 41, 54, 56 ] 43 1620 Z_{3} ≀ 10T 3 [ 55 ]

44 2430  15T 33 ×_{2T 1} 3T 2  Z_{3} ≀ 10T 1
45 2430  15T 35 ×_{2T 1} 3T 2  Z_{3} ≀ 10T 2
46 2430  3T 1 ≀ 5T 2  Z_{3} ≀ 5T 2

47 2520* 7T6 48 3000 Z_{5} ≀ 6T 8

49 3000  15T 27 ×_{4T 1} 5T 3  Z_{5} ≀ 12T 11
50 3000  5T 2 ≀ 3T 1  Z_{5} ≀ 6T 6

51 3000 Z_{5} ≀ 4T 5 52 3240 Z_{3} ≀ 10T 5 [ 64 ] 53 4860 Z_{3} ≀ 5T 4 [ 69 ]

54 4860  15T 42 ×_{2T 1} 3T 2  Z_{3} ≀ 10T 4
55 4860  15T 43 ×_{2T 1} 3T 2  Z_{3} ≀ 10T 3
56 4860  3T 1 ≀ 5T 3  Z_{3} ≀ 5T 3

57 6000 Z_{5} ≀ 12T 31 58 6000 Z_{5} ≀ 12T 27 59 6000 Z_{5} ≀ 12T 29 60 6000 5T 2 ≀ 3T 2 Z_{5} ≀ 6T 11 61 9720 Z_{3} ≀ 10T 11 [ 76 ] 62 9720 Z_{3} ≀ 10T 12 [ 77 ] 63 9720 Z_{3} ≀ 5T 5 [ 78 ] 64 9720 15T 52 ×_{2T 1} 3T 2 Z_{3} ≀ 10T 5 65 12000 Z_{5} ≀ 12T 63 66 12000 Z_{5} ≀ 12T 62 67 12000 Z_{5} ≀ 12T 55 68 12000 Z_{5} ≀ 12T 53 69 14580 3T 1 ≀ 5T 4 Z_{3} ≀ 5T 4 70 19440 Z_{3} ≀ 10T 22 [ 83 ] 71 19440 Z_{3} ≀ 10T 8 72 20160* 8T49 73 24000 Z_{5} ≀ 12T 98 74 24000 Z_{5} ≀ 12T 95 75 24000 5T 3 ≀ 3T 1 Z_{5} ≀ 12T 94 76 29160 15T 61 ×_{2T 1} 3T 2 Z_{3} ≀ 10T 11 77 29160 15T 62 ×_{2T 1} 3T 2 Z_{3} ≀ 10T 12 78 29160 3T 1 ≀ 5T 5 Z_{3} ≀ 5T 5 79 38880 Z_{3} ≀ 10T 16 80 38880 Z_{3} ≀ 10T 15 81 38880 3T 2 ≀ 5T 1 Z_{3} ≀ 10T 14 82 48000 5T 3 ≀ 3T 2 Z_{5} ≀ 12T 150 83 58320 15T 70 ×_{2T 1} 3T 2 Z_{3} ≀ 10T 22 84 77760 Z_{3} ≀ 10T 25 85 77760 Z_{3} ≀ 10T 24 86 77760 3T 2 ≀ 5T 2 Z_{3} ≀ 10T 23 87 155520 3T 2 ≀ 5T 3 Z_{3} ≀ 10T 29 88 233280 Z_{3} ≀ 10T 34 89 466560 Z_{3} ≀ 10T 38 90 466560 3T 2 ≀ 5T 4 Z_{3} ≀ 10T 36 91 466560 Z_{3} ≀ 10T 37 92 648000 5T 4 ≀ 3T 1 93 933120 3T 2 ≀ 5T 5 Z_{3} ≀ 10T 39 94 1296000 95 1296000 96 1296000 5T 4 ≀ 3T 2 97 2592000 98 2592000 99 5184000 100 5184000 101 5184000 5T 5 ≀ 3T 1 102 10368000 5T 5 ≀ 3T 2 103 15!/2* 104 15!*

Explicit Galois realization 29

6. The polynomials

Degree 12

S_{12} x^{12} − x + 1
A_{12} x^{12} − 866052x + 2381643
T_{299} x^{12} − x^{9} − x^{7} − x^{6} − x^{4} + x^{3} + x^{2} + x + 1
T_{298} x^{12} − 72x^{2} − 120x − 50
T_{297} x^{12} − 24x^{7} + 144x^{2} + 200
T_{296} x^{12} − 12x^{11} + 36x^{10} + 6251x^{6} − 37506x^{5} + 9768751
M_{12} x^{12} − 375x^{8} − 3750x^{6} − 75000x^{3} + 228750x^{2} − 750000x + 1265625
T_{294} x^{12} − x^{11} + x^{10} + x^{9} + x^{7} + x^{6} + 3x^{4} − x^{3} + x^{2} + 1
T_{293} x^{12} − x^{6} − x^{2} − 1
T_{292} x^{12} − 3x^{11} + 5x^{9} − 3x^{8} + 3x^{7} + 2x^{6} − 6x^{5} − 3x^{4} + 1
T_{291} x^{12} − 12x^{9} − 9x^{8} − 64x^{3} − 144x^{2} − 108x − 27
T_{290} x^{12} − 2x^{10} − 6x^{9} + x^{8} + 8x^{7} − 10x^{5} − 7x^{4} + 20x^{3} + 2x^{2} − 2x − 4
T_{289} x^{12} − x^{11} − x^{10} − x^{7} − x^{5} + 3x^{4} − x^{3} + 2x^{2} − x + 1
T_{288} x^{12} − 4x^{11} + 4x^{10} − 50x^{4} + 120x^{3} − 112x^{2} + 48x − 8
T_{287} x^{12} − 3x^{10} − 3x^{8} + 2x^{4} + 2x^{2} + 2
T_{286} x^{12} − x^{6} − 3x^{4} − 1
T_{285} x^{12} − x^{10} − 2x^{4} + 1
T_{284} x^{12} − 12x^{9} − 9x^{8} + 64x^{3} + 144x^{2} + 108x + 27
T_{283} x^{12} − 8x^{9} + 24x^{6} + 144x^{5} + 96x^{3} + 144x^{2} + 48
T_{282} x^{12} − 7x^{10} − 8x^{9} + 7x^{8} + 16x^{7} − x^{6} − 16x^{5} − 16x^{4} − 16x^{3} + 24x^{2} + 40x − 8
T_{281} x^{12} − 12x^{10} + 54x^{8} − 129x^{6} + 207x^{4} − 70x^{3} − 189x^{2} + 210x − 153
T_{280} x^{12} − 4x^{11} + 6x^{10} − 2x^{9} − 5x^{8} + 6x^{7} − 4x^{5} + 2x^{4} + 2
T_{279} x^{12} − 4x^{11} + 4x^{10} + 4x^{7} − 6x^{6} − 4x^{5} + 36x^{2} + 36x + 9
T_{278} x^{12} + 20x^{8} − 80x^{6} + 50x^{4} − 320x^{3} − 912x^{2} + 1280x + 800
T_{277} x^{12} + 3x^{6} + 3x^{2} + 4
T_{276} x^{12} + 192x^{6} − 288x^{5} + 108x^{4} + 256x^{3} − 576x^{2} + 432x − 108
T_{275} x^{12} − 9x^{8} + 288x^{6} − 756x^{5} + 324x^{4} + 1728x^{3} − 1296x^{2} − 1944x + 2916
T_{274} x^{12} + x^{11} + x^{10} − x^{9} + x^{6} + 3x^{5} + x^{3} − x + 1
T_{273} x^{12} − 12x^{9} + 9x^{8} + 192x^{3} − 432x^{2} + 324x − 81
M_{11} x^{12} − x^{11} − 16x^{10} + 15x^{9} + 145x^{8} − 8x^{7} − 392x^{6} + 88x^{5} + 415x^{4} − 255x^{3}
−64x^{2} + 89x − 41
T_{271} x^{12} − 135x^{8} − 180x^{7} + 399x^{6} + 918x^{5} + 693x^{4} + 352x^{3} + 216x^{2} + 96x + 16
T_{270} x^{12} − 2x^{8} − 4x^{2} + 8
T_{269} x^{12} − 5x^{10} − 11x^{8} − 5x^{7} − 3x^{6} − 60x^{5} − 21x^{4} − 20x^{3} + 4x^{2} + 70x − 29
T_{268} x^{12} + 4x^{9} − 3x^{8} − 64x^{3} + 144x^{2} − 108x + 27
T_{267} x^{12} + 12x^{10} − 8x^{9} + 54x^{8} − 48x^{7} + 132x^{6} − 72x^{5} − 33x^{4} − 32x^{3} + 8
T_{266} x^{12} − 8x^{9} − 9x^{8} + 24x^{6} + 36x^{5} − 81x^{4} − 32x^{3} − 36x^{2} + 16
T_{265} x^{12} + 36x^{10} − 24x^{9} + 333x^{8} − 288x^{7} + 810x^{6} − 72x^{5} − 486x^{4} + 80x^{3}
+162x^{2} − 72x + 9
T_{264} x^{12} − 4x^{11} + 6x^{10} − 3x^{9} − 2x^{8} + 3x^{7} − 2x^{5} + x^{4} + x^{3} − x^{2} + 1
T_{263} x^{12} − 162x^{4} − 432x^{3} − 432x^{2} − 192x − 32
T_{262} x^{12} − 72x^{8} − 96x^{7} + 184x^{6} + 432x^{5} + 369x^{4} + 280x^{3} + 216x^{2} + 96x + 16
T_{261} x^{12} − 4x^{11} + 6x^{10} − 4x^{9} + x^{8} + 1
T_{260} x^{12} − 3x^{2} + 3
T_{259} x^{12} − 12x^{10} + 54x^{8} − 110x^{6} + 93x^{4} − 4x^{3} − 18x^{2} + 12x − 8
T_{258} x^{12} − 2x^{9} + 2x^{3} + 3
T_{257} x^{12} + 3x^{8} − 2x^{6} + 6x^{4} + 1

T_{256} x^{12} − 3x^{8} + 6x^{4} − 8x^{2} + 2
T_{255} x^{12} − 2x^{8} − 6x^{6} + 9x^{4} − 1
T_{254} x^{12} − 12x^{11} − 2x^{10} + 316x^{9} + 381x^{8} − 2760x^{7} − 11742x^{6} − 26260x^{5}
−42490x^{4} − 52184x^{3} − 48664x^{2} − 32640x − 11943
T_{253} x^{12} − 4x^{9} − 3x^{8} − 32x^{6} − 48x^{5} − 18x^{4} + 64x^{3} + 144x^{2} + 108x + 27
T_{252} x^{12} − 12x^{9} + 27x^{8} + 12x^{6} − 36x^{5} + 27x^{4} − 16x^{3} + 36x^{2} + 9
T_{251} x^{12} + 48x^{6} − 72x^{5} + 27x^{4} + 64x^{3} − 144x^{2} + 108x − 27
T_{250} x^{12} + x^{10} + x^{8} − x^{6} + 2
T_{249} x^{12} − 12x^{10} + 54x^{8} − 108x^{6} + 81x^{4} − 8x^{3} + 24x + 8
T_{248} x^{12} + 324x^{6} − 648x^{5} + 675x^{4} − 744x^{3} + 648x^{2} − 288x + 48
T_{247} x^{12} − 8x^{9} + 24x^{6} + 162x^{4} − 32x^{3} + 16
T_{246} x^{12} + 81x^{4} − 216x^{3} + 216x^{2} − 96x + 16
T_{245} x^{12} − 12x^{10} − 54x^{8} − 72x^{7} + 96x^{6} + 9x^{4} + 200x^{3} + 108x^{2} − 4
T_{244} x^{12} − 10x^{9} + 60x^{7} + 195x^{6} + 180x^{5} + 105x^{4} − 200x^{3} + 600x + 400
T_{243} x^{12} − 9x^{8} − 12x^{7} − 4x^{6} − 81x^{4} − 216x^{3} − 216x^{2} − 96x − 16
T_{242} x^{12} − 4x^{9} + 18x^{8} − 4x^{6} − 36x^{5} + 81x^{4} + 16x^{3} + 108x^{2} + 16
T_{241} x^{12} + x^{10} − 3x^{8} − x^{6} + 6x^{4} − 3
T_{240} x^{12} + 4x^{8} − 2x^{6} + 4x^{4} − x^{2} + 7
T_{239} x^{12} − 12x^{9} + 9x^{8} − 32x^{6} + 48x^{5} − 18x^{4} − 64x^{3} + 144x^{2} − 108x + 27
T_{238} x^{12} − 2x^{10} − 3x^{8} + 4x^{6} + 2x^{4} + 4x^{2} − 2
T_{237} x^{12} − 2x^{10} − x^{8} − 4x^{6} + 2
T_{236} x^{12} + x^{10} + x^{8} − x^{6} + 4x^{4} + 1
T_{235} x^{12} − 2x^{10} − x^{8} + 4x^{4} + 4x^{2} + 2
T_{234} x^{12} − x^{9} − 3x^{3} + 4
T_{233} x^{12} − 4x^{3} − 6
T_{232} x^{12} − 13x^{8} − 26x^{7} − 11x^{6} + 6x^{5} + 25x^{4} + 78x^{3} + 114x^{2} + 76x + 19
T_{231} x^{12} + 36x^{8} − 48x^{7} − 65x^{6} + 162x^{5} + 459x^{4} − 1488x^{3} + 1512x^{2} − 672x + 112
T_{230} x^{12} + x^{10} − 3x^{8} + 4x^{4} + 1
T_{229} x^{12} − 24x^{9} + 108x^{8} − 720x^{6} + 324x^{5} + 2349x^{4} − 1728x^{3} − 1296x^{2}
+5832x + 5103
T_{228} x^{12} − 6x^{10} − 24x^{9} − 15x^{8} + 96x^{7} + 786x^{6} − 912x^{5} + 1974x^{4} − 6992x^{3}
+16896x^{2} − 19728x + 12609
T_{227} x^{12} − 3x^{10} − 3x^{8} − x^{6} + 2
T_{226} x^{12} + x^{10} − x^{8} − 2x^{6} + x^{2} + 1
T_{225} x^{12} − 3x^{10} + 2x^{6} + 2x^{4} − 3
T_{224} x^{12} − 2x^{10} + x^{8} + 3x^{4} − 2x^{2} + 3
T_{223} x^{12} − 3x^{10} − 5x^{6} + 6x^{4} + 3x^{2} − 3
T_{222} x^{12} − 4x^{6} + 3x^{2} − 1
T_{221} x^{12} − 2x^{10} − x^{8} + 6x^{6} − x^{4} − 4x^{2} − 1
T_{220} x^{12} − 4x^{9} − 12x^{8} + 34x^{6} − 12x^{5} + 45x^{4} + 42x^{2} + 10
T_{219} x^{12} − x^{6} + 2x^{4} + x^{2} + 1
T_{218} x^{12} − 2x^{11} + 22x^{9} − 88x^{7} + 176x^{5} − 176x^{3} + 64x + 4
T_{217} x^{12} − 6x^{6} − 8x^{3} − 4
T_{216} x^{12} − 12x^{10} − 8x^{9} + 162x^{4} + 432x^{3} + 432x^{2} + 192x + 32
T_{215} x^{12} − 12x^{10} − 12x^{9} + 54x^{8} + 108x^{7} − 84x^{6} − 324x^{5} − 63x^{4} + 436x^{3}
+216x^{2} − 336x − 304
T_{214} x^{12} − 12x^{10} + 8x^{9} + 216x^{6} − 432x^{5} + 207x^{4} + 152x^{3} − 216x^{2} + 96x − 16
T_{213} x^{12} − 2x^{9} − 1
T_{212} x^{12} − 16x^{9} − 72x^{8} − 192x^{7} − 800x^{6} − 1824x^{4} + 4608x^{2} − 2048
T_{211} x^{12} − 135x^{8} − 180x^{7} + 210x^{6} + 540x^{5} + 765x^{4} + 1160x^{3} + 1080x^{2}
+480x + 80

T_{210} x^{12} − 4x^{9} + 8x^{6} − 36x^{5} + 105x^{4} − 120x^{3} + 90x^{2} − 36x + 9
T_{209} x^{12} − 8x^{9} + 18x^{8} − 24x^{7} + 24x^{6} − 33x^{4} − 16x^{3} − 48x − 8
T_{208} x^{12} − 3x^{10} + 3x^{6} + 3x^{4} + 3
T_{207} x^{12} − 6x^{10} − 8x^{9} + 9x^{8} − 60x^{6} + 207x^{4} − 256x^{3} − 1494x^{2} − 1848x − 793
T_{206} x^{12} − 12x^{9} + 15x^{8} − 12x^{5} + 18x^{4} − 64x^{3} + 96x^{2} − 36x + 9
T_{205} x^{12} − 208x^{6} − 312x^{5} − 117x^{4} − 832x^{3} − 1872x^{2} − 1404x − 351
T_{204} x^{12} + 260x^{9} + 63x^{8} − 648x^{7} − 780x^{6} + 108x^{5} + 2133x^{4} − 32x^{3}
−900x^{2} + 125
T_{203} x^{12} − x^{10} − x^{4} + x^{2} + 1
T_{202} x^{12} − 2x^{10} − 4x^{6} + 6x^{4} + 4x^{2} + 4
T_{201} x^{12} − 3x^{10} + 3x^{6} + 6x^{4} + 3x^{2} + 3
T_{200} x^{12} + 6x^{10} + 9x^{8} − 8x^{6} − 24x^{4} + 52
T_{199} x^{12} − 2x^{10} − 4x^{8} − x^{6} + x^{4} + 4
T_{198} x^{12} − 2x^{10} − x^{8} + 6x^{4} − 4x^{2} + 2
T_{197} x^{12} − 14x^{10} + 70x^{8} − 152x^{6} + 144x^{4} − 50x^{2} + 5
T_{196} x^{12} − 2x^{8} − 4x^{6} + 6x^{4} + 4x^{2} − 1
T_{195} x^{12} − x^{8} + 4x^{6} − 5x^{4} + 4x^{2} + 1
T_{194} x^{12} − 4x^{11} − 32x^{9} + 198x^{8} + 216x^{7} + 1032x^{6} − 384x^{5} + 801x^{4}
−452x^{3} + 32x^{2} + 4
T_{193} x^{12} + 6x^{6} + 6x^{4} + 3
T_{192} x^{12} − 6x^{10} + x^{8} + 36x^{6} − 30x^{4} − 28x^{2} + 18
T_{191} x^{12} + x^{10} + 2x^{8} − x^{6} + 2x^{4} − 3x^{2} + 1
T_{190} x^{12} + 2x^{10} − 13x^{8} + 36x^{6} + 15x^{4} − 38x^{2} − 19
T_{189} x^{12} − 6x^{10} + 7x^{8} + 12x^{6} − 16x^{4} − 8x^{2} + 5
T_{188} x^{12} − 2x^{10} + 5x^{6} + 5x^{2} − 1
T_{187} x^{12} − 6x^{8} − 2x^{6} − 3x^{2} + 1
T_{186} x^{12} − 2x^{8} − 3x^{2} − 1
T_{185} x^{12} − x^{8} − x^{4} + 2
T_{184} x^{12} − 2x^{10} − 2x^{8} + 4x^{6} − x^{4} + 6x^{2} + 1
T_{183} x^{12} − 6x^{10} + 49x^{8} − 2464x^{6} + 388x^{4} + 80x^{2} + 4
T_{182} x^{12} − 8x^{9} + 6x^{8} + 20x^{6} − 24x^{5} + 18x^{4} − 16x^{3} + 24x^{2} + 8
T_{181} x^{12} − 18x^{8} − 36x^{6} − 72x^{5} + 54x^{4} − 144x^{3} − 216x^{2} − 72
T_{180} x^{12} − 2x^{10} + 5x^{8} − 8x^{6} + 6x^{4} − 4x^{2} + 1
L_{2}(11) x^{12} − 4x^{11} + 396x^{8} − 1056x^{7} + 2112x^{6} + 52272x^{4} − 69696x^{3} + 278784x^{2}
−211968x + 2336832
T_{178} x^{12} − x^{9} + 4x^{3} − 1
T_{177} x^{12} − 4x^{9} + 4x^{3} + 2
T_{176} x^{12} + 4x^{6} − 8x^{3} + 8
T_{175} x^{12} − 12x^{10} − 8x^{9} + 36x^{8} + 48x^{7} − 65x^{6} − 162x^{5} + 135x^{4} + 624x^{3}
+648x^{2} + 288x + 48
T_{174} x^{12} − 12x^{10} − 8x^{9} − 120x^{6} + 432x^{5} − 828x^{4} − 32x^{3} − 864x^{2} − 1920x − 128
T_{173} x^{12} − 36x^{8} − 48x^{7} − 32x^{6} + 162x^{4} − 288x^{2} + 128
T_{172} x^{12} − 2x^{11} + 16x^{10} − 68x^{9} − 530x^{8} − 300x^{7} + 5380x^{6} + 19304x^{5} + 27280x^{4}
+19880x^{3} + 10476x^{2} + 2704x + 676
T_{171} x^{12} − 8x^{9} − 36x^{8} − 72x^{5} + 81x^{4} + 64x^{3} − 144x^{2} + 64
T_{170} x^{12} − x^{9} + 2x^{6} + 4x^{3} + 3
T_{169} x^{12} − 8x^{3} + 18
T_{168} x^{12} − 10x^{6} − 12x^{3} − 2
T_{167} x^{12} − 3x^{3} + 3
T_{166} x^{12} + 18x^{10} + 135x^{8} + 348x^{6} + 63x^{4} − 512x^{3} − 270x^{2} + 729
T_{165} x^{12} − 16x^{9} + 12x^{8} + 256x^{3} − 576x^{2} + 432x − 108

T_{164} x^{12} + 3x^{11} − 6x^{10} − 33x^{9} − 30x^{8} + 54x^{7} + 155x^{6} + 180x^{5} + 192x^{4} + 272x^{3}
+300x^{2} + 192x + 64
T_{163} x^{12} − x^{8} − 2x^{6} + x^{4} − 2x^{2} + 1
T_{162} x^{12} − 2x^{8} − 8x^{6} + 14x^{4} − 16x^{2} + 4
T_{161} x^{12} − x^{8} + 2x^{6} + x^{4} + 2x^{2} + 1
T_{160} x^{12} + x^{10} + x^{8} + x^{6} − 4x^{4} + 5
T_{159} x^{12} + 4x^{10} − 4x^{8} − 24x^{6} − x^{4} + 32x^{2} + 8
T_{158} x^{12} − x^{8} − 2x^{6} + 2x^{2} + 1
T_{157} x^{12} − 8x^{9} + 24x^{7} + 44x^{6} − 51x^{4} + 48x^{3} − 72x^{2} + 16
T_{156} x^{12} − 6x^{6} − 8x^{3} − 1
T_{155} x^{12} − 2x^{10} − 3x^{8} + 2
T_{154} x^{12} + 10x^{8} − 4x^{6} + 49x^{4} + 52x^{2} + 104
T_{153} x^{12} − 3x^{10} + 5x^{8} − 8x^{6} + 10x^{4} − 12x^{2} + 8
T_{152} x^{12} − 4x^{8} − 2x^{6} + 4x^{4} − 1
T_{151} x^{12} − 3x^{8} − 2
T_{150} x^{12} − x^{6} − 3x^{4} + 2x^{2} + 2
T_{149} x^{12} − 9x^{4} − 6
T_{148} x^{12} − 2x^{10} + 2x^{8} − 2x^{6} − 2x^{4} − 2x^{2} − 1
T_{147} x^{12} − 9x^{4} − 12
T_{146} x^{12} − 2x^{10} − x^{8} − 2x^{6} − 2x^{4} − 8x^{2} + 8
T_{145} x^{12} + 6x^{8} + 4x^{6} − 18x^{4} − 24x^{2} − 8
T_{144} x^{12} + 6x^{10} + 4x^{8} − 24x^{6} − 21x^{4} + 22x^{2} + 4
T_{143} x^{12} − 6x^{10} + 24x^{8} − 56x^{6} + 93x^{4} − 90x^{2} + 51
T_{142} x^{12} + 3x^{8} + 4x^{6} + 6x^{4} + 3
T_{141} x^{12} + 3x^{8} − 3
T_{140} x^{12} + 4x^{4} − 4
T_{139} x^{12} + x^{8} − 3x^{6} + x^{4} + 1
T_{138} x^{12} − x^{4} + 1
T_{137} x^{12} + x^{10} + x^{8} − x^{4} + x^{2} − 1
T_{136} x^{12} − 2x^{10} − 5x^{4} − 2x^{2} + 4
T_{135} x^{12} − 18x^{8} − 24x^{6} + 27x^{4} + 36x^{2} − 6
T_{134} x^{12} − 7x^{10} + 14x^{8} − 21x^{4} + 7x^{2} + 7
T_{133} x^{12} + 54x^{8} − 72x^{7} + 204x^{6} + 216x^{5} + 585x^{4} + 8x^{3} + 972x^{2} − 216x + 216
T_{132} x^{12} − 4x^{10} − 36x^{7} − 12x^{6} + 144x^{5} + 228x^{4} + 208x^{3} + 360x^{2} + 464x + 216
T_{131} x^{12} − 3x^{10} − 2x^{9} + 54x^{8} + 72x^{7} + 402x^{6} + 756x^{5} + 5445x^{4} + 13288x^{3}
+13176x^{2} + 5856x + 976
T_{130} x^{12} − x^{9} + 5x^{6} − 8x^{3} + 4
T_{129} x^{12} − 24x^{9} + 72x^{7} + 2x^{6} − 60x^{5} + 9x^{4} + 64x^{3} + 30x^{2} − 24x − 17
T_{128} x^{12} − 32x^{9} + 50x^{8} + 248x^{6} − 768x^{5} + 579x^{4} + 64x^{3} − 160x^{2} + 40x + 76
T_{127} x^{12} − 54x^{9} − 315x^{8} + 4372x^{6} + 3996x^{5} − 22005x^{4} − 13176x^{3} + 17484x^{2}
−22518x + 1775
T_{126} x^{12} + x^{8} + x^{6} − 2x^{4} − x^{2} + 1
T_{125} x^{12} − 2x^{8} − 2x^{6} + x^{4} + 2x^{2} − 1
T_{124} x^{12} − 2x^{10} − 5x^{8} + 35x^{4} − 30x^{2} + 5
T_{123} x^{12} − 2x^{10} + 10x^{6} − 8x^{2} + 1
T_{122} x^{12} − 2x^{11} − 3x^{10} − 6x^{9} + 21x^{8} − 32x^{7} + 37x^{6} − 16x^{5} + 11x^{4} + 32x^{3}
−x^{2} + 20x + 1
T_{121} x^{12} − x^{9} + 2x^{3} + 1
T_{120} x^{12} − 2x^{9} − 6x^{3} + 9
T_{119} x^{12} − 8x^{6} − 8x^{3} − 2
T_{118} x^{12} + 8x^{6} − 8x^{3} + 2

T_{117} x^{12} − 2x^{9} + x^{6} + 5
T_{116} x^{12} − 2x^{9} + 4x^{3} + 4
T_{115} x^{12} − 2x^{8} + 3x^{4} − 4
T_{114} x^{12} − x^{4} − 1
T_{113} x^{12} − x^{8} + 4
T_{112} x^{12} − 3x^{8} + 9x^{4} + 9
T_{111} x^{12} − 6x^{8} + 68x^{6} + 105x^{4} + 36x^{2} + 12
T_{110} x^{12} + x^{8} − x^{6} − x^{4} − 1
T_{109} x^{12} + x^{10} − 4x^{2} + 1
T_{108} x^{12} − 3x^{8} − 4x^{6} + 6x^{4} + 4
T_{107} x^{12} + 6x^{10} + 3x^{8} − 28x^{6} − 21x^{4} + 30x^{2} + 5
T_{106} x^{12} + 3x^{10} − 2x^{8} − 9x^{6} + 5x^{2} + 1
T_{105} x^{12} − 7x^{10} + 7x^{8} + 14x^{6} − 16x^{4} − 5x^{2} + 5
T_{104} x^{12} + 6x^{10} + 12x^{8} + 8x^{6} − 3x^{4} − 6x^{2} − 1
T_{103} x^{12} − 2x^{8} + 5x^{6} − 2x^{4} + 1
T_{102} x^{12} − 5x^{10} + 20x^{8} − 70x^{6} + 145x^{4} − 280x^{2} + 208
T_{101} x^{12} − x^{8} + x^{6} − x^{4} + 1
T_{100} x^{12} − x^{10} + x^{8} + 4x^{6} − x^{4} − x^{2} − 1
T_{99} x^{12} + 12x^{10} + 60x^{8} + 160x^{6} + 228x^{4} + 144x^{2} + 8
T_{98} x^{12} + 214x^{10} − 1046693x^{8} + 18491564x^{6} − 29606993x^{4} + 937950x^{2} + 8560357
T_{97} x^{12} + x^{8} + 9x^{4} + 1
T_{96} x^{12} − 3x^{4} − 4
T_{95} x^{12} − x^{10} + 3x^{6} − 2x^{4} − 3x^{2} + 1
T_{94} x^{12} − 57x^{8} − 38x^{6} + 318x^{4} − 204x^{2} + 17
T_{93} x^{12} + 10x^{10} + 28x^{8} + 6x^{6} − 43x^{4} + 6x^{2} + 3
T_{92} x^{12} − 9x^{4} − 9
T_{91} x^{12} + 5x^{10} + 9x^{8} + 8x^{6} + 2x^{4} − 12x^{2} + 16
T_{90} x^{12} + 2x^{10} − x^{6} + 2x^{2} + 1
T_{89} x^{12} − 3x^{4} + 1
T_{88} x^{12} − 6x^{8} − 4x^{6} − 3x^{4} − 18x^{2} + 3
T_{87} x^{12} + 6x^{10} + 9x^{8} − 4x^{6} − 12x^{4} + 1
T_{86} x^{12} + 2x^{8} − 2
T_{85} x^{12} − 3x^{11} − 3x^{10} + 15x^{9} − 15x^{8} − 33x^{7} + 29x^{6} + 15x^{5} − 30x^{4} − 128x^{3}
−30x^{2} + 198x + 48
T_{84} x^{12} − 4x^{11} + 2x^{10} + 12x^{9} − 20x^{8} + 16x^{6} − 6x^{4} − 8x^{3} + 4x^{2} + 8x + 4
T_{83} x^{12} + 3x^{6} − x^{3} + 3
T_{82} x^{12} − 12x^{10} + 54x^{8} − 116x^{6} + 129x^{4} − 72x^{2} − 16
T_{81} x^{12} + x^{6} + 2
T_{80} x^{12} − 90x^{8} + 160x^{6} − 135x^{4} + 7200x^{2} − 80
T_{79} x^{12} − 4x^{8} + 4x^{6} + 5x^{4} − 4x^{2} + 2
T_{78} x^{12} − x^{9} + x^{3} + 1
T_{77} x^{12} − 2x^{10} + x^{8} + 6x^{6} − 6x^{4} + 1
T_{76} x^{12} + 2x^{8} − 2x^{6} + 5x^{4} − 6x^{2} + 1
T_{75} x^{12} − 2x^{10} − 2x^{8} + 6x^{6} + x^{4} − 6x^{2} + 1
T_{74} x^{12} − x^{10} + 2x^{8} + 4x^{6} − 3x^{4} − 3x^{2} + 1
T_{73} x^{12} + 12x^{10} − 3x^{9} + 54x^{8} − 27x^{7} + 122x^{6} − 81x^{5} + 165x^{4} − 93x^{3} + 126x^{2}
−36x + 31
T_{72} x^{12} + 12x^{10} − 40x^{9} + 414x^{8} − 1416x^{7} + 3388x^{6} − 6552x^{5} + 8001x^{4} − 7448x^{3}
+7056x^{2} + 4704x + 2744
T_{71} x^{12} − 4x^{9} + 4x^{6} + 3
T_{70} x^{12} + 9x^{6} − 18x^{3} + 9

T_{69} x^{12} − 3x^{10} − 2x^{8} + 9x^{6} − 5x^{2} + 1
T_{68} x^{12} + x^{10} + 6x^{8} + 3x^{6} + 6x^{4} + x^{2} + 1
T_{67} x^{12} − x^{8} − x^{6} − x^{4} + 1
T_{66} x^{12} + 6x^{10} + 12x^{8} + 8x^{6} − 3
T_{65} x^{12} − 3x^{4} + 4
T_{64} x^{12} − x^{8} + 9x^{4} − 1
T_{63} x^{12} − 6x^{10} + 104x^{6} + 93x^{4} + 18x^{2} + 4
T_{62} x^{12} + 6x^{10} − 104x^{6} + 93x^{4} − 18x^{2} + 4
T_{61} x^{12} − 3x^{4} − 1
T_{60} x^{12} − 14x^{8} − 7x^{4} + 4
T_{59} x^{12} − 6x^{10} + 6x^{8} − 4x^{6} − 3x^{4} + 3
T_{58} x^{12} − 12x^{8} − 14x^{6} + 9x^{4} + 12x^{2} + 1
T_{57} x^{12} − 69x^{10} − 2091x^{8} − 7571x^{6} + 134691x^{4} + 960267x^{2} + 1545049
T_{56} x^{12} − 2x^{10} + x^{6} − 2x^{2} + 1
T_{55} x^{12} − 30x^{10} + 348x^{8} − 1960x^{6} + 5505x^{4} − 7050x^{2} + 3025
T_{54} x^{12} − 6x^{8} + 9x^{4} + 12
T_{53} x^{12} + 2x^{8} − 16x^{6} + 4x^{4} + 8
T_{52} x^{12} + 12x^{4} − 12
T_{51} x^{12} + 6x^{8} + 9x^{4} + 3
T_{50} x^{12} − 3x^{4} + 6
T_{49} x^{12} + 3x^{8} − 4x^{6} − 3x^{4} − 1
T_{48} x^{12} − x^{8} + 3x^{4} + 1
T_{47} x^{12} − 6x^{10} + 20x^{9} − 72x^{7} + 128x^{6} − 96x^{5} + 45x^{4} − 8x^{3} − 18x^{2} + 12x − 2
T_{46} x^{12} + 12x^{10} − 19x^{9} + 54x^{8} − 171x^{7} + 169x^{6} − 513x^{5} + 447x^{4} − 573x^{3}
+549x^{2} − 180x + 16
T_{45} x^{12} − 3x^{9} − 18x^{8} − 24x^{6} − 9x^{5} + 69x^{4} − x^{3} + 3x − 1
T_{44} x^{12} − 6x^{6} − 10x^{3} − 6
T_{43} x^{12} − 6x^{9} + 10x^{6} + 4x^{3} + 2
T_{42} x^{12} − x^{6} + 7
T_{41} x^{12} − x^{9} − 9x^{6} − x^{3} + 1
T_{40} x^{12} − 7x^{10} + 24x^{8} − 36x^{6} + 24x^{4} + 13x^{2} + 1
T_{39} x^{12} − 5x^{3} + 5
T_{38} x^{12} + x^{6} − 3
T_{37} x^{12} + x^{6} + 9
T_{36} x^{12} − x^{9} − x^{6} − x^{3} + 1
T_{35} x^{12} − x^{9} − x^{6} + x^{3} + 1
T_{34} x^{12} + 12x^{10} + 54x^{8} + 108x^{6} + 81x^{4} + 16
T_{33} x^{12} + 2x^{8} + 58x^{6} + 301x^{4} + 174x^{2} + 25
T_{32} x^{12} + 7x^{10} − x^{8} − 23x^{6} − x^{4} + 7x^{2} + 1
T_{31} x^{12} − 30x^{10} + 343x^{8} − 1860x^{6} + 4760x^{4} − 4600x^{2} + 225
T_{30} x^{12} − 7x^{10} − 14x^{8} + 115x^{6} − 70x^{4} − 175x^{2} + 125
T_{29} x^{12} − 36x^{8} − 40x^{6} + 36x^{4} + 48x^{2} + 8
T_{28} x^{12} + 2
T_{27} x^{12} + 12x^{10} + 68x^{8} + 220x^{6} + 392x^{4} + 360x^{2} + 148
T_{26} x^{12} − 9x^{8} − 8x^{6} − 9x^{4} + 1
T_{25} x^{12} + 5x^{8} + 6x^{4} + 1
T_{24} x^{12} + 4x^{10} + 7x^{8} + 4x^{6} − x^{4} − 2x^{2} + 1
T_{23} x^{12} − 4x^{4} + 4
T_{22} x^{12} − 5x^{10} + 7x^{8} − 6x^{7} − 17x^{6} − 6x^{5} + 7x^{4} − 5x^{2} + 1
T_{21} x^{12} + 3x^{8} − 4x^{6} + 3x^{4} + 1
T_{20} x^{12} − 4x^{9} + 72x^{8} − 84x^{7} + 236x^{6} − 144x^{5} + 324x^{4} − 192x^{3} + 72x^{2} + 8

Explicit Galois realization 35

T_{19} x^{12}− 2x^{11} + 4x^{10} + x^{9} + 5x^{8} + 8x^{7} + 53x^{6} + 44x^{5} + 59x^{4} + 19x^{3} + 13x^{2} + 5x + 1^{T_{18} x^{12} − 4x^{6} + 16}^{T_{17} x^{12} + 4x^{8} + 4x^{6} + 5x^{4} + 12x^{2} + 2}^{T_{16} x^{12} − x^{6} + 4}^{T_{15} x^{12} + 3}^{T_{14} x^{12} − 9x^{6} + 27}^{T_{13} x^{12} − 12}^{T_{12} x^{12} + x^{6} − 27}^{T_{11} x^{12} + 10x^{6} + 5}^{T_{10} x^{12} + 16}^{T_{9} x^{12} + 3x^{8} + 4x^{6} + 3x^{4} + 1}^{T_{8} x^{12} − 6x^{10} − 8x^{9} + 9x^{8} + 12x^{7} − 20x^{6} + 9x^{4} − 24x^{3} − 4}^{T_{7} x^{12} + 4x^{10} − x^{8} − x^{4} + 4x^{2} + 1}^{T_{6} x^{12} + 2x^{10} − 6x^{8} + 2x^{6} − 6x^{4} + 2x^{2} + 1}^{T_{5} x^{12} − 80x^{10} + 1820x^{8} − 13680x^{6} + 29860x^{4} − 2720x^{2} + 32}^{T_{4} x^{12} + 6x^{8} + 26x^{6} − 63x^{4} + 162x^{2} + 81}^{T_{3} x^{12} + 36}^{T_{2} x^{12} − x^{6} + 1}^{T_{1} x^{12} − x^{11} + x^{10} − x^{9} + x^{8} − x^{7} + x^{6} − x^{5} + x^{4} − x^{3} + x^{2} − x + 1} Degree 13

S_{13} x^{13} − x^{12} + x^{4} − x + 1
A_{13} x^{13} + 156x − 144
T_{7} x^{13} − 3x^{12} + 3x^{11} + 13x^{10} − 24x^{9} + 36x^{8} − 99x^{7} + 63x^{6} − 18x^{5} + 66x^{4}
+9x^{3} − 9x^{2} − 21x − 9
T_{6} x^{13} − 2
T_{5} x^{13} − x^{12} + 20x^{10} − 50x^{9} + 141x^{8} − 225x^{7} + 420x^{6} − 456x^{5} + 470x^{4}
−101x^{3} + 90x^{2} − 8x + 11
T_{4} x^{13} + 13x^{10} − 26x^{8} + 13x^{7} + 52x^{6} − 39x^{4} + 26x^{2} + 13x + 2
T_{3} x^{13} − 39x^{11} + 507x^{9} − 156x^{8} − 2925x^{7} + 1872x^{6} + 7605x^{5} − 7488x^{4} − 6435x^{3}
+10062x^{2} − 2691x − 306
T_{2} x^{13} − 3x^{12} − 278x^{11} + 205x^{10} + 24414x^{9} + 55909x^{8} − 638959x^{7} − 3888668x^{6}
−8795208x^{5} − 9304040x^{4} − 4272496x^{3} − 446336x^{2} + 199680x + 41600
T_{1} x^{13} − x^{12} − 24x^{11} + 19x^{10} + 190x^{9} − 116x^{8} − 601x^{7} + 246x^{6} + 738x^{5} − 215x^{4}
−291x^{3} + 68x^{2} + 10x − 1

36 J. Klüners and G. Malle

Degree 14 \begin{aligned}S_{14} & x^{14} − x + 2 \\ A_{14} & x^{14} − 154x^{11} − 643076643 \\ T_{61} & x^{14} + 49x^{2} − 84x + 36 \\ T_{60} & x^{14} + 147x^{2} − 252x + 108 \\ T_{59} & x^{14} − 14x^{13} + 49x^{12} − 4353564672 \\ T_{58} & x^{14} + 10976x^{2} − 18816x + 8064 \\ T_{57} & x^{14} + 2x^{4} + x^{2} + 1 \\ T_{56} & x^{14} + 2x^{12} + 2x^{2} − 2 \\ T_{55} & x^{14} − 2x^{4} + x^{2} − 1 \\ T_{54} & x^{14} − 2x^{13} − 10x^{12} + 73x^{11} − 133x^{10} − 108x^{9} + 418x^{8} − 300x^{7} − 605x^{6} + 772x^{5} \\ & +717x^{4} − 499x^{3} + 444x^{2} + 42x − 37 \\ T_{53} & x^{14} − x^{8} − x^{6} + 3x^{4} − 1 \\ T_{52} & x^{14} − 6x^{13} + 11x^{12} − 17x^{10} + 6x^{9} + x^{8} + 24x^{7} − 40x^{5} − 8x^{4} + 32x^{3} + 8x^{2} \\ & −16x + 8 \\ T_{51} & x^{14} + 2x^{8} − 5x^{6} − 3x^{2} + 4 \\ T_{50} & x^{14} − 2x^{8} − 5x^{6} − 3x^{2} − 4 \\ T_{49} & x^{14} − 3x^{8} + 2x^{4} + 1 \\ T_{48} & x^{14} − 7x^{6} + 14x^{4} − 7x^{2} + 8 \\ T_{47} & x^{14} + 2x^{12} − 2x^{10} + x^{6} − 8x^{4} + 5x^{2} + 2 \\ T_{46} & x^{14} + 189x^{8} − 234x^{7} + 28224x^{2} + 24192x + 41472 \\ T_{45} & x^{14} − x^{13} + 3x^{12} + x^{11} − 8x^{10} + 2x^{9} + 12x^{8} − 35x^{7} + 52x^{6} + 26x^{5} − 59x^{4} \\ & −39x^{3} + 99x^{2} − 64x + 19 \\ T_{44} & x^{14} − 14x^{10} + 56x^{6} − 56x^{2} + 20 \\ T_{43} & x^{14} − 5x^{8} − 5x^{6} + 9x^{4} + 2x^{2} + 1 \\ T_{42} & x^{14} − 42 x^{12} − 2268 x^{10} + 110376 x^{8} + 650916 x^{6} − 62338248 x^{4} \\ & +446777856 x^{2} − 859963392 \\ T_{41} & x^{14} − 7x^{10} + 14x^{6} − 7x^{2} − 4 \\ T_{40} & x^{14} − x^{12} − 3x^{10} − 3x^{8} + 12x^{6} − 9x^{2} + 9 \\ T_{39} & x^{14} − x^{13} + 26x^{10} + 65x^{6} + 13x^{5} + 52x^{2} + 12x + 1 \\ T_{38} & x^{14} + 4x^{10} + 2x^{8} + 2x^{6} − 3x^{2} − 2 \\ T_{37} & x^{14} − 28x^{11} − 14x^{10} − 28x^{9} + 280x^{8} + 180x^{7} + 833x^{6} − 2548x^{5} + 1400x^{4} \\ & −728x^{3} + 1316x^{2} − 672x + 136 \\ T_{36} & x^{14} − 7x^{12} − 147x^{11} + 399x^{10} − 2282x^{9} + 5068x^{8} + 5510x^{7} + 120393x^{6} \\ & −706293x^{5} + 1918966x^{4} − 2714152x^{3} + 2804872x^{2} − 1751568x + 551872 \\ T_{35} & x^{14} + 6x^{12} − 12x^{10} − 64x^{8} + 16x^{6} + 128x^{4} − 64 \\ T_{34} & x^{14} + 5x^{8} − 5x^{6} − 9x^{4} + 2x^{2} − 1 \\ T_{33} & x^{14} + 84x^{12} + 1848x^{10} + 11872x^{8} + 359730x^{6} + 2314872x^{4} + 43964424x^{2} \\ & −746496 \\ T_{32} & x^{14} − 42x^{10} − 16x^{7} + 392x^{6} − 560x^{5} − 448x^{3} − 392x^{2} − 1120x − 736 \\ T_{31} & x^{14} + 14x^{12} − 7x^{11} + 259x^{10} − 21x^{9} + 1680x^{8} − 1885x^{7} + 9114x^{6} − 13391x^{5} \\ & +27853x^{4} − 31885x^{3} + 21322x^{2} − 20370x − 13275 \\ T_{30} & x^{14} − 6x^{13} + 13x^{12} − 338x^{9} + 845x^{8} + 17576x^{4} + 70304x + 35152 \\ T_{29} & x^{14} + 7x^{12} − 49x^{8} + 98x^{4} − 49x^{2} − 7 \\ T_{28} & x^{14} + 7x^{6} + 7x^{4} + 7x^{2} − 1 \\ T_{27} & x^{14} + 9x^{12} + 37x^{10} + 85x^{8} + 163x^{6} + 267x^{4} + 439x^{2} + 151 \\ T_{26} & x^{14} + 714x^{12} + 16100x^{11} + 238623x^{10} + 5068476x^{9} + 85951551x^{8} + 40905306x^{7} \\ & +3304719936x^{6} + 129890941166x^{5} − 423192245805x^{4} − 3704820336210x^{3} \\ & +42231389433079x^{2} − 241689758907660x + 670952296880775 \\ T_{25} & x^{14} + 42x^{12} − 56x^{11} + 371x^{10} − 840x^{9} + 1834x^{8} − 5008x^{7} + 7987x^{6} − 6384x^{5} \\ & +22358x^{4} − 3416x^{3} + 18753x^{2} − 5544x + 7702 \end{aligned}

T_{24} x^{14} − 3x^{7} + 6
T_{23} x^{14} − 56x^{12} + 245x^{11} + 2534x^{10} − 1372x^{9} − 24528x^{8} − 257782x^{7}

−1470049x^{6} + 1229802x^{5} + 12092913x^{4} + 48481531x^{3} + 478963261x^{2} +704643912x − 1494810659

T_{22} x^{14} − 63x^{12} − 9555x^{11} + 118671x^{10} − 708246x^{9} − 17922660x^{8} + 859373823x^{7}

+2085856500x^{6} − 117366985106x^{5} − 335941176396x^{4} + 4638317668005x^{3} +17926524826973x^{2} + 7429846568445x + 91264986397629

T_{21} x^{14} − x^{12} − 12x^{10} + 7x^{8} + 28x^{6} − 14x^{4} − 9x^{2} − 1
T_{20} x^{14} + 210x^{12} − 3164x^{11} + 63455x^{10} − 534016x^{9} + 7977046x^{8} − 27661364x^{7}

+1002627612x^{6} + 6022284016x^{5} − 28570776528x^{4} + 138886748224x^{3} −3146649429952x^{2} + 4701085568256x − 59618052726016

T_{19} x^{14} + 6x^{12} + 16x^{10} + 28x^{8} + 32x^{6} + 26x^{4} + 12x^{2} + 2
T_{18} x^{14} − 5x^{12} − 39x^{10} − 7x^{8} + 41x^{6} + 7x^{4} − 9x^{2} + 1
T_{17} x^{14} −18x^{12} − 964x^{10} + 32592x^{8} − 353912x^{6} + 1736792x^{4} − 3987152x^{2} + 3495368
T_{16} x^{14} − 7x^{12} + 42x^{10} − 98x^{8} + 441x^{6} − 196x^{5} − 343x^{4} + 392x^{3} + 1372x^{2}

+588x + 112

T_{15} x^{14} − 105x^{12} − 147x^{11} + 5271x^{10} + 19838x^{9} − 94150x^{8} − 607634x^{7} + 570164x^{6}

+12260920x^{5} + 42847770x^{4} + 95169270x^{3} + 197804880x^{2} + 348280352x +336238208

T_{14} x^{14} − 31958x^{7} + 656356768
T_{13} x^{14} − 2x^{13} − 29x^{12} − 222x^{11} − 352x^{10} + 3498x^{9} + 18163x^{8} + 46467x^{7}

+92188x^{6} + 128405x^{5} + 96637x^{4} + 31142x^{3} + 7064x^{2} + 6304x + 2432

T_{12} x^{14} + 28x^{12} − 189x^{11} + 756x^{10} − 4004x^{9} + 15953x^{8} − 48856x^{7} + 129262x^{6}

−251559x^{5} + 330764x^{4} − 272986x^{3} − 123305x^{2} + 739662x − 577916

T_{11} x^{14} + 7x^{12} − 21x^{10} − 147x^{8} + 91x^{6} + 693x^{4} + 385x^{2} − 9
T_{10} x^{14} + 14x^{8} − 84x^{6} + 84x^{4} + 21x^{2} − 9
T_{9} x^{14} + 7x^{12} − 49x^{10} − 245x^{8} + 588x^{6} + 294x^{4} − 7
T_{8} x^{14} + 28x^{11} + 28x^{10} − 28x^{9} + 140x^{8} + 360x^{7} + 147x^{6} + 196x^{5} + 336x^{4}

−546x^{3} − 532x^{2} + 896x + 823

T_{7} x^{14} + 2
T_{6} x^{14} + 28x^{12} + 56x^{10} − 245x^{8} − 322x^{6} + 406x^{4} − 56x^{2} − 1
T_{5} x^{14} + 28x^{12} + 308x^{10} + 1680x^{8} + 4704x^{6} + 6272x^{4} + 3136x^{2} + 484
T_{4} x^{14} + 7
T_{3} x^{14} − 8x^{12} + 22x^{10} − 8x^{8} − 55x^{6} + 48x^{4} + 64x^{2} − 71
T_{2} x^{14} + 9x^{12} + 53x^{10} + 333x^{8} + 1251x^{6} + 731x^{4} + 5415x^{2} + 8591
T_{1} x^{14} + 25x^{12} + 214x^{10} + 767x^{8} + 1194x^{6} + 686x^{4} + 53x^{2} + 1

38 J. Klüners and G. Malle

Degree 15 \begin{aligned}S_{15} & x^{15} − x − 1 \\ A_{15} & x^{15} − 240x − 224 \\ T_{102} & x^{15} + 375x^{3} − 900x^{2} + 720x − 192 \\ T_{101} & x^{15} − 15x^{11} − 12x^{10} + 375x^{3} + 900x^{2} + 720x + 192 \\ T_{100} & x^{15} + 500x^{3} − 1200x^{2} + 960x − 256 \\ T_{99} & x^{15} − 15x^{11} + 75x^{7} − 125x^{3} − 32 \\ T_{98} & x^{15} − 5x^{11} + 4x^{10} − 1100x^{7} + 1760x^{6} − 704x^{5} + 8000x^{3} − 19200x^{2} \\ & +15360x − 4096 \\ T_{97} & x^{15} − 500x^{3} − 1200x^{2} − 960x − 256 \\ T_{96} & x^{15} + 15x^{11} − 12x^{10} + 125x^{3} − 300x^{2} + 240x − 64 \\ T_{95} & x^{15} + 15x^{11} − 12x^{10} − 125x^{3} + 300x^{2} − 240x + 64 \\ T_{94} & x^{15} − 15x^{14} + 75x^{13} − 125x^{12} − 180x^{5} + 900x^{4} + 15360 \\ T_{93} & x^{15} + 5x^{12} + 8x^{9} + 6x^{6} − x^{5} + 3x^{3} + 1 \\ T_{92} & x^{15} + 20x^{11} − 8x^{10} + 75x^{7} − 60x^{6} − 16x^{5} − 125x^{3} − 200x^{2} + 80x + 64 \\ T_{91} & x^{15} − 1620x^{7} + 4320x^{6} − 432x^{5} − 11040x^{4} + 16960x^{3} − 11520x^{2} + 3840x − 512 \\ T_{90} & x^{15} + 5x^{12} + x^{11} + 13x^{9} + 3x^{8} + 16x^{6} + 8x^{3} + x^{2} + 1 \\ T_{89} & x^{15} − 405x^{11} − 4824x^{10} − 25600x^{9} − 76800x^{8} − 137040x^{7} − 138240x^{6} \\ & −62208x^{5} + 400x^{3} + 1920x^{2} + 3840x + 2048 \\ T_{88} & x^{15} + 5x^{14} + 5x^{13} − 5x^{12} + 7x^{11} + 31x^{10} − 5x^{9} − 23x^{8} + 13x^{7} + 9x^{6} + 45x^{5} \\ & +19x^{4} − 53x^{3} − 5x^{2} − 13x + 1 \\ T_{87} & x^{15} + 2x^{5} − 10x^{4} + 20x^{3} − 20x^{2} + 10x − 2 \\ T_{86} & x^{15} + 5x^{12} + 5x^{11} + 5x^{9} + 15x^{8} + 5x^{7} + 18x^{5} + 5x^{4} + 5x^{2} + 1 \\ T_{85} & x^{15} − 15x^{13} + 90x^{11} − 270x^{9} + 405x^{7} − 243x^{5} − 16 \\ T_{84} & x^{15} − 765x^{11} − 510x^{10} + 2075x^{9} + 2160x^{8} + 15705x^{7} + 20060x^{6} − 27432x^{5} \\ & −60150x^{4} − 41395x^{3} − 13860x^{2} − 2310x − 154 \\ T_{83} & x^{15} − 2x^{12} + 2x^{6} − 2 \\ T_{82} & x^{15} − 3x^{13} − 4x^{12} + 27x^{11} + 6x^{10} − 79x^{9} − 342x^{8} + 858x^{7} − 632x^{6} − 504x^{5} \\ & +420x^{4} + 716x^{3} − 1728x^{2} + 1296x − 432 \\ T_{81} & x^{15} + 5x^{13} + 10x^{11} − 25x^{7} − 5x^{6} − 29x^{5} − 10x^{4} − 5x^{2} + 10x − 1 \\ T_{80} & x^{15} − 15x^{13} + 90x^{11} − 280x^{9} + 495x^{7} − 5x^{6} − 513x^{5} + 30x^{4} + 330x^{3} \\ & −45x^{2} − 180x − 16 \\ T_{79} & x^{15} − 5x^{13} + 7x^{12} + 10x^{11} − 28x^{10} + 16x^{9} + 42x^{8} − 73x^{7} + 31x^{6} + 77x^{5} \\ & −111x^{4} + 43x^{3} + 59x^{2} − 69x + 29 \\ T_{78} & x^{15} − 44x^{13} + 782x^{11} − x^{10} − 7005x^{9} + 5x^{8} + 31605x^{7} − 10x^{6} − 57406x^{5} \\ & +10x^{4} − 692x^{3} − 5x^{2} − 9x + 1 \\ T_{77} & x^{15} − 952x^{12} − 512732x^{9} − 8487340x^{6} + 3418039096x^{3} + 369608289704 \\ T_{76} & x^{15} + 2x^{9} − 2x^{6} − x^{3} + 2 \\ T_{75} & x^{15} − x^{14} − 16x^{13} + 36x^{12} + 159x^{11} − 981x^{10} − 487x^{9} + 8972x^{8} − 1724x^{7} \\ & −41608x^{6} + 49632x^{5} + 9600x^{4} − 41728x^{3} + 27648x − 13824 \\ T_{74} & x^{15} − 4x^{14} + 29x^{13} − 70x^{12} + 249x^{11} − 336x^{10} + 267x^{9} − 154x^{8} + 698x^{7} \\ & −400x^{6} − 1620x^{5} + 1548x^{4} + 1628x^{3} − 3744x^{2} + 2016x − 432 \\ T_{73} & x^{15} + 30x^{13} + 30x^{12} + 255x^{11} + 674x^{10} + 150x^{9} + 3580x^{8} − 2295x^{7} − 4070x^{6} \\ & +17912x^{5} − 14800x^{4} − 2705x^{3} + 25520x^{2} − 24180x + 10942 \\ T_{72} & x^{15} + x^{14} − 37x^{13} − 81x^{12} + 394x^{11} + 906x^{10} − 1650x^{9} − 3878x^{8} + 1548x^{7} \\ & +3724x^{6} + 28484x^{5} − 42844x^{4} − 18504x^{3} + 59864x^{2} − 36216x + 7288 \\ T_{71} & x^{15} + 3x^{14} − 33x^{13} + 18x^{12} + 414x^{11} − 1011x^{10} − 1662x^{9} + 5289x^{8} + 6717x^{7} \\ & −16690x^{6} − 14886x^{5} + 29238x^{4} + 13966x^{3} − 24117x^{2} − 4551x + 7303 \\ T_{70} & x^{15} − x^{3} − 1 \\ T_{69} & x^{15} − 47x^{13} − 2x^{12} + 871x^{11} + 44x^{10} − 7925x^{9} − 286x^{8} + 35415x^{7} + 616x^{6} \\ & −62625x^{5} − 542x^{4} + 1569x^{3} + 172x^{2} − 27x − 2 \end{aligned}

T_{68} x^{15} − 25x^{13} − 30x^{12} + 370x^{11} + 148x^{10} − 2560x^{9} + 1120x^{8} + 9000x^{7} − 12880x^{6}

−8272x^{5} + 32800x^{4} − 21760x^{3} − 23040x^{2} + 34560x − 13824

T_{67} x^{15} + 27x^{13} − 76x^{12} + 2640x^{11} − 11358x^{10} − 21029x^{9} + 130602x^{8} − 22668x^{7}

−465624x^{6} + 645456x^{5} − 210144x^{4} − 151552x^{3} + 36864x^{2} + 34560x − 13824

T_{66} x^{15} − 114x^{14} + 282185319x^{13} + 1247857228852x^{12} − 35114805704965233x^{11}

−141524337796433387826x^{10} + 2604584980442264028744009x^{9} +14153948932132918272984150384x^{8}−178273077248353369941327628479552x^{7} −1142953506821390914419260564494304768x^{6} +15975069142211276963134599495014990639616x^{5} +33516684438303088018217308253251277376159744x^{4} −617589777108203716232396372453619309554471256064x^{3} −397561412445066919545461762354884631501806174863360x^{2} +2266657182908547570648245464215192357802047101628186624x −1302222456532760256406916223259306960561657428777814196224

T_{65} x^{15} − 9x^{14} − 777124356x^{13} − 1133827391276x^{12} + 248667491445971640x^{11}

+1982470037108913200760x^{10} − 20251674921386743233089664x^{9} −477582096907660950523117014816x^{8} −3308886107335264486810139708481072x^{7} +782196413071317769675085549386001392x^{6} +136771384654763125975628635944913787162688x^{5} +495305752715364614800204906020616494062534208x^{4} −2654636123517266162930575109745213496431971332096x^{3} −26690849479086638949750226189970304815661445420472832x^{2} −84762887552754551927936865646583259242697034492208111616x −102487167144832305896578849873538581080313524947316321767424

T_{64} x^{15} + 5x^{9} + 5x^{3} − 2
T_{63} x^{15} − 15x^{13} + 90x^{11} − 247x^{9} + 198x^{7} − 20x^{6} + 378x^{5} + 120x^{4} − 600x^{3} − 180x^{2}

−63x + 146

T_{62} x^{15} + 23x^{12} + 130x^{9} + 18x^{6} − 27x^{3} + 27
T_{61} x^{15} + 2x^{9} − 5x^{6} + 2x^{3} − 1
T_{60} x^{15} − 5x^{12} + 10x^{11} − 12x^{10} + 10x^{9} − 20x^{8} + 40x^{7} − 60x^{6} + 73x^{5} − 65x^{4}

+35x^{3} − 35x^{2} + 30x − 9

T_{59} x^{15} + 20x^{13} + 125x^{11} − 53x^{10} + 250x^{9} − 275x^{8} − 75x^{7} − 245x^{6} − 1009x^{5}

−1025x^{4} + 1825x^{3} − 1150x^{2} + 225x + 2059

T_{58} x^{15} + 30x^{13} + 45x^{12} + 255x^{11} + 986x^{10} + 525x^{9} + 4870x^{8} + 2355x^{7} − 6605x^{6}

+21707x^{5} − 7825x^{4} − 12005x^{3} + 38780x^{2} − 30480x + 14578

T_{57} x^{15} − 6x^{14} − 185929x^{13} − 7490711x^{12} + 10447793997x^{11} + 850215157784x^{10}

−183592676667989x^{9} − 22599650147469301x^{8} + 268871342200307043x^{7} +111386427537228333570x^{6} + 2235951723532365786373x^{5} −151310638511676685186017x^{4} − 5086537813630840677351345x^{3} −5767418822154378301922100x^{2} + 338165560497864890767793817x +612968886770859386964763021

T_{56} x^{15} − 45x^{13} + 810x^{11} − 2x^{10} − 7290x^{9} + 10x^{8} + 32805x^{7} − 20x^{6} − 59049x^{5}

+20x^{4} − 10x^{2} + 2

T_{55} x^{15} + 3x^{9} − x^{6} + 3x^{3} + 3
T_{54} x^{15} + 6x^{14} − 60x^{13} − 1074x^{12} − 4950x^{11} + 5454x^{10} + 90400x^{9} + 136296x^{8}

−428418x^{7} − 1326320x^{6} − 295140x^{5} + 2170494x^{4} + 2415630x^{3} − 135012x^{2} −6410634x − 5989054

T_{53} x^{15} + 7x^{14} + 22x^{13} + 44x^{12} + 67x^{11} + 90x^{10} + 133x^{9} + 198x^{8} + 198x^{7} + 102x^{6}

+36x^{5} + 21x^{4} − 15x^{3} − 24x^{2} + 3

T_{52} x^{15} + 5x^{9} + 5x^{3} − 1
T_{51} x^{15} − 50x^{13} − 5x^{12} + 330x^{11} + 266x^{10} + 9150x^{9} − 12230x^{8} − 55090x^{7} + 142500x^{6}

−156072x^{5} + 136600x^{4} − 150040x^{3} + 126420x^{2} − 48320x + 1024

T_{50} x^{15} − 5x^{13} − 10x^{12} + 139x^{10} + 50x^{9} − 1670x^{8} − 1950x^{7} + 6980x^{6} + 16836x^{5}

+5450x^{4} − 28890x^{3} − 58420x^{2} − 39745x − 7279

T_{49} x^{15} − x^{5} − 2
T_{48} x^{15} − x^{14} − 42x^{13} + 576x^{12} + 6248x^{11} + 1220x^{10} − 123132x^{9} − 275290x^{8}

+902478x^{7} + 3670908x^{6} − 147896x^{5} − 15604028x^{4} − 10561837x^{3} +25131303x^{2} + 22337416x − 19837924

T_{47} x^{15} − 2x^{14} − 6x^{13} − 18x^{12} + 823x^{11} − 4627x^{10} + 12741x^{9} − 15872x^{8} − 6962x^{7}

+59365x^{6} − 91720x^{5} + 43297x^{4} + 53515x^{3} − 108581x^{2} + 80509x − 28454

T_{46} x^{15} − 50x^{13} + 965x^{11} − 4x^{10} − 8910x^{9} + 20x^{8} + 39435x^{7} − 40x^{6} − 68134x^{5}

+40x^{4} + 4015x^{3} − 20x^{2} − 90x + 4

T_{45} x^{15} − 6x^{14} + 28x^{13} + 133x^{12} − 17x^{11} + 508x^{10} + 14373x^{9} + 24160x^{8} − 119521x^{7}

+95861x^{6} + 681292x^{5} + 97666x^{4} − 2852755x^{3} + 4387594x^{2} − 2726259x +777069

T_{44} x^{15} − 5x^{12} + 15x^{6} − 5x^{3} − 7
T_{43} x^{15} − 2x^{12} + 2x^{9} − x^{6} + 1
T_{42} x^{15} − 6x^{13} + 31x^{12} − 270x^{10} − 674x^{9} − 54x^{8} + 5670x^{7} + 8818x^{6} + 11790x^{5}

−372x^{4} + 4043x^{3} − 774x^{2} − 2670x + 1

T_{41} x^{15} − 3x^{14} + 24x^{13} − 4x^{12} − 315x^{11} + 453x^{10} + 818x^{9} − 2964x^{8} + 2370x^{7}

+5990x^{6} − 13470x^{5} + 2028x^{4} + 13804x^{3} − 13104x^{2} + 7320x − 2872

T_{40} x^{15} − 5x^{14} + 15x^{13} + 45x^{12} − 935x^{11} + 2619x^{10} + 17025x^{9} − 19445x^{8} − 171320x^{7}

+344390x^{6} + 3336719x^{5} + 2746335x^{4} − 5890980x^{3} − 1407410x^{2}

+2396000x + 798080

T_{39} x^{15} + 5x^{14} − 675x^{13} − 5500x^{12} + 83550x^{11} + 1101150x^{10} + 4157575x^{9}

+3236100x^{8} − 11798025x^{7} − 21372125x^{6} − 63158850x^{5} − 281431250x^{4} −135052500x^{3} + 1343230000x^{2} + 2535100000x + 1313680000

T_{38} x^{15} − 5x^{10} + 4x^{5} + 5
T_{37} x^{15} − 6x^{10} + 2
T_{36} x^{15} − 55x^{13} − 5x^{12} + 1110x^{11} + 104x^{10} − 10400x^{9} − 685x^{8} + 45645x^{7} + 1480x^{6}

−76639x^{5} − 1295x^{4} + 7660x^{3} + 400x^{2} − 90x + 1

T_{35} x^{15} − 5x^{14} − 8x^{13} + 73x^{12} − 55x^{11} − 382x^{10} + 1017x^{9} + 482x^{8} − 3994x^{7}

+1543x^{6} + 2690x^{5} − 4868x^{4} + 3487x^{3} − 1350x^{2} + 279x − 27

T_{34} x^{15} − 5x^{14} + 7x^{13} + 6x^{11} − 44x^{10} + 67x^{9} − 53x^{8} + 52x^{7} − 48x^{6} + 10x^{5} + 12x^{4}

−11x^{3} + 16x^{2} − 12x + 1

T_{33} x^{15} − 10x^{9} − 5x^{6} + 10x^{3} − 1
T_{32} x^{15} − 3x^{14} − 9x^{13} + 26x^{12} − 3x^{11} − 60x^{10} + 14x^{9} + 72x^{8} + 63x^{7} − 190x^{6}

+150x^{5} − 117x^{4} + 79x^{3} − 42x^{2} + 15x − 1

T_{31} x^{15} − 615x^{13} + 8650x^{12} + 196335x^{11} − 1724326x^{10} − 10046015x^{9} + 511745770x^{8}

+2281327270x^{7} − 30001270850x^{6} + 7799981467x^{5} + 2779038445300x^{4} +8058450933185x^{3} − 52551636276610x^{2} − 201477388078380x +674662192576472

T_{30} x^{15} + 5x^{14} − 55x^{13} − 220x^{12} + 700x^{11} + 2791x^{10} − 1980x^{9} − 14270x^{8} − 8030x^{7}

+23050x^{6} + 35952x^{5} + 13270x^{4} − 8695x^{3} − 9555x^{2} − 3200x − 377

T_{29} x^{15} − 4x^{12} + 6x^{9} − 6x^{6} + 4x^{3} − 2
T_{28} x^{15} + x^{12} − x^{11} + x^{10} − 2x^{9} + x^{8} − 2x^{6} + 2x^{5} + x^{3} + 1
T_{27} x^{15} − x^{5} − 1
T_{26} x^{15} − 150x^{13} − 520x^{12} + 2400x^{11} + 12366x^{10} − 1700x^{9} − 73410x^{8} − 60675x^{7}

+161150x^{6} + 214578x^{5} − 119280x^{4} − 247825x^{3} − 3750x^{2} + 93525x + 23255

T_{25} x^{15} − 110x^{13} + 130x^{12} + 4425x^{11} − 10351x^{10} − 71925x^{9} + 269785x^{8} + 212240x^{7}

−2211090x^{6} + 3656145x^{5} − 2402600x^{4} + 722420x^{3} − 99015x^{2} + 5125x − 41

T_{24} x^{15} − x^{14} − 2x^{13} + x^{12} + 16x^{10} + 22x^{9} − 13x^{8} − 57x^{5} + 26x^{4} − 1
T_{23} x^{15} − 4x^{12} + 10x^{9} − 12x^{6} + 10x^{3} − 4

T_{22} x^{15} + 6x^{14} + 27x^{13} − 65x^{12} − 1818x^{11} − 4800x^{10}+ 15172x^{9}+ 83334x^{8} + 100197x^{7}

−95461x^{6} − 358221x^{5} − 215697x^{4} + 185503x^{3} + 195837x^{2} − 101025x − 138979

T_{21} x^{15} − 96x^{13} + 59x^{12} + 5031x^{11} − 13182x^{10} − 109799x^{9} + 487485x^{8} + 821655x^{7}
−6975764x^{6} + 2066256x^{5} + 46961715x^{4} − 80656366x^{3} − 8511147x^{2}
+57236934x + 26273539
T_{20} x^{15} + 3x^{13} − 9x^{11} − 6x^{10} − 32x^{9} − 9x^{8} + 5x^{6} + 18x^{5} + 6x^{4} − 3x − 1
T_{19} x^{15} + x^{10} − 2x^{5} − 1
T_{18} x^{15} − 15x^{13} − 15x^{12} + 55x^{11} + 149x^{10} + 150x^{9} − 165x^{8} − 740x^{7} − 1105x^{6}
−1153x^{5} − 875x^{4} − 460x^{3} − 160x^{2} − 25x + 1
T_{17} x^{15} + 45x^{13} + 630x^{11} + 162x^{10} + 3625x^{9} + 1350x^{8} + 9000x^{7} + 1800x^{6}
+7305x^{5} − 1375x^{3} − 2250x − 450
T_{16} x^{15} − x^{14} + 3x^{13} + 8x^{12} + 36x^{11} − 17x^{10} − 40x^{9} − 3x^{8} + 140x^{7} − 90x^{6} − 32x^{5}
+46x^{4} − 8x^{3} − 13x^{2} − 2x + 1
T_{15} x^{15} − 30x^{13} − 12x^{12} + 369x^{11} + 171x^{10} − 2769x^{9} − 234x^{8} + 16218x^{7} + 2328x^{6}
−58374x^{5} − 50094x^{4} + 18009x^{3} + 57132x^{2} − 42849x + 4761
T_{14} x^{15} + 5x^{14} + 50x^{13} + 220x^{12} + 970x^{11} + 1142x^{10} + 7935x^{9} − 2815x^{8} − 156480x^{7}
+1127980x^{6} − 3389737x^{5} + 7033715x^{4} − 9360980x^{3} + 9369840x^{2}
−5090420x + 1432484
T_{13} x^{15} − 60x^{13} − 90x^{12} + 1095x^{11} + 1224x^{10} − 14450x^{9} − 26775x^{8} + 59640x^{7}

+177525x^{6} + 12810x^{5} − 335025x^{4} − 386555x^{3} − 246465x^{2} − 192375x − 92889

T_{12} x^{15} − 235x^{13} + 70x^{12} + 15930x^{11} − 14493x^{10} − 325950x^{9} + 112750x^{8}
+2876560x^{7} + 708890x^{6} − 11702794x^{5} − 8278925x^{4} + 19171805x^{3}
+20087950x^{2} − 6362585x − 9179941
T_{11} x^{15} − 2
T_{10} x^{15} − 5x^{14} − 5x^{13} + 65x^{12} − 55x^{11} − 301x^{10} + 550x^{9} + 505x^{8} − 1925x^{7}
+400x^{6} + 3450x^{5} − 3670x^{4} − 1690x^{3} + 6335x^{2} − 5195x + 1547
T_{9} x^{15} − 470x^{13} − 305x^{12} + 71840x^{11} + 85357x^{10} − 4292700x^{9} − 3714805x^{8}

+119761820x^{7} + 25284495x^{6} − 1542190154x^{5} + 717324725x^{4} + 7178878600x^{3}

−5452953875x^{2} − 7998223215x + 4461221029
T_{8} x^{15} + 32x^{10} − 228x^{5} − 8
T_{7} x^{15} − 2x^{12} + 36x^{9} + 304x^{3} − 32
T_{6} x^{15} − 30x^{10} − 3708x^{5} − 2
T_{5} x^{15} − 4x^{14} − x^{13} + 9x^{12} + 2x^{11} − 2x^{10} − 32x^{9} + 36x^{8} + 39x^{7} − 88x^{6} + 57x^{5}
+7x^{4} − 33x^{3} + 30x^{2} − 15x + 5
T_{4} x^{15} − 8x^{12} − 124x^{9} + 480x^{6} + 672x^{3} − 32
T_{3} x^{15} + 2x^{14} + 2x^{13} − 2x^{11} − 12x^{10} + 37x^{9} − 2x^{8} + 37x^{7} + 37x^{6} − 39x^{5}
−47x^{4} − 22x^{3} − 6x^{2} + 1
T_{2} x^{15} − 10x^{12} + 15x^{11} − 24x^{10} + 95x^{9} − 90x^{8} + 90x^{7} − 125x^{6} + 27x^{5} + 90x^{4}
+120x^{3} + 195x^{2} + 90x + 33
T_{1} x^{15} − 27x^{13} − 4x^{12} + 252x^{11} + 60x^{10} − 976x^{9} − 288x^{8} + 1473x^{7} + 384x^{6}
−765x^{5} − 168x^{4} + 150x^{3} + 27x^{2} − 9x − 1

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