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Sato–Tate distributions of twists of the Fermat and the Klein quartics
 Francesc Fité^{1},
 Elisa Lorenzo García^{2} and
 Andrew V. Sutherland^{3}Email authorView ORCID ID profile
https://doi.org/10.1007/s4068701801620
© The Author(s) 2018
 Received: 25 May 2018
 Accepted: 18 September 2018
 Published: 15 October 2018
Abstract
We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is \({\overline{\mathbb Q}}\)isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato–Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.
1 Introduction
 (ST)
The sequence \(\{s(\mathfrak {p})\}_\mathfrak {p}\) is equidistributed on X with respect to the measure \(\mu \).
 (ST\({}^\prime \)):

The sequence \(\{a(\mathfrak {p})\}_\mathfrak {p}\) is equidistributed on I with respect to \(\mu _I\).
 (ST\({}^{\prime \prime }\)):

The sequences \(\{a_j(\mathfrak {p})\}_\mathfrak {p}\) are equidistributed on \(I_j\) with respect to \(\mu _{I_j}\), for \(1\le j \le g\).
Theorem 1
 (i)
There are 54 distinct Sato–Tate groups of twists of the Fermat quartic. These give rise to 54 (resp. 48) distinct joint (resp. independent) coefficient measures.
 (ii)
There are 23 distinct Sato–Tate groups of twists of the Klein quartic. These give rise to 23 (resp. 22) distinct joint (resp. independent) coefficient measures.
 (iii)
There are 60 distinct Sato–Tate groups of twists of the Fermat or the Klein quartics. These give rise to 60 (resp. 54) distinct joint (resp. independent) coefficient measures.
One motivation for our work is a desire to extend the classification of Sato–Tate groups that is known for dimensions \(g\le 2\) to dimension 3. Of the 52 Sato–Tate groups that arise for abelian surfaces (see [10, Table 10] for a list), 32 can be realized as the Sato–Tate group of the Jacobian of a twist of one of the two genus 2 curves with the largest automorphism groups, as shown in [12]; these groups were the most difficult to treat in [10] and notably include cases missing from the candidate list of trace distributions identified in [21, Table 13]. While the classification of Sato–Tate groups in dimension 3 remains open, the 60 Sato–Tate groups identified in Theorem 1 and explicitly described in Sect. 3.3 are likely to include many of the most delicate cases and represent significant progress toward this goal.
Overview of the paper. This article can be viewed as a genus 3 analog of [12], where the Sato–Tate groups of the Jacobians of twists of the curves \(y^2=x^5x\) and \(y^2=x^6+1\) were computed. However, there are two important differences in the techniques we use here; these are highlighted in the paragraphs below that outline our approach. We also note [13], where the Sato–Tate groups of the Jacobians of certain twists of the genus 3 curves \(y^2=x^7x\) and \(y^2=x^8+1\) are computed, and [2], where the Sato–Tate groups of the Jacobians of twists of the curve \(y^2=x^814x^4+1\) are determined. Like the Fermat and Klein quartics we consider here, these three curves represent extremal points in the moduli space of genus 3 curves, but they are all hyperelliptic, and their automorphism groups are smaller (of order 24, 32, 48, respectively).
As noted above, the Sato–Tate conjecture is known for abelian varieties that are \({\overline{\mathbb Q}}\)isogenous to a product of CM abelian varieties [19, Cor. 15]. It follows that we can determine the set of independent coefficient measures \(\{\mu _{I_j}\}_j\) by computing the sequences \(\{\mathrm {M }_n[a_j]\}_{j,n}\), and similarly for \(\mu _I\) and the sequences \(\{M_{n_1,\ldots ,n_g}[a]\}_{n_1,\ldots ,n_g}\). Closed formulas for these sequences are determined in Sect. 2 in the more general setting of abelian threefolds defined over a number field k that are \({\overline{\mathbb Q}}\)isogenous to the cube of an elliptic curve defined over k; see Proposition 2.2 and Corollary 2.4. This analysis closely follows the techniques developed in [12, §3].
In Sect. 3, we specialize to the case of Jacobians of twists of the Fermat and Klein quartics. In Sect. 3.2, we obtain a complete list of possibilities for \(\{\mathrm {M }_n[a_j]\}_{j,n}\): there are 48 in the Fermat case, 22 in the Klein case, and 54 when combined; see Corollary 3.12. We also compute lower bounds on the number of possibilities for \(\{\mathrm {M }_{n_1,n_2,n_3}[a]\}_{n_1,n_2,n_3}\) by computing the number of possibilities for the first several terms (up to a certain conveniently chosen bound) of this sequence. These lower bounds are 54 in the Fermat case, 23 in the Klein case, and 60 when combined; see Proposition 3.13.
The first main difference with [12] arises in Sect. 3.3, where we compute the Sato–Tate groups of the twists of the Fermat and Klein quartics using the results of [3]. Such an analysis would have been redundant in [12], since a complete classification of Sato–Tate groups of abelian surfaces was already available from [10]. We show that there are at most 54 in the Fermat case, and at most 23 in the Klein case; see Corollaries 3.23 and 3.26. Combining the implications in (1.3) together with the lower bounds of Sect. 3.2 and upper bounds of Sect. 3.3 yields Theorem 1.
The second main difference with [12] arises in Sect. 3.4, where we provide explicit equations of twists of the Fermat and Klein quartics that realize each of the possible Sato–Tate groups. Here, the computational search used in [12] is replaced by techniques developed in [22, 23] that involve the resolution of certain Galois embedding problems, and a moduli interpretation of certain twists \(X_E(7)\) of the Klein quartic as twists of the modular curve X(7), following [14]. In order to apply the latter approach, which also plays a key role in [25], we obtain a computationally effective description of the minimal field over which the automorphisms of \(X_E(7)\) are defined (see Propositions 3.34 and 3.35), a result that may have other applications.
Finally, in Sect. 3.5, we give an algorithm for the efficient computation of the Lpolynomials of twists of the Fermat and Klein quartics. This algorithm combines an average polynomialtime for computing Hasse–Witt matrices of smooth plane quartics [18] with a result specific to our setting that allows us to easily derive the full Lpolynomial at \(\mathfrak {p}\) from the Frobenius trace using the splitting behavior of \(\mathfrak {p}\) in certain extensions; see Proposition 3.38. Our theoretical results do not depend on this algorithm, but it played a crucial role in our work by allowing us to check our computations and may be of independent interest.
Notation. Throughout this paper, k denotes a number field contained in a fixed algebraic closure \(\overline{\mathbb Q}\) of \(\mathbb Q\). All the field extensions of k, we consider are algebraic and assumed to lie in \({\overline{\mathbb Q}}\). We denote by \(G_k\) the absolute Galois group \(\mathrm {Gal}({\overline{\mathbb Q}}/k)\). For an algebraic variety X defined over k and a field extension L / k, write \(X_L\) for the algebraic variety defined over L obtained from X by base change from k to L. For abelian varieties A and B defined over k, we write \(A\sim B\) if there is an isogeny from A to B that is defined over k. We use \(M^\mathrm {T}\) to denote the transpose of a matrix M. We label the isomorphism class ID\((H)=\langle n, m\rangle \) of a finite group H according to the Small Groups Library [4], in which n is the order of H and m distinguishes the isomorphism class of H from all other isomorphism classes of groups of order n.
2 Equidistribution results for cubes of CM elliptic curves
Let A be an abelian variety over k of dimension 3 such that \(A_{\overline{\mathbb Q}}\sim E_{\overline{\mathbb Q}}^3\), where E is an elliptic curve defined over k with complex multiplication (CM) by an imaginary quadratic field M. Let L / k be the minimal extension over which all the homomorphisms from \(E_{\overline{\mathbb Q}}\) to \(A_{\overline{\mathbb Q}}\) are defined. We note that \(kM\subseteq L\), and we have \({\text {Hom}}(E_{\overline{\mathbb Q}},A_{\overline{\mathbb Q}})\simeq {\text {Hom}}(E_{L} ,A_{L} )\) and \(A_{L}\, {\sim }\, E_{L} ^3\).
Definition 2.1
Let \(\theta :=\theta _{M,\sigma }(E,A)\) (resp. \(\theta _{M,\sigma }(A)\)) denote the representation afforded by the module \({\text {Hom}}(E_{L} ,A_{L} )\otimes _{M,\sigma }{\overline{\mathbb Q}}\) (resp. \({\text {End}}(A_{L} )\otimes _{M,\sigma }{\overline{\mathbb Q}}\)), and let us similarly define \(\overline{\theta }:=\theta _{M,\overline{\sigma }}(E,A)\) and \(\theta _{M,\overline{\sigma }}(A)\). Let \(\theta _\mathbb Q:=\theta _\mathbb Q(E,A)\) (resp. \(\theta _\mathbb Q(A)\)) denote the representation afforded by the \(\mathbb Q[\mathrm {Gal}({L} /k)]\)module \({\text {Hom}}(E_{L},A_{L})\otimes \mathbb Q\) (resp. \({\text {End}}(A_L)\otimes \mathbb Q\)).
Proposition 2.2
Proof
Remark 2.3
In the statement of the proposition, we included the hypothesis that \(a_3(\theta )\) is rational, which is satisfied for Jacobians of twists of the Fermat and Klein curves (see Section 3.1), because this makes the formulas considerably simpler. This hypothesis is not strictly necessary; one can similarly derive a more general formula without it.
Corollary 2.4
Proof
 (a)
\(\mathfrak {p}\) has absolute residue degree 1, that is, \(N(\mathfrak {p})=p\) is prime.
 (b)
\(\mathfrak {p}\) is of good reduction for both A and E.
 (c)
\(\mathbb Q(\sqrt{p})\cap \mathbb Q(\zeta _{4N})=\mathbb Q\), where \(\zeta _{4N}\) is a primitive 4Nth root of unity.
 (i)
The polynomial \(\overline{L}_\mathfrak {p}(A,T)\) divides the Rankin–Selberg polynomial \(\overline{L}_\mathfrak {p}(E,\theta _\mathbb Q(E,A),T)\).
 (ii)
The roots of \(D(T,\tau )\) are quotients of roots of \(\overline{L}_\mathfrak {p}(A,T)\) and \(\overline{L}_\mathfrak {p}(E,T)\).
If \({\text {ord}}(\tau )=4\), then all the roots of \(\overline{L}_\mathfrak {p}(E,\theta _\mathbb Q(E,A),T)\) are of order dividing 4. Thus so are the roots of \(P_a(T)\), which leaves the two possibilities \(a=1\) or \(a=3\). But (ii) implies that the latter is not possible: if \(a=3\), then the roots of \(D(T,\tau )\) would all be of order dividing 2 and this contradicts the fact that \(\tau \) has order 4. Thus \(a=1\).
If \({\text {ord}}(\tau )=6\), then by (ii) we have \(a\not =1,1,3\) (otherwise the order of \(\tau \) would not be divisible by 3). If \(a=2\), then again by (ii) the polynomial \(D(T,\tau )\) would have a root of order at least 12, which is impossible for \({\text {ord}}(\tau )=6\). Thus \(a=0\).
If \({\text {ord}}(\tau )=8\), then \(\overline{L}_\mathfrak {p}(E,\theta _\mathbb Q(E,A),T)\) has at least 8 roots of order 8 and thus \(P_a(T)\) has at least a root of order 8. Thus \(a=1\).
If \({\text {ord}}(\tau )=12\), then by (ii) we have \(a\not =1,1,3\) (otherwise the order of \(\tau \) would not be divisible by 3). If \(a=0\), then again by (ii) the polynomial \(D(T,\tau )\) would only have roots of orders 1, 2, 3 or 6, which is incompatible for \({\text {ord}}(\tau )=12\). Thus, \(a=2\).\(\square \)
3 The Fermat and Klein quartics
Proposition 3.1
For \(d=1\) or 7, the Jacobian of \(C^0_d\) is \(\mathbb Q\)isogenous to the cube of \(E_d^0\).
Proof
Remark 3.2
To simplify notation, for the remainder of this article d is either 1 or 7, and we write \(C^0\) for \(C^0_ d\), \(E^0\) for \(E^0_d\), M for \(\mathbb Q(\sqrt{d})\), and s, t, u for \(s_d\), \(t_d\), \(u_d\). When d is not specified, it means we are considering both values of d simultaneously.
3.1 Twists
Let C be a ktwist^{2} of \(C^0\), a curve defined over k that is \({\overline{\mathbb Q}}\)isomorphic to \(C^0\). The set of ktwists of \(C^0\), up to kisomorphism, is in onetoone correspondence with \(H^1(G_k,{\text {Aut}}(C^0_M))\). Given an isomorphism \(\phi :C_{\overline{\mathbb Q}}\overset{\sim }{\rightarrow } C^0_{\overline{\mathbb Q}}\), the 1cocycle defined by \(\xi (\sigma ):=\phi ({}^\sigma \phi )^{1}\), for \(\sigma \in G_k\), is a representative of the cohomology class corresponding to C.
Let K / k (resp. L / k) denote the minimal extension over which all endomorphisms of \({\text {Jac}}(C)_{\overline{\mathbb Q}}\) (resp. all homomorphisms from \({\text {Jac}}(C)_{\overline{\mathbb Q}}\) to \(E^0_{\overline{\mathbb Q}}\)) are defined. Let \(\tilde{K}/k\) (resp. \(\tilde{L}/k\)) denote the minimal extension over which all automorphisms of \(C_{\overline{\mathbb Q}}\) (resp. all isomorphisms from \(C_{\overline{\mathbb Q}}\) to \(C^0_{\overline{\mathbb Q}}\)) are defined.
Lemma 3.3
Proof
The inclusion \(M\subseteq \tilde{K}\) follows from the fact that \({\text {Tr}}(A_u)\in M\setminus \mathbb Q\), where \(A_u\) is as in (3.12) and (3.13). From the proof of Proposition 3.1, we know that \({\text {Jac}}(C)_{\tilde{K}}\sim E^3\), where E is an elliptic curve defined over \(\tilde{K}\) with CM by M. This implies \(K=\tilde{K} M\) and \(L=\tilde{L}M\), as in the proof of [12, Lem. 4.2].\(\square \)
We now associate with \(C^0\) a finite group \(G^0\) that will play a key role in the rest of the article.
Definition 3.4
Lemma 3.5
Proof
Remark 3.6
Observe that since \(\det (\theta _{E^0,C^0})\) is a rational character of \({\text {Aut}}(C^0_M)\), by (3.5) so is \(a_3(\theta )=\det \theta _{M,\sigma }(E^0,{\text {Jac}}(C))\). Thus, Corollary 2.4 can be used to compute the moments of \(a_i({\text {Jac}}(C))\) for \(i=1,2,3\).
Proposition 3.7
The fields K and L coincide.^{3}
Proof
Note that L / K is the minimal extension over which an isomorphism between \(E^0_{\overline{\mathbb Q}}\) and \(E_{\overline{\mathbb Q}}\) is defined. It follows that \(L=K(\gamma ^{\nicefrac {1}{n}})\) for some \(\gamma \in K\), with \(n=4\) for \(d=1\) and \(n=2\) for \(d=7\); see [28, Prop. X.5.4]. In either case, \(\mathrm {Gal}(L/K)\) is cyclic of order dividing 4 (note that \(\mathbb Q(\zeta _n)\subseteq K\)). Suppose that \(L\not =K\), let \(\omega \) denote the element in \(\mathrm {Gal}(L/K)\) of order 2, and write \(K^0=L^{\langle \omega \rangle }\). Fix an isomorphism \(\psi _1:E^0_L\rightarrow E_L\) and an isogeny \(\psi _2:(E_{K^0})^3 \rightarrow {\text {Jac}}(C)_{K^0}\). For \(i=1,2,3\), let \(\iota _i:E_{K^0}\rightarrow (E_{K^0})^3\) denote the natural injection to the ith factor. Then, \(\{\psi _2\circ \iota _i\circ \psi _1\}_{i=1,2,3}\) constitute a basis of the \({\overline{\mathbb Q}}[\mathrm {Gal}(L/M)]\)module \({\text {Hom}}(E^0_L,{\text {Jac}}(C)_L)\otimes _{M,\sigma }{\overline{\mathbb Q}}\). Since \({}^{\omega }\psi _1=\psi _1\), \({}^{\omega }\psi _2=\psi _2\), and \({}^{\omega }\iota _i=\iota _i\), we have \({{\mathrm{Trace}}}\theta _{M,\sigma }(E^0,{\text {Jac}}(C))(\omega )=3\). But this contradicts (3.5), because there is no \(\alpha \) in \({\text {Aut}}(C^0_M)\) for which \({{\mathrm{Trace}}}\theta _{E^0,C^0}(\alpha )=3\).\(\square \)
Remark 3.8
By Proposition 3.7, and the identities (2.3) and (3.5), the independent and joint coefficient measures of \({\text {Jac}}(C)\) depend only on the conjugacy class of \(\lambda _\phi (\mathrm {Gal}(K/k))\) in \(G_{C^0}\). In Proposition 3.22, we will see that this also applies to the Sato–Tate group of \({\text {Jac}}(C)\). For this reason, henceforth, subgroups \(H\subseteq G_{C^0}\) will be considered only up to conjugacy.
Definition 3.9
Let \(G_0:={\text {Aut}}(C_M^0)\times \langle 1\rangle \subseteq G_{C^0}\), and for subgroups \(H\subseteq G_{C^0}\), let \(H_0:=H \cap \mathrm {G}_0\). We may view \(H_0\) as a subgroup of \({\text {Aut}}(C_M^0)\simeq G_0\) whenever it is convenient to do so.
 (c\(_1\)):

\(H\subseteq G_0\), in which case \([H:H_0]=1\);
 (c\(_2\)):

\(H\not \subseteq G_0\), in which case \([H:H_0]=2\).
Remark 3.10
 (i)
In Sect. 3.4, we will show that for each subgroup \(H\subseteq G_{C^0}\), there is a twist C of \(C^0\) such that \(H=\lambda _\phi (\mathrm {Gal}(K/k))\). From the definition of \(\lambda _\phi \), we must then have \(H_0=\lambda _\phi (\mathrm {Gal}(K/kM))\). The case (c\(_1\)) corresponds to \(kM = k\), and the case (c\(_2\)) corresponds to \(k\not = kM\).
 (ii)
There are 83 subgroups \(H \subseteq G_{C^0_1}\) up to conjugacy, of which 24 correspond to case (c\(_1\)) and 59 correspond to case (c\(_2\)). In Table 4 we list the subgroups in case (c\(_2\)). For any subgroup H in case (c\(_1\)), there exists a subgroup \(H'\) in case (c\(_2\)) such that \(H'_0=H\); thus the subgroups in case (c\(_1\)) can be recovered from Table 4 by looking at the column for \(H_0\).
 (iii)
There are 23 subgroups \(H \subseteq G_{C^0_7}\) up to conjugacy, of which 12 correspond to case (c\(_1\)) and 11 correspond to case (c\(_2\)). For all but 3 exceptional subgroups H in case (c\(_1\)), there exists a subgroup \(H'\) in case (c\(_2\)) such that \(H'_0=H\). In Table 5, we list the subgroups in case (c\(_2\)) as well as the 3 exceptional subgroups, which appear in rows #3, #8, and #12 of Table 5. As in (ii), the nonexceptional subgroups in case (c\(_1\)) can be recovered from Table 5 by looking at the column for \(H_0\), which for the exceptional groups is equal to H.
 (iv)The subgroups \(H\subseteq G_{C^0}\) in Tables 4 and 5 are presented as follows. First, generators of \(H_0\subseteq G_0\simeq {\text {Aut}}(C^0_M)\) are given in terms of the generators s, t, u for \({\text {Aut}}(C^0_M)\) listed in (3.2) and (3.3). For the 3 exceptional subgroups of Table 5 we necessarily have \(H=H_0\), and for the others, H is identified by listing an element \(h\in {\text {Aut}}(C^0_M)\) such thatwhere \(\tau \) is the generator of \(\mathrm {Gal}(M/\mathbb Q)\).$$\begin{aligned} H=H_0\cup H_0\cdot (h,\tau ) \subseteq G_{C^0}, \end{aligned}$$(3.6)
3.2 Moment sequences
Using Lemma 3.5 and (3.5), we can apply Corollary 2.4 to compute the moments \(\mathrm {M }_n[a_j(C)]\) for any n, and as explained in Sect. 3.2.2, it is easy to compute \(M_{n_1,n_2,n_3}[a(C)]\) for any particular values of \(n_1,n_2,n_3\). Magma scripts [6] to perform these computations are available at [11], which we note depend only on the pairs \((H,H_0)\) (or just \(H_0\) when \(k=kM\)) listed in Tables 4 and 5, and are otherwise independent of the choice of C.
3.2.1 Independent coefficient moment sequences
We now show that for any twist of the Fermat or Klein quartic, each of the independent coefficient moment sequences (and hence the corresponding measures) is determined by the first several moments.
Proposition 3.11
Proof
For the sake of brevity, we assume \(k\not =kM\) (the case \(k=kM\) is analogous and easier). It follows from Corollary 2.4 that, for \(i=1,2,3\), the sequence \(\{\mathrm {M }_{n}[a_i(C)]\}_{n\ge 0}\) is determined by \(a_1(C)\), \(a_2(C)\), \(a_3(C) a_2(C)\bar{a}_1(C)\), and \(\bar{o}(2)\), \(\bar{o}(4)\), \(\bar{o}(8)\) (note that \(G_{C_1^0}\) and \(G_{C_7^0}\) contain no elements of order 12, so we ignore the \(\bar{o}(12)\) term in the formula for \(\mathrm {M }_n[a_2(C)]\)). We consider the Fermat and Klein cases separately.
For the Fermat case, with the notation for conjugacy classes as in Table 3a, let \(x_1\) (resp. \(x_2\), \(x_3\), \(x_4\), \(x_5\)) denote the proportion of elements in \(\mathrm {Gal}(L/k)\) lying in the conjugacy class 1a (resp. \(2a \cup 2b \cup 4c \cup 4d\), 3a, \(4a \cup 4b\), \(8a\cup 8b\)); note that by Lemma 3.5, we are interested in the representation with character \(\chi _8\) listed in Table 3a, which motivates this partitioning of conjugacy classes.
For \(\mathrm {M }_n[a_2(C)]\), one similarly obtains an invertible linear system in \(x_1\), \(x_2+x_5\), \(x_4\), \(y_1\), \(y_2\), \(y_3\) of dimension 6, and it follows that the moments \(\mathrm {M }_n[a_2(C)]\) for \(n\le 6\) determine all the \(\mathrm {M }_{n}[a_2(C)]\).
For \(\mathrm {M }_{2n}[a_3(C)]\), one obtains an invertible linear system in \(x_1\), \(x_2\), \(x_3\), \(x_4\), \(x_5\) of dimension 5, and it follows that the moments \(\mathrm {M }_{2n}[a_2(C)]\) for \(n\le 5\) determine all the \(\mathrm {M }_{2n}[a_3(C)]\).
In the Klein case one proceeds analogously. With the notation of Table 3b, let \(x_1\) (resp. \(x_2\), \(x_3\), \(x_4\)) denote the proportion of elements in \(\mathrm {Gal}(L/k)\) lying in the conjugacy class 1a (resp. \(2a \cup 4a\), 3a, \(7a\cup 7b\)), and let \(y_1\), \(y_2\), \(y_3\) be as in the Fermat case. Now \(x_1\), \(x_2\), \(x_4\) determine \(\mathrm {M }_n[a_1(C)]\); \(x_1\), \(x_2\), \(x_4\) and \(y_1\), \(y_2\), \(y_3\) determine \(\mathrm {M }_{n}[a_2(C)]\); and \(x_1\), \(x_2\), \(x_3\), \(x_4\) determine \(\mathrm {M }_{n}[a_3(C)]\). These proportions are, as before, determined by the first several moments (never more than are needed in the Fermat case), and the result follows. We spare the reader the lengthy details.\(\square \)
With Proposition 3.11 in hand we can completely determine the moment sequences \(\mathrm {M }_n[a_i(C)]\) that arise among ktwists C of \(C^0\) by computing the moments \(\mathrm {M }_n[a_i(C)]\) for \(n\le N_i\) for the 59 pairs \((H,H_0)\) listed in Table 4 in the case \(C^0=C^0_1\), and for the 14 pairs \((H,H_0)\) listed in Table 5 (as described in Remark 3.10). Note that each pair \((H,H_0)\) with \(H\ne H_0\) gives rise to two moments sequences \(\mathrm {M }_n[a_i(C)]\) for each i, one with \(k\ne kM\) and one with \(k=kM\).
After doing so, one finds that in fact Proposition 3.11 remains true with \(N_2=4\) and \(N_3=4\). The value of \(N_1\) cannot be improved, but one also finds that the sequences \(\mathrm {M }_n[a_2(C)]\) and \(\mathrm {M }_n[a_3(C)]\) together determine the sequence \(\mathrm {M }_n[a_1(C)]\), and in fact just two wellchosen moments suffice.
Corollary 3.12
There are 48 (resp. 22) independent coefficient measures among twists of \(C^0_1\) (resp. \(C^0_7\)). In total, there are 54 independent coefficient measures among twists C of either \(C_1^0\) or \(C_7^0\), each of which is uniquely distinguished by the moments \(\mathrm {M }_3[a_2(C)]\) and \(\mathrm {M }_4[a_3(C)]\).
The moments \(\mathrm {M }_3[a_2(C)]\) and \(\mathrm {M }_4[a_3(C)]\) correspond to the joint moments \(\mathrm {M }_{0,3,0}[a(C)]\) and \(\mathrm {M }_{0,0,4}[a(C)]\) whose values are listed in Table 6 for each of the 60 distinct joint coefficient moment measures obtained in the next section (this includes all the independent coefficient measures).
3.2.2 Joint coefficient moments
Proposition 3.13
There are at least 54 (resp. 23) joint coefficient measures (and hence Sato–Tate groups) of twists of the Fermat (resp. Klein) quartic, and at least 60 in total. These 60 joint coefficient measures are listed in Table 6, in which each is uniquely distinguished by the three moments \(\mathrm {M }_{1,0,1}[a(C)]\), \(\mathrm {M }_{0,3,0}[a(C)]\), and \(\mathrm {M }_{2,0,2}[a(C)]\).
Computing \(\mathrm {M }_\mathrm{joint}^{\le N}(C)\) with \(N=5,6,7,8,\) does not increase any of the lower bounds in Proposition 3.13, leading one to believe they are tight. We will prove this in the next section, but an affirmative answer to the following question would make it easy to directly verify such a claim.
Question 3.14
Lacking an answer to Question, 3.14, in order to determine the exact number of distinct joint coefficient measures, we take a different approach. In the next section, we will classify the possible Sato–Tate groups of twists of the Fermat and Klein quartics. This classification yields an upper bound that coincides with the lower bound of Proposition 3.13.
Table 6 also lists \(z_1=[z_{1,0}]\), \(z_2=[z_{2,1},z_{2,0},z_{2,1},z_{2,2},z_{2,3}]\), and \(z_3=[z_{3,0}]\), where \(z_{i,j}\) denotes the density of the set of primes \(\mathfrak {p}\) for which \(a_i(C)(\mathfrak {p})=j\). For these, we record the following lemma.
Lemma 3.15
Proof
The formula for \(z_1\) is immediate from (2.3) and the study of the polynomial (2.6) in the proof of Corollary 2.4, together from the fact that \(\tau \in \mathrm {Gal}(L/k)\) satisfies \(a_1(\theta )(\tau )=0\) if and only if \(\tau \) has order 3 (as can be seen from Table 3).
The formula for \(z_2\) follows from a similar reasoning, once one observes that again \(a_2(\theta )(\tau )=0\) if and only if \(\tau \) has order 3, and the discussion of the end of Corollary 2.4. Note also that, as \(G_{C^0}\) contains no elements of order 12, we have \(\bar{o}(12)=0\).
For \(z_3\), it suffices to note that \(a_3(\tau )\) does not vanish.\(\square \)
3.3 Sato–Tate groups
Remark 3.16
Remark 3.17
From now on, we fix the following notation: denote by \(A_3\) the 3diagonal embedding of a subset A of \(\mathrm {GL}_2\) in \(\mathrm {GL}_6\). Throughout this section, we consider the general symplectic group \(\mathrm {GSp}_6/\mathbb Q\), the symplectic group \(\mathrm {Sp}_6/\mathbb Q\), and the unitary symplectic group \(\mathrm {USp}(6)\) with respect to the symplectic form given by the 3diagonal embedding \((J_2)_3\).
Lemma 3.18
Proof
Remark 3.19
Note that since \({\text {Aut}}(C^0_M)\) has finite order, the image of \(\iota \) is contained in \(\mathrm {USp}_6(\mathbb Q)\).
Remark 3.20
It is easy to check that the matrices \(\iota (s_1)\), \(\iota (t_1)\), \(\iota (u_1)\) (resp. \(\iota (s_7)\), \(\iota (t_7)\), \(\iota (u_7)\)) are symplectic with respect to \(J:=(J_2)_3\).
Theorem 3.21
 (i)The monomorphism of Lemma 3.18 extends to a monomorphismby defining$$\begin{aligned} \iota :G_{C^0}\hookrightarrow \mathrm {USp}(6)/\langle 1\rangle \, \end{aligned}$$where \(\tau \) denotes the nontrivial element of \(\mathrm {Gal}(M/\mathbb Q)\).$$\begin{aligned} \iota ((1,\tau )):={\left\{ \begin{array}{ll} \frac{1}{\sqrt{2}} \begin{pmatrix} i &{} i\\ i &{} i \end{pmatrix}_3 &{} \text { if } C^0=C^0_1,\\ \begin{pmatrix} i &{} i\\ 0 &{} i \end{pmatrix}_3&\text { if } C^0=C^0_7, \end{array}\right. } \end{aligned}$$
 (ii)Let \(\phi :C_{\overline{\mathbb Q}}\rightarrow C^ 0_{\overline{\mathbb Q}}\) denote a ktwist of \(C^0\) and write \(H:=\lambda _\phi (\mathrm {Gal}(K/k))\subseteq G_{C^0}\). The Sato–Tate group of \({\text {Jac}}(C)\) is given bywhere$$\begin{aligned} \mathrm {ST}(C)=\mathrm {ST}(E^0,1)_3\cdot \iota (H), \end{aligned}$$if \(C^0=C^0_1\), and$$\begin{aligned} \mathrm {ST}(E^0,1)=&\left\{ \begin{pmatrix} \cos (2\pi r) &{} \sin (2\pi r)\\ \sin (2\pi r) &{} \cos (2\pi r) \end{pmatrix}\,\, r\in [0,1] \right\} \, \end{aligned}$$if \(C^0=C^0_7\).$$\begin{aligned} \mathrm {ST}(E^0,1)= \left\{ \begin{pmatrix} \cos (2\pi r)\frac{1}{\sqrt{7}}\sin (2\pi r) &{} \frac{4}{\sqrt{7}}\sin (2\pi r) \\ \frac{2}{\sqrt{7}} \sin (2\pi r) &{} \cos (2\pi r)+\frac{1}{\sqrt{7}}\sin (2\pi r) \end{pmatrix}\,\,r\in [0,1] \right\} \end{aligned}$$
Proof
\(\square \)
The previous theorem describes the Sato–Tate group of a twist C of \(C^0\). Now suppose that C and \(C'\) are both twists of \(C^0\). The next proposition gives an effective criterion to determine when \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C')\) coincide. Let \(H\subseteq G_{C^0}\) (resp. \(H'\)) be attached to C (resp. \(C'\)) as in Remark 3.10.
Proposition 3.22
If H and \(H'\) are conjugate in \(G_{C^0}\), then \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C')\) coincide.
Proof
Corollary 3.23
There are at most 23 Sato–Tate groups of twists of the Klein quartic \(C_7^0\).
Proof
There are 23 subgroups of \(G_{C^0_7}\), up to conjugacy.\(\square \)
In the Fermat case, \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C')\) may coincide when H and \(H'\) are not conjugate in \(G_{C^0}\). We thus require a sharper criterion.
Definition 3.24
Proposition 3.25
Let C and \(C'\) be twists of \(C^0\) and \(C^{0\prime }\). If H and \(H'\) are equivalent, then \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C')\) coincide.
Proof
\(\square \)
Corollary 3.26
 (i)
There are at most 54 distinct Sato–Tate groups of twists of the Fermat quartic.
 (ii)
There are at most 60 distinct Sato–Tate groups of twists of the Fermat and Klein quartics.
Proof
Determining whether two subgroups H and \(H'\) are equivalent is a finite problem. Using the computer algebra program [6], one can determine a set of representatives for equivalence classes of subgroups H that turn out to have size 54 in case (i), and of size 60 in case (ii). For the benefit of the reader, here we give a direct proof of (ii), assuming (i).
The 6 Sato–Tate groups of a twist of the Klein quartic that do not show up as the Sato–Tate group of a twist of the Fermat quartic are precisely those ruled out by the fact that \(H_0\) contains an element of order 7 (those in rows #10, #13, #14 of Table 5), since 7 does not divide \(\#G_{C_1^0}\).
 (1)
There exists an isomorphism \(\Psi :H\rightarrow H'\) such that \(\Psi (H_0)=H_0'\);
 (2)
\({\text {Tr}}(j(h))=1\) for every \(h\in H_0'\) such that \({\text {ord}}(h)=4\).
Combining the lower and upper bounds proved in this section yields our main theorem, which we restate for convenience.
Theorem 1
 (i)
There are 54 distinct Sato–Tate groups of twists of the Fermat quartic. These give rise to 54 (resp. 48) distinct joint (resp. independent) coefficient measures.
 (ii)
There are 23 distinct Sato–Tate groups of twists of the Klein quartic. These give rise to 23 (resp. 22) distinct joint (resp. independent) coefficient measures.
 (iii)
There are 60 distinct Sato–Tate groups of twists of the Fermat or the Klein quartics. These give rise to 60 (resp. 54) distinct joint (resp. independent) coefficient measures.
Corollary 3.27
If C and \(C'\) are twists of \(C^0\) corresponding to H and \(H'\), respectively, then \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C')\) coincide if and only if H and \(H'\) are equivalent.
Remark 3.28
One could have obtained the lower bounds of Proposition 3.13 by computing the joint coefficient measures \(\mu _I\) of the Sato–Tate groups explicitly described in Theorem 3.21. This is a lengthy but feasible task that we will not inflict on the reader. We note that this procedure also allows for casebycase verifications of the equalities \(\mathrm {M }_{n_1,n_2,n_3}[\mu _I]=\mathrm {M }_{n_1,n_2,n_3}[a]\) and thus of the Sato–Tate conjecture in the cases considered.
Remark 3.29
In virtue of the above remark, one might ask about conditions on twists C and \(C'\) corresponding to distinct but equivalent groups H and \(H'\) that ensure their Jacobians have the same Sato–Tate group. One such condition is that \({\text {Jac}}(C)\) and \({\text {Jac}}(C')\) are isogenous (recall that the Sato–Tate group of an abelian variety is an isogeny invariant). The next proposition shows that, under the additional hypothesis that K and \(K'\) coincide, the previous statement admits a converse.
Proposition 3.30
Let C and \(C'\) be ktwists of \(C^0\). Suppose that the corresponding subgroups H and \(H'\) of \(G_{C^0}\) are equivalent and that the corresponding fields K and \(K'\) coincide. Then, \({\text {Jac}}(C)\) and \({\text {Jac}}(C')\) are isogenous.
Proof
Remark 3.31
Let H and \(H'\) be any two equivalent pairs attached to twists C and \(C'\) of the same curve \(C^0\). As one can read from Tables 4 and 5 (and as we will see in the next section), one can choose C and \(C'\) so that K and \(K'\) coincide. It follows from Proposition 3.30 that on Table 4 (resp. Table 5) two curves C and \(C'\) satisfy \(\mathrm {ST}(C)=\mathrm {ST}(C')\) if and only if \({\text {Jac}}(C)\sim {\text {Jac}}(C')\).
Proposition 3.32
The curves \(C_5\) and \(C_8\) are not isomorphic (over \(\mathbb Q\)), but their reductions \(\tilde{C}_5\) and \(\tilde{C}_8\) modulo p are isomorphic (over \(\mathbb F_p\)) for every prime \(p>3\).
Proof
3.4 Curve equations
In this section, we construct explicit twists of the Fermat and the Klein quartics realizing each of the subgroups \(H\subseteq G_{C^0}\) described in Remark 3.10. Recall that each H has an associated subgroup \(H_0:=H\cap G_0\) of index at most 2 (see Definition 3.9), and there exists a twist corresponding to H with \(k=kM\) if and only if \(H=H_0\), where, as always, M denotes the CM field of \(E^0\) (the elliptic curve for which \({\text {Jac}}(C^0)\sim (E^0)^3\)).
Equations for these twists are listed in Tables 4 and 5 in Sect. 4. As explained in Remark 3.10, in the Fermat case every subgroup \(H\subseteq G_{C^0_1}\) with \([H:H_0]=1\) (case (c\(_1\)) of Definition 3.9) arises as \(H_0'\) for some subgroup \(H'\subseteq G_{C^0_1}\) for which \([H':H_0']=2\) (case (c\(_2\)) of Definition 3.9), and a twist corresponding to H can thus be obtained as the base change to kM of a twist corresponding to \(H'\). We thus only list twists for the 59 subgroups H in case (c\(_2\)), since base changes of these twists to kM then address the 24 subgroups H in case (c\(_1\)). In the Klein case, we list twists for the 11 subgroups H in case (c\(_2\)) and also the 3 exceptional subgroups H in case (c\(_1\)) that cannot be obtained as base changes of twists corresponding to subgroups in case (c\(_2\)); see Remark 3.10.
Our twists are all defined over base fields k of minimal possible degree, never exceeding 2. For the 3 exceptional subgroups H in the Klein case noted above, we must have \(k=kM\), and we use \(k=M=\mathbb Q(\sqrt{7})\). In all but 5 of the remaining cases with \([H:H_0]=2\), we use \(k=\mathbb Q\). These 5 exceptions are all explained by Lemma 3.33 below (the second of the 4 pairs listed in Lemma 3.33 arises in both the Fermat and Klein cases, leading to 5 exceptions in total). In each of these 5 exceptions with \([H:H_0]=2\), the subgroup \(H_0\) also arises as \(H_0'\) for some subgroup \(H'\subseteq G_{C_1^0}\) with \([H':H'_0]=2\) that is realized by a twist with \(k=\mathbb Q\), allowing \(H_0\) to be realized over a quadratic field as the base change to M of a twist defined over \(\mathbb Q\).
Lemma 3.33
Proof
If k is totally real, then complex conjugation acts trivially on k but not on kM, giving an involution in \(H=\mathrm {Gal}(K/k)\) with nontrivial image in \(H/H_0=\mathrm {Gal}(K/k)/\mathrm {Gal}(K/kM)\). For the four pairs \((H,H_0)\) listed in the lemma, no such involution exists.\(\square \)
In addition to listing equations and a field of definition k for a twist C associated with each subgroup H, in Tables 4 and 5 we also list the minimal field K over which all the endomorphisms of \({\text {Jac}}(C)\) are defined, and we identify the conjugacy class of \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C_{kM})\), which depends only on H, not the particular choice of C. As noted in Remark 3.31, we have chosen twists C so that twists with the same Sato–Tate group have the same fields K and thus have isogenous Jacobians, by Proposition 3.30 (thereby demonstrating that the hypotheses of the proposition can always be satisfied).
3.4.1 Constructing the Fermat twists
The twists of the Fermat curve over any number field are parametrized in [23], and we specialize the parameters in Theorems 4.1, 4.2, 4.5 of [23] to obtain the desired examples. In every case, we are able to obtain equations with coefficients in \(\mathbb {Q}\), but as explained above, we cannot always take \(k=\mathbb Q\); the exceptions can be found in rows #4, #13, #27, #33 of Table 4. The parameterizations in [23] also allow us to determine the field L over which all the isomorphisms to \((C_7^0)_{\overline{\mathbb Q}}\) are defined, which by Lemma 3.3 and Proposition 3.7, this is the same as the field K over which all the endomorphisms of \({\text {Jac}}(C_7^0)_{\overline{\mathbb Q}}\) are defined.
3.4.2 Constructing the Klein twists
Twists of the Klein curve over arbitrary number fields are parametrized in Theorems 6.1 and 6.8 of [23], following the method described in [22], which is based on the resolution of certain Galois embedding problems. However, in the most difficult case, in which \(H=G_{C_7^0}\) has order 336, this Galois embedding problem is computationally difficult to resolve explicitly. This led us to pursue an alternative approach that exploits the moduli interpretation of twists of the Klein curve as twists of the modular curve X(7). As described in [14, §3] and [25, §4], associated with each elliptic curve \(E/\mathbb Q\) is a twist \(X_E(7)\) of the Klein quartic defined over \(\mathbb Q\) that parameterizes isomorphism classes of 7torsion Galois modules isomorphic to E[7], as we recall below. With this approach, we can easily treat the case \(H=G_{C_7^0}\), and we often obtain twists with nicer equations. In one case, we also obtain a better field of definition k, allowing us to achieve the minimal possible degree \([k:\mathbb Q]\) in every case.
However, as noted in [25, §4.5], not every twist of the Klein curve can be written as \(X_E(7)\) for some elliptic curve \(E/\mathbb Q\), and there are several subgroups \(H\subseteq G_{C_7^0}\) for which the parameterizations in [23] yield a twist of the Klein quartic defined over \(\mathbb Q\), but no twists of the form \(X_E(7)\) corresponding to H exist. We are thus forced to use a combination of the two approaches. For twists of the form \(X_E(7)\), we need to determine the minimal field over which the endomorphisms of \({\text {Jac}}(X_E(7))\) are defined; this is addressed by Propositions 3.34 and 3.35.
Proposition 3.34
 (i)
The minimal extension over which all endomorphisms of \({\text {Jac}}(X_E(7))\) are defined;
 (ii)
The minimal extension over which all automorphisms of \(X_E(7)\) are defined;
 (iii)
The field \(\overline{\mathbb Q}^{\ker \overline{\varrho }_{E,7}}\);
 (iv)
The minimal extension over which all 7isogenies of E are defined.
Proof
The equivalence of (i) and (ii) follows from 3.3, since \(X_E(7)\) is a twist of \(C^0_7\).
To explicitly determine the field over which all the 7isogenies of E are defined, we rely on Proposition 3.35 below, in which \(\Phi _7(X,Y)\in \mathbb Z[X,Y]\) denotes the classical modular polynomial; the equation for \(\Phi _7(X,Y)\) is too large to print here, but it is available in [6] and can be found in the tables of modular polynomials listed in [33] that were computed via [5]; it is a symmetric in X and Y, and has degree 8 in both variables.
The equation \(\Phi _7(X,Y)=0\) is a canonical (singular) model for the modular curve \(Y_0(7)\) that parameterizes 7isogenies. If \(E_1\) and \(E_2\) are elliptic curves related by a 7isogeny then \(\Phi _7(j(E_1),j(E_2))=0\), and if \(j_1,j_2\in F\) satisfy \(\Phi _7(j_1,j_2)=0\), then there exist elliptic curves \(E_1\) and \(E_2\) with \(j(E_1)=j_1\) and \(j(E_2)=j_2\) that are related by a 7isogeny. However, this 7isogeny need not be defined over F! The following proposition characterizes the relationship between F and the minimal field K over which all the 7isogenies of E are defined.
Proposition 3.35
Let E be an elliptic curve over a number field k with \(j(E)\ne 0,1728\). Let F be the splitting field of \(\Phi _7(j(E),Y)\in k[X]\), and let K be the minimal field over which all the 7isogenies of E are defined. The fields K and F coincide.
Proof
If E does not have complex multiplication, then \(m(r)=1\) for all \(r\in S\), since otherwise over \({\overline{\mathbb Q}}\) we would have two 7isogenies \(\alpha ,\beta :E_{{\overline{\mathbb Q}}}\rightarrow E'\) with distinct kernels, and then \((\hat{\alpha }\circ \beta )\in {\text {End}}(E_{{\overline{\mathbb Q}}})\) is an endomorphism of degree 49 which is not \(\pm 7\), contradicting \({\text {End}}(E_{{\overline{\mathbb Q}}})\simeq \mathbb Z\). It follows that \(\varphi \) and therefore \(\phi \) is injective, so \(\ker \bar{\varrho }_{E,7}=\ker (\phi \circ \bar{\varrho }_{E,7})\) and \(K=F\).
If E does have complex multiplication, then \({\text {End}}(E_{{\overline{\mathbb Q}}})\) is isomorphic to an order in an imaginary quadratic field M. We now consider the isogeny graph whose vertices are jinvariants of elliptic curves \(E'/FM\) with edges \((j_1,j_2)\) present with multiplicity equal to the multiplicity of \(j_2\) as a root of \(\Phi _7(j_1,Y)\). Since \(j(E)\ne 0,1728\), the component of \(j(E_{FM})\) in this graph is an isogeny volcano, as defined in [31]. In particular, there are at least 6 distinct edges \((j(E_{FM}),j_2)\) (edges with multiplicity greater than 1 can occur only at the surface of an isogeny volcano and the subgraph on the surface is regular of degree at most 2). It follows that \(m(r)>1\) for at most one \(r\in S\).
The image of \(\bar{\varrho }_{E,7}\) is isomorphic to a subgroup of \(\mathrm {PGL}_2(\mathbb F_7)\), and this implies that if \(\bar{\varrho }_{E,7}(\sigma )\) fixes more than 2 elements of \(\mathbb {P}(E[7])\) then \(\sigma \in \ker \bar{\varrho }_{E,7}\). This necessarily applies whenever \(\bar{\varrho }_{E,7}(\sigma )\) lies in \(\ker \phi \), since it must fix 6 elements, thus \(\ker \bar{\varrho }_{E,7}=\ker (\phi \circ \bar{\varrho }_{E,7})\) and \(K=F\).\(\square \)
Corollary 3.36
Let E be an elliptic curve over a number field k with \(j(E)\ne 0,1728\). The minimal field K over which all the 7isogenies of E are defined depends only on j(E).
Remark 3.37
The first part of the proof of Proposition 3.35 also applies when j(E) is 0 or 1728, thus we always have \(F\subseteq K\). Equality does not hold in general, but a direct computation finds that \([K\!:\!F]\) must divide 6 (resp. 2) when \(j(E)=0\) (resp. 1728), and this occurs when \(k=\mathbb Q\).
We now fix E as the elliptic curve \(y^{2}=x^{3}+6x+7\) with Cremona label 144b1. Note that \(\varrho _{E,7}\) is surjective; this can be seen in the entry for this curve in the Lfunctions and Modular Forms Database [35] and was determined by the algorithm in [32]. It follows that \(\mathrm {Gal}(\mathbb Q(E[7])/\mathbb Q)\simeq \mathrm {GL}_2(\mathbb F_7)\), and Proposition 3.34 implies that we then have \(H:=\mathrm {Gal}(K/\mathbb Q)\simeq \mathrm {PGL}_2(\mathbb F_7)\).
For our chosen curve E, we have \(a=6\) and \(b=7\). Plugging these values into equation (3.22) and applying the algorithm of [30] to simplify the result yields the curve listed in entry #14 of Table 5 for \(H=G_{C_7^0}\). To determine the field K, we apply Proposition 3.35. Plugging \(j(E)=48384\) into \(\Phi _7(x,j(E))\) and using PARI/GP to simplify the resulting polynomial, we find that K is the splitting field of the polynomial \({x^8 + 4x^7 + 21x^4 + 18x + 9}\).
We applied the same procedure to obtain the equations for C and the polynomials defining K that are listed in rows #4, #6, #9, #10 of Table 5 using the elliptic curves E with Cremona labels 2450ba1, 64a4, 784h1, 36a1, respectively, with appropriate adjustments for the cases with \(j(E)=0,1728\) as indicated in Proposition 3.35. The curve in row #11 is a base change of the curve in row #6, and for the remaining 7 curves we used the parameterizations in [23].
3.5 Numerical computations

Explicit equations for twists C of \(C^0\) corresponding to subgroups H of \(G_{C^0}\);

Defining polynomials for the minimal field K for which \({\text {End}}({\text {Jac}}(C)_K)={\text {End}}({\text {Jac}}(C)_{\overline{\mathbb Q}})\);

Independent and joint coefficient moments of the Sato–Tate groups \(\mathrm {ST}({\text {Jac}}(C))\).
3.5.1 Naïve pointcounting
A simple but effective way to test the compatibility of a twist C / k and endomorphism field K is to verify that for the first several degree one primes \(\mathfrak {p}\) of k of good reduction for C that split completely in K, the reduction of \({\text {Jac}}(C)\) modulo \(\mathfrak {p}\) is isogenous to the cube of the reduction of \(E^0\) modulo the prime \(p:=N(\mathfrak {p})\). By a theorem of Tate, it suffices to check that \(L_\mathfrak {p}({\text {Jac}}(C),T)=L_p(E^0,T)^3\). For this task, we used optimized brute force pointcounting methods adapted from [20, §3]. The Lpolynomial \(L_\mathfrak {p}({\text {Jac}}(C),T)\) is the numerator of the zeta function of C, a genus 3 curve, so it suffices to compute \(\#C(\mathbb F_p)\), \(\#C(\mathbb F_{p^2})\), \(\#C(\mathbb F_{p^3})\), reducing the problem to counting points on smooth plane quartics and elliptic curves over finite fields.
To count projective points (x : y : z) on a smooth plane quartic \(f(x,y,z)=0\) over a finite field \(\mathbb F_q\), one first counts affine points (x : y : 1) by iterating over \(a\in \mathbb F_q\), computing the number r of distinct \(\mathbb F_q\)rational roots of \(g_a(x):=f(x,a,1)\) via \(r=\deg (\gcd (x^qx,g_a(x)))\), and then determining the multiplicity of each rational root by determining the least \(n\ge 1\) for which \(\gcd (g_a(x),g^{(n)}_a(x))=1\), where \(g^{(n)}_a\) denotes the nth derivative of \(g_a\in \mathbb F_q[x]\); note that to compute \(\gcd (x^qx,g_a(x))\) one first computes \(x^q\bmod g_a(x)\) using a squareandmultiply algorithm. Having counted affine points (x : y : 1), one then counts the \(\mathbb F_q\)rational roots of f(x, 1, 0) and finally checks whether (1 : 0 : 0) is a point on the curve.
To optimize this procedure, one first seeks a linear transformation of f(x, y, z) that ensures \(g_a(x)=f(x,a,1)\) has degree at most 3 for all \(a\in \mathbb F_q\); for this, it suffices to translate a rational point to (1 : 0 : 0), which is always possible for \(q\ge 37\) (by the Weil bounds). This yields an \(O(p^3(\log p)^{2+o(1)})\)time algorithm to compute \(L_\mathfrak {p}({\text {Jac}}(C),T)\) that is quite practical for p up to \(2^{12}\), enough to find several (possibly hundreds) of degree one primes \(\mathfrak {p}\) of k that split completely in K.
Having computed \(L_\mathfrak {p}({\text {Jac}}(C),T)\), one compares this to \(L_p(E^0,T)^3\); note that the polynomial \(L_p(E^0,T)=pT^2a_pT+1\) is easily computed via \(a_p=p+1\#E^0(\mathbb F_p)\). If this comparison fails, then either C is not a twist of \(C^0\), or not all of the endomorphisms of \({\text {Jac}}(C)_{\overline{\mathbb Q}}\) are defined over K. The converse is of course false, but if this comparison succeeds for many degree1 primes \(\mathfrak {p}\) it gives one a high degree of confidence in the computations of C and K.^{4} Note that this test will succeed even when K is not minimal, but we also check that \(H\simeq \mathrm {Gal}(K/k)\), which means that so long as C is a twist of \(C^0\) corresponding to the subgroup \(H\subseteq G_{C_0}\), the field K must be minimal.
3.5.2 An average polynomialtime algorithm
In order to numerically test our computations of the Sato–Tate groups \(\mathrm {ST}(C)\), and to verify our computation of the coefficient moments, we also computed Sato–Tate statistics for all of our twists C / k of the Fermat and Klein quartics. This requires computing the Lpolynomials \(L_\mathfrak {p}({\text {Jac}}(C),T)\) at primes \(\mathfrak {p}\) of good reduction for C up to some bound N, and it suffices to consider only primes \(\mathfrak {p}\) of prime norm \(p=N(\mathfrak {p})\), since nearly all the primes of norm less than N are degree1 primes. In order to get statistics that are close to the values predicted by the Sato–Tate group one needs N to be fairly large. We used \(N=2^{26}\), which is far too large for the naive \(O(p^3(\log p)^{2+o(1)})\)time algorithm described above to be practical, even for a single prime \(p\approx N\), let alone all good \(p\le N\).
In [18], Harvey and Sutherland give an average polynomialtime algorithm to count points on smooth plane quartics over \(\mathbb Q\) that allows one to compute \(L_p({\text {Jac}}(C),T)\bmod p\) for all good primes \(p\le N\) in time \(O(N(\log N)^{3+o(1)})\), which represents an average cost of \(O((\log p)^{4+o(1)})\) per prime \(p\le N\). This is achieved by computing the Hasse–Witt matrices of the reductions of C modulo p using a generalization of the approach given in [16, 17] for hyperelliptic curves. In [18], they also give an \(O(\sqrt{p}(\log p)^{1+o(1)})\)time algorithm to compute \(L_p({\text {Jac}}(C),T)\bmod p\) for a single good prime p, which allows one to handle reductions of smooth plane quartics C defined over number fields at degree one primes; this increases the total running time for \(p\le N\) to \(O(N^{3/2+o(1)})\), which is still feasible with \(N=2^{26}\).
Having computed \(L_p({\text {Jac}}(C),T)\bmod p\), we need to lift this polynomial for \((\mathbb Z/p\mathbb Z)[T]\) to \(\mathbb Z[T]\), which is facilitated by Proposition 3.38 below. It follows from the Weil bounds that the linear coefficient of \(L_p({\text {Jac}}(C),T)\) is an integer of absolute value at most \(6\sqrt{p}\). For \(p>144\), the value of this integer is uniquely determined by its value modulo p, and for \(p<144\) we can apply the naive approach described above. This uniquely determines the value in the column labeled \(F_1(x)\) in Tables 1 and 2 of Proposition 3.38, which then determines the values in columns \(F_2(x)\) and \(F_3(s)\), allowing the integer polynomial \(L_p({\text {Jac}}(C),T)\in \mathbb Z[T]\) to be completely determined.
Proposition 3.38
 (i)
The pair (s, t) is one of the 9 pairs listed on Table 1 if \(C^0=C^0_1\), and one of the 9 pairs listed on Table 2 if \(C^0=C^0_7\).
 (ii)For each pair (s, t), letbe the map defined in Table 1 if \(C_0=C_0^1\) or Table 2 if \(C_0=C_0^7\). For every prime \(\mathfrak {p}\) unramified in K of good reduction for both \({\text {Jac}}(C)\) and \(E^0\), we have$$\begin{aligned} F_{(s,t)}:=F_{1}\times F_{2}\times F_{3}:[2,2]\rightarrow [6,6]\times [1,15]\times [20,20]\subseteq \mathbb R^3 \end{aligned}$$where \(f_L(\mathfrak {p})\) (resp. \(f_{kM}(\mathfrak {p})\)) is the residue degree of \(\mathfrak {p}\) in L (resp. kM).$$\begin{aligned} F_{(f_L(\mathfrak {p}),f_{kM}(\mathfrak {p}))}(a_1(E^0)(\mathfrak {p}))=\bigl ( a_1({\text {Jac}}(C))( \mathfrak {p}), a_2({\text {Jac}}(C))(\mathfrak {p}), a_3({\text {Jac}}(C))(\mathfrak {p}) \bigr ), \end{aligned}$$(3.26)
For each possible pair (s, t), the corresponding values of \(F_{(s,t)}(x)\) if \(C^0=C^0_1\). In the table below, y denotes \(\pm \sqrt{4x^2}\)
(s, t)  \(F_{1}(x)\)  \(F_{2}(x)\)  \(F_{3}(x)\) 

(1, 1)  3x  \(3x^2+3\)  \(x^3+6x\) 
(2, 1)  or \({\left\{ \begin{array}{ll} x\\ x \end{array}\right. }\)  \(\begin{array}{l} x^2+3\\ x^2+3\end{array}\)  \(\begin{array}{l}x^32x\\ x^3+2x\end{array}\) 
(3, 1)  0  0  \(x^33x\) 
(4, 1)  or \({\left\{ \begin{array}{ll}x+2y \\ x \\ x\end{array}\right. } \)  \(\begin{array}{l}x^2+72xy \\ x^21\\ x^21\end{array}\)  \(\begin{array}{l}x^36x+4y \\ x^32x\\ x^3+2x\end{array}\) 
(8, 1)  y  \(xy+1\)  \(x^34x\) 
(2, 2)  0  3  0 
(4, 2)  0  \(1\)  0 
(6, 2)  0  0  0 
(8, 2)  0  1  0 
For each possible pair (s, t), the corresponding values of \(F_{(s,t)}(x)\) if \(C^0=C^0_7\). In the table below, y denotes \(\pm \sqrt{(4x^2)/7}\)
(s, t)  \(F_{1}(x)\)  \(F_{2}(x)\)  \(F_{3}(x)\) 

(1, 1)  3x  \(3x^2+3\)  \(x^3+6x\) 
(2, 1)  \(x\)  \(x^2+3\)  \(x^32x\) 
(3, 1)  0  0  \(x^33x\) 
(4, 1)  x  \(x^21\)  \(x^32x\) 
(7, 1)  \(\frac{x}{2}+ \frac{ 7}{2}y\)  \(\frac{x^2}{2}+3\frac{ 7}{2}xy\)  \(x^3\frac{9}{2}x+\frac{7}{2}y\) 
(2, 2)  0  3  0 
(4, 2)  0  \(1\)  0 
(6, 2)  0  0  0 
(8, 2)  0  1  0 
(A) \({\text {Aut}}((C_1^0)_M)\simeq \langle 96,64\rangle \)  
Class  1a  2a  2b  3a  4a  4b  4c  4d  8a  8b 
Repr.  1  \(u^2\)  \(u^2t\)  s  u  \(u^3\)  tut  t  tu  \(tu^3\) 
Order  1  2  2  3  4  4  4  4  8  8 
Size  1  3  12  32  3  3  6  12  12  12 
\(\chi _1\)  1  1  1  1  1  1  1  1  1  1 
\(\chi _2\)  1  1  \(\)1  1  1  1  1  \(\)1  \(\)1  \(\)1 
\(\chi _3\)  2  2  0  \(\)1  2  2  2  0  0  0 
\(\chi _4\)  3  3  \(\)1  0  \(\)1  \(\)1  \(\)1  \(\)1  1  1 
\(\chi _5\)  3  3  1  0  \(\)1  \(\)1  \(\)1  1  \(\)1  \(\)1 
\(\chi _6\)  3  \(\)1  1  0  \(12i\)  \(1+2i\)  1  \(\)1  i  \(i\) 
\(\chi _7\)  3  \(\)1  1  0  \(1+2i\)  \(12i\)  1  \(\)1  \(i\)  i 
\(\chi _8\)  3  \(\)1  \(\)1  0  \(1+2i\)  \(12i\)  1  1  i  \(i\) 
\(\chi _9\)  3  \(\)1  \(\)1  0  \(12i\)  \(1+2i\)  1  1  \(i\)  i 
\(\chi _{10}\)  6  \(\)2  0  0  2  2  \(\)2  0  0  0 
(B) \({\text {Aut}}((C_7^0)_M)\simeq \langle 168,42\rangle \)  

Class  1a  2a  3a  4a  7a  7b  
Repr.  1  t  s  \(u^2tu^3tu^2\)  u  \(u^3\)  
Order  1  2  3  4  7  7  
Size  1  21  56  42  24  24  
\(\chi _1\)  1  1  1  1  1  1  
\(\chi _2\)  3  \(\)1  0  1  a  \(\overline{a}\)  
\(\chi _3\)  3  \(\)1  0  1  \(\overline{a}\)  a  
\(\chi _4\)  6  2  0  0  \(\)1  \(\)1  
\(\chi _5\)  7  \(\)1  1  \(\)1  0  0  
\(\chi _6\)  8  0  \(\)1  0  1  1 
Twists of the Fermat quartic corresponding to subgroups \(H \subseteq G_{C^0_1}\). See (3.2) for the definitions of s, t, u and (3.6) for the definition of h. We identify \(\mathrm {ST}_k=\mathrm {ST}(C)\) and \(\mathrm {ST}_{kM}=\mathrm {ST}(C_{kM})\) by row numbers in Table 6; here \(M=\mathbb Q(i)\)
#  Gen\((H_0)\)  h  ID(H)  ID\((H_0)\)  k  K  \(\mathrm {ST}_k\)  \(\mathrm {ST}_{kM}\) 

1  id  id  \(\langle 2,1 \rangle \)  \(\langle 1,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(i)\)  3  1 
\({C_1^0}\)  \(x^4+y^4+z^4\)  
2  id  \(u^2t\)  \(\langle 2,1 \rangle \)  \(\langle 1,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(i)\)  3  1 
\(x^46x^2y^2+y^42z^4\)  
3  id  \(t^3utu\)  \(\langle 2,1 \rangle \)  \(\langle 1,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(i)\)  3  1 
\(4x^4y^4z^4\)  
4  \(t^2\)  t  \(\langle 4,1 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q(\sqrt{5})\)  \(\mathrm {Gal}(x^4\!\!x^2\!\!1)\)  5  2 
\(12x^4+40x^3y100xy^375y^42z^4\)  
5  \(t^2\)  \(t^3utu\)  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},i)\)  9  2 
\(9x^4+9y^44z^4\)  
6  \(t^2\)  \(u^2t\)  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},i)\)  9  2 
\(9x^454x^2y^2+9y^42z^4\)  
7  \(t^2\)  id  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},i)\)  9  2 
\(9x^4+y^4+z^4\)  
8  \(t^2\)  u  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},i)\)  9  2 
\(9x^44y^4+z^4\)  
9  \(u^2t\)  id  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},i)\)  9  2 
\(2x^4+36x^2y^2+18y^4+z^4\)  
10  \(u^2t\)  \(t^3utu\)  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},i)\)  9  2 
\(9x^4+18x^2y^2+y^42z^4\)  
11  s  \(u^2t\)  \(\langle 6,1 \rangle \)  \(\langle 3,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^3\!\!3x\!\!4)\)  11  4 
\(x^4+4x^3y+12x^2y^212x^2yz6x^2z^2+36xyz^2+6y^436y^2z^212yz^3+9z^4\)  
12  s  id  \(\langle 6,2 \rangle \)  \(\langle 3,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^6\!+\!5x^4\!+\!6x^2\!+\!1)\)  12  4 
\(5x^4+8x^3y4x^3z+6x^2y^2+12x^2z^2+12xyz^2+4xz^3+2y^4+4y^3z+6y^2z^2+4yz^3+2z^4\)  
13  \(t^3utu^3\)  tu  \(\langle 8,1 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q(\sqrt{5})\)  \(\mathrm {Gal}(x^8\!\!2x^4\!\!4)\)  14  6 
\(x^410x^3z+30x^2z^22y^4100z^4\)  
14  t  \(t^3utu\)  \(\langle 8,2 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!15x^4\!+\!25)\)  16  6 
\(3x^44x^3y+12x^2y^2+4xy^3+3y^45z^4\)  
15  \(t^3utu^3\)  \(u^2t\)  \(\langle 8,2 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!15x^4\!+\!25)\)  16  6 
\(12x^4+40x^3y100xy^375y^4+10z^4\)  
16  t  id  \(\langle 8,2 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!15x^4\!+\!25)\)  16  6 
\(3x^4+4x^3y+12x^2y^24xy^3+3y^4+20z^4\)  
17  \(u^2t,t^2\)  \(t^3utu^3\)  \(\langle 8,3 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!10)\)  19  8 
\(11x^4+12x^3y+54x^2y^212xy^3+11y^42z^4\)  
18  \(t^2,u^2\)  \(u^2t\)  \(\langle 8,3 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!10)\)  19  8 
\(x^4+5x^3y25xy^325y^4+z^4\)  
19  \(u^2t,t^2\)  \(u^2\)  \(\langle 8,3 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!10)\)  19  8 
\(19x^412x^3y+6x^2y^2+12xy^3+19y^4+2z^4\)  
20  t  \(u^2\)  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  20  6 
\(9x^418x^2y^212xy^32y^4+12z^4\)  
21  t  \(t^3utu^3\)  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  20  6 
\(9x^418x^2y^212xy^32y^43z^4\)  
22  \(t^3utu^3\)  u  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  20  6 
\(9x^4+3y^44z^4\)  
23  \(t^3utu^3\)  id  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  20  6 
\(9x^4+3y^4+z^4\)  
24  \(t^3utu\)  \(u^2t\)  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  21  7 
\(x^46x^2y^2+y^46z^4\)  
25  \(t^3utu\)  u  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  21  7 
\(3x^44y^4+z^4\)  
26  \(t^3utu\)  id  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!6x^2\!+\!12)\)  21  7 
\(3x^4+y^4+z^4\)  
27  \(t^3utu\)  t  \(\langle 8,4 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q(\sqrt{2})\)  \(\mathrm {Gal}(x^8\!+\!9)\)  23  7 
\(3x^3y3xy^32z^4\)  
28  \(t^2,u^2\)  id  \(\langle 8,5 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},\sqrt{5},i)\)  24  8 
\(9x^4+25y^4+z^4\)  
29  \(t^2,u^2\)  u  \(\langle 8,5 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},\sqrt{5},i)\)  24  8 
\(9x^4+25y^44z^4\)  
30  \(u^2t,t^2\)  \(t^3utu\)  \(\langle 8,5 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},\sqrt{5},i)\)  24  8 
\(x^4+30x^2y^2+25y^418z^4\)  
31  \(u^2t,t^2\)  id  \(\langle 8,5 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\sqrt{3},\sqrt{5},i)\)  24  8 
\(2x^4+60x^2y^2+50y^4+9z^4\)  
32  \(s,u^2t\)  id  \(\langle 12,4 \rangle \)  \(\langle 6,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^6\!+\!2x^3\!+\!2)\)  26  10 
\(4x^3y3x^2z^2+12xy^2z2y^42yz^3\)  
33  \(t^3utu,u^2\)  ut  \(\langle 16,6 \rangle \)  \(\langle 8,2 \rangle \)  \(\mathbb Q(\sqrt{5})\)  \(\mathrm {Gal}(x^8\!\!2x^4\!+\!5)\)  29  17 
\(x^430x^2y^280xy^355y^42z^4\)  
34  \(t^3utu^3,u^2t\)  u  \(\langle 16,7 \rangle \)  \(\langle 8,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!6x^4\!\!8x^2\!\!1)\)  31  18 
\(x^412x^2y^232xy^328y^4+z^4\)  
35  \(tu^2tut\)  \(tutu^2\)  \(\langle 16,7 \rangle \)  \(\langle 8,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!8x^4\!\!2)\)  32  15 
\(2x^3yxy^3z^4\)  
36  \(u^2tu\)  u  \(\langle 16,8 \rangle \)  \(\langle 8,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!2)\)  34  15 
\(x^3y+2xy^3+z^4\)  
37  \(t^3utu^3,t\)  u  \(\langle 16,8 \rangle \)  \(\langle 8,4 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!10x^4\!\!100)\)  33  22 
\(x^4+10x^3y+30x^2y^2100y^410z^4\)  
38  \(t,u^2\)  utu  \(\langle 16,11 \rangle \)  \(\langle 8,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!2x^4\!+\!9)\)  35  18 
\(4x^4+4x^3y+6x^2y^22xy^3+y^42z^4\)  
39  \(t,u^2\)  id  \(\langle 16,11 \rangle \)  \(\langle 8,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!2x^4\!+\!9)\)  35  18 
\(4x^4+4x^3y+6x^2y^22xy^3+y^4+2z^4\)  
40  \(t^3utu^3,u^2t\)  id  \(\langle 16,11 \rangle \)  \(\langle 8,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!2x^4\!+\!9)\)  35  18 
\(5x^48x^3y+12x^2y^2+16xy^3+20y^4+2z^4\)  
41  \(t^3utu,u^2\)  id  \(\langle 16,11 \rangle \)  \(\langle 8,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!5x^4\!+\!25)\)  36  17 
\(9x^4+5y^4+z^4\)  
42  \(t^3utu,u^2\)  u  \(\langle 16,11 \rangle \)  \(\langle 8,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!5x^4\!+\!25)\)  36  17 
\(9x^4+5y^44z^4\)  
43  \(utu,t^3\)  \(u^2\)  \(\langle 16,11 \rangle \)  \(\langle 8,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!5x^4\!+\!25)\)  36  17 
\(7x^4+8x^3y+6x^2y^2+8xy^3+7y^4+10z^4\)  
44  \(t^3utu,u^2\)  \(u^2t\)  \(\langle 16,13 \rangle \)  \(\langle 8,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!8x^4\!+\!25)\)  39  17 
\(x^4+5x^3y25xy^325y^4+2z^4\)  
45  t, utu  id  \(\langle 16,13 \rangle \)  \(\langle 8,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!8x^4\!+\!25)\)  39  17 
\(19x^4+32x^3y+21x^2y^2+8xy^3+3y^4+2y^3z+6y^2z^2+8yz^3+4z^4\)  
46  \(t^3utu^3,t\)  id  \(\langle 16,13 \rangle \)  \(\langle 8,4 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!12x^4\!+\!9)\)  37  22 
\(x^42x^3y+6x^2y^2+4xy^3+4y^4+3z^4\)  
47  \(s,u^2\)  \(u^2t\)  \(\langle 24,12 \rangle \)  \(\langle 12,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!16x\!\!24)\)  42  25 
\(x^43x^3z12x^2yz+16xy^3xz^3+9y^4+12y^3z+6y^2z^2\)  
48  \(s,u^2\)  id  \(\langle 24,13 \rangle \)  \(\langle 12,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^6\!\!x^4\!\!2x^2\!+\!1)\)  43  25 
\(3x^4+4x^3y+4x^3z+6x^2y^2+6x^2z^2+8xy^3+12xyz^2+5y^4+4y^3z+12y^2z^2+z^4\)  
49  \(tu^2tut,u^2\)  id  \(\langle 32,7 \rangle \)  \(\langle 16,6 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!10x^4\!+\!20)\)  44  30 
\(4x^48x^3y+12x^2y^2+2y^4+5z^4\)  
50  \(u,u^2tu^3t\)  \(u^2t\)  \(\langle 32,11 \rangle \)  \(\langle 16,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!2x^4\!+\!5)\)  45  28 
\(x^430x^2y^280xy^355y^42z^4\)  
51  \(u,t^3ut\)  id  \(\langle 32,34 \rangle \)  \(\langle 16,2 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^{16}\!\!4x^{12}\!+\!6x^8\!+\!20x^4\!+\!1)\)  47  28 
\(2x^4+3y^4+z^4\)  
52  \(t^3utu,u^2,t\)  u  \(\langle 32,43 \rangle \)  \(\langle 16,13 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!6x^4\!\!9)\)  48  38 
\(3x^436x^2y^296xy^384y^4+2z^4\)  
53  \(tu^2tut,u^2\)  u  \(\langle 32,43 \rangle \)  \(\langle 16,6 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!10x^4\!+\!45)\)  49  30 
\(9x^4+36x^3y24xy^34y^410z^4\)  
54  \(utu,u^2,t\)  id  \(\langle 32,49 \rangle \)  \(\langle 16,13 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!8x^4\!+\!9)\)  50  38 
\(x^4+x^3y+24x^2y^2+67xy^3+79y^4+2z^4\)  
55  s, t  id  \(\langle 48,48 \rangle \)  \(\langle 24,12 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^6\!\!x^4\!+\!5x^2\!+\!1)\)  53  41 
\(3x^4+2x^3z+6x^2yz+12xy^330xy^2z+2xz^327y^4+38y^3z+18y^2z^210yz^3\)  
56  t, u  id  \(\langle 64,134 \rangle \)  \(\langle 32,11 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!\!4x^4\!\!14)\)  54  46 
\(x^442x^2y^2168xy^3203y^4+z^4\)  
57  s, u  \(u^2t\)  \(\langle 96,64 \rangle \)  \(\langle 48,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^{12}\!+\!6x^4\!+\!4)\)  55  52 
\(6x^3z+3x^2y^2+27x^2z^2+6xy^3+18xyz^2+4y^4+2y^3z+18yz^336z^4\)  
58  s, u  id  \(\langle 96,72 \rangle \)  \(\langle 48,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^{12}\!+\!x^8\!\!2x^4\!\!1)\)  57  52 
\(x^3yx^3z+3x^2y^2+28xy^384xy^2z+84xyz^2+28y^456y^3z+98z^4\)  
59  s, t, u  id  \(\langle 192,956 \rangle \)  \(\langle 96,64 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^{12}\!+\!48x^4\!+\!64)\)  59  56 
\(44x^4+120x^3y+36x^3z+60x^2yz+9x^2z^2200xy^3+xz^3150y^415y^2z^2\) 
Twists of the Klein quartic corresponding to subgroups \(H\subseteq G_{C^0_7}\). See (3.3) for the definitions of s, t, u and see (3.6) for the definition of h. We identify \(\mathrm {ST}_k=\mathrm {ST}(C)\) and \(\mathrm {ST}_{kM}=\mathrm {ST}(C_{kM})\) by row numbers in Table 6; here \(M=\mathbb Q(\sqrt{7})\) and \(a:=(1+\sqrt{7})/2\)
#  Gen\((H_0)\)  h  ID(H)  ID\((H_0)\)  k  K  \(\mathrm {ST}_k\)  \(\mathrm {ST}_{kM}\) 

1  id  id  \(\langle 2,1 \rangle \)  \(\langle 1,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(a)\)  3  1 
\({C_7^0}\)  \(x^4+y^4+z^4+6(xy^3+yz^3+zx^3)3(x^2y^2+y^2z^2+z^2x^2)+3xyz(x+y+z)\)  
2  t  id  \(\langle 4,2 \rangle \)  \(\langle 2,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(a,i)\)  9  2 
\(3x^4+28x^3y+105x^2y^221x^2z^2+196xy^3+147y^4+147y^2z^249z^4\)  
3  \(ustu^6,sutu^6s^2\)  –  \(\langle 4,2 \rangle \)  \(\langle 4,2 \rangle \)  \(\mathbb Q(a)\)  \(\mathbb Q(\sqrt{2},\sqrt{3},a)\)  8  8 
\(x^4+9ax^2y^2+6ax^2z^2+9y^4+18ay^2z^2+4z^4\)  
4  s  t  \(\langle 6,1 \rangle \)  \(\langle 3,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^3\!\!x^2\!+\!2x\!\!3)\)  11  4 
\(x^4+3x^3y9x^3z+9x^2y^26x^2z^2+18xy^3+3xy^2z3xyz^2+y^4+4y^3z3y^2z^2+7yz^3\)  
5  s  id  \(\langle 6,2 \rangle \)  \(\langle 3,1 \rangle \)  \(\mathbb Q\)  \(\mathbb Q(\zeta _7)\)  12  4 
\(x^3y+xz^3+y^3z\)  
6  \(u^2tu^3tu^2\)  \(u^5tu^2\)  \(\langle 8,1 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q(i)\)  \(\mathrm {Gal}(x^8\!+\!2x^7\!\!14x^4\!+\!16x\!+\!4)\)  14  6 
\(x^4+3x^2y^23x^2z^2+2y^3z+3y^2z^2+2yz^3\)  
7  \(u^2tu^3tu^2\)  id  \(\langle 8,3 \rangle \)  \(\langle 4,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^4\!\!4x^2\!\!14)\)  20  6 
\(12x^480x^3y+60x^2y^224x^2z^2104xy^3+24xyz^2+83y^4+36y^2z^22z^4\)  
8  su, tu  –  \(\langle 12,3 \rangle \)  \(\langle 12,3 \rangle \)  \(\mathbb Q(a)\)  \(\mathrm {Gal}(x^6\!\!147x^2\!+\!343)\)  25  25 
\(3x^4+(18a+12)x^3y+(12a+4)x^3z+(27a+36)x^2y^2+(9a+6)x^2z^2+36xy^2z+27y^4\) \(+\,(54a36)y^3z+(54a+36)y^2z^2+(18a12)yz^3+(3a+2)z^4\)  
9  s, t  id  \(\langle 12,4 \rangle \)  \(\langle 6,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^3\!\!x^2\!+\!5x\!+\! 1)\cdot \mathbb Q(a)\)  26  10 
\(7x^3z+3x^2y^26xyz^2+2y^3z4z^4\)  
10  u  id  \(\langle 14,1 \rangle \)  \(\langle 7,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^7\!+\!7x^3\!\!7x^2\!+\!7x\!+\!1)\)  27  13 
\(x^3y21x^2z^2+xy^342xyz^2147xz^3+2y^4+21y^3z+63y^2z^2196z^4\)  
11  \(u^2tu^3tu^2,u^5tu^2\)  id  \(\langle 16,7 \rangle \)  \(\langle 8,3 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!2x^7\!\!14x^4\!+\!16x\!+\!4)\)  31  18 
\(x^4+3x^2y^23x^2z^2+2y^3z+3y^2z^2+2yz^3\)  
12  \(sust,su^6s^2tu^2\)  –  \(\langle 24,12 \rangle \)  \(\langle 24,12 \rangle \)  \(\mathbb Q(a)\)  \(\mathrm {Gal}(x^4\!+\! 2x^3\!+\!6x^2\!\!6)\cdot \mathbb Q(a)\)  41  41 
\((3a2)x^4+(30a20)x^3y+(90a60)x^2y^2+(9a+6)x^2z^2+(150a100)xy^3+60xy^2z\) \(+\,(12a+4)xz^3+(150a25)y^4+(45a+60)y^2z^2+(30a20)yz^3+3z^4\)  
13  u, s  id  \(\langle 42,1 \rangle \)  \(\langle 21,1 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^7\!\!2)\)  51  40 
\(2x^3y+xz^3+y^3z\)  
14  t, u, s  id  \(\langle 336,208 \rangle \)  \(\langle 168,42 \rangle \)  \(\mathbb Q\)  \(\mathrm {Gal}(x^8\!+\!4x^7\!+\!21x^4\!+\!18x\!+\!9)\)  60  58 
\(2x^3y2x^3z3x^2z^22xy^32xz^34y^3z+3y^2z^2yz^3\) 
The 60 Sato–Tate distributions arising for Fermat and Klein twists
\(\#\)  ID  \(\mathrm {M }_{101}\)  \(\mathrm {M }_{030}\)  \(\mathrm {M }_{202}\)  \(\mathrm {M }_{200}\)  \(\mathrm {M }_{400}\)  \(\mathrm {M }_{010}\)  \(\mathrm {M }_{020}\)  \(\mathrm {M }_{002}\)  \(\mathrm {M }_{004}\)  \(z_1\)  \(z_2\)  \(z_3\)  

1  \(\langle 1,1\rangle \)  54  1215  4734  18  486  9  99  164  47148  0  0  0  0  0  0  0  
2  \(\langle 2,1\rangle \)  26  611  2374  10  246  5  51  84  23596  0  0  0  0  0  0  0  
3  \(\langle 2,1\rangle \)  27  621  2367  9  243  6  54  82  23574  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
4  \(\langle 3,1\rangle \)  18  405  1578  6  162  3  33  56  15720  \(\nicefrac {2}{3}\)  0  \(\nicefrac {2}{3}\)  0  0  0  0  
5  \(\langle 4,1\rangle \)  13  305  1187  5  123  2  26  42  11798  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  
6  \(\langle 4,1\rangle \)  14  309  1194  6  126  3  27  44  11820  0  0  0  0  0  0  0  a 
7  \(\langle 4,1\rangle \)  24  443  1614  10  198  5  43  68  14444  0  0  0  0  0  0  0  
8  \(\langle 4,2\rangle \)  12  309  1194  6  126  3  27  44  11820  0  0  0  0  0  0  0  a 
9  \(\langle 4,2\rangle \)  13  319  1187  5  123  4  30  42  11798  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
10  \(\langle 6,1\rangle \)  8  206  796  4  84  2  18  30  7882  \(\nicefrac {1}{3}\)  0  \(\nicefrac {1}{3}\)  0  0  0  0  
11  \(\langle 6,1\rangle \)  9  216  789  3  81  3  21  28  7860  \(\nicefrac {5}{6}\)  0  \(\nicefrac {1}{3}\)  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
12  \(\langle 6,2\rangle \)  9  207  789  3  81  2  18  28  7860  \(\nicefrac {5}{6}\)  0  \(\nicefrac {2}{3}\)  0  0  \(\nicefrac {1}{6}\)  \(\nicefrac {1}{2}\)  
13  \(\langle 7,1\rangle \)  12  201  732  6  90  3  21  32  6936  0  0  0  0  0  0  0  
14  \(\langle 8,1\rangle \)  7  155  597  3  63  2  14  22  5910  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{2}\)  
15  \(\langle 8,1\rangle \)  12  225  812  6  102  3  23  36  7236  0  0  0  0  0  0  0  
16  \(\langle 8,2\rangle \)  7  161  597  3  63  2  16  22  5910  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{4}\)  0  0  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  b 
17  \(\langle 8,2\rangle \)  12  225  814  6  102  3  23  36  7244  0  0  0  0  0  0  0  
18  \(\langle 8,3\rangle \)  6  158  604  4  66  2  15  24  5932  0  0  0  0  0  0  0  c 
19  \(\langle 8,3\rangle \)  6  161  597  3  63  2  16  22  5910  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{4}\)  0  0  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  b 
20  \(\langle 8,3\rangle \)  7  168  597  3  63  3  18  22  5910  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  d 
21  \(\langle 8,3\rangle \)  12  235  807  5  99  4  26  34  7222  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
22  \(\langle 8,4\rangle \)  8  158  604  4  66  2  15  24  5932  0  0  0  0  0  0  0  c 
23  \(\langle 8,4\rangle \)  12  221  807  5  99  2  22  34  7222  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  
24  \(\langle 8,5\rangle \)  6  168  597  3  63  3  18  22  5910  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  d 
25  \(\langle 12,3\rangle \)  4  103  398  2  42  1  9  16  3944  \(\nicefrac {2}{3}\)  0  \(\nicefrac {2}{3}\)  0  0  0  0  
26  \(\langle 12,4\rangle \)  4  112  398  2  42  2  12  15  3941  \(\nicefrac {2}{3}\)  0  \(\nicefrac {1}{3}\)  0  0  \(\nicefrac {1}{3}\)  \(\nicefrac {1}{2}\)  
27  \(\langle 14,1\rangle \)  6  114  366  3  45  3  15  16  3468  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
28  \(\langle 16,2\rangle \)  12  183  624  6  90  3  21  32  4956  0  0  0  0  0  0  0  
29  \(\langle 16,6\rangle \)  6  113  407  3  51  2  12  18  3622  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{2}\)  
30  \(\langle 16,6\rangle \)  6  116  412  4  54  2  13  20  3636  0  0  0  0  0  0  0  
31  \(\langle 16,7\rangle \)  3  86  302  2  33  2  10  12  2966  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  e 
32  \(\langle 16,7\rangle \)  6  126  406  3  51  3  16  18  3618  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
33  \(\langle 16,8\rangle \)  4  86  302  2  33  2  10  12  2966  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  e 
34  \(\langle 16,8\rangle \)  6  119  406  3  51  2  14  18  3618  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{4}\)  0  0  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  
35  \(\langle 16,11\rangle \)  3  89  302  2  33  2  11  12  2966  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{8}\)  0  0  0  \(\nicefrac {3}{8}\)  \(\nicefrac {1}{2}\)  f 
36  \(\langle 16,11\rangle \)  6  126  407  3  51  3  16  18  3622  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
37  \(\langle 16,13\rangle \)  4  89  302  2  33  2  11  12  2966  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{8}\)  0  0  0  \(\nicefrac {3}{8}\)  \(\nicefrac {1}{2}\)  f 
38  \(\langle 16,13\rangle \)  6  116  414  4  54  2  13  20  3644  0  0  0  0  0  0  0  
39  \(\langle 16,13\rangle \)  6  119  407  3  51  2  14  18  3622  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{4}\)  0  0  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  
40  \(\langle 21,1\rangle \)  4  67  244  2  30  1  7  12  2316  \(\nicefrac {2}{3}\)  0  \(\nicefrac {2}{3}\)  0  0  0  0  
41  \(\langle 24,12\rangle \)  2  55  206  2  24  1  6  10  1994  \(\nicefrac {1}{3}\)  0  \(\nicefrac {1}{3}\)  0  0  0  0  
42  \(\langle 24,12\rangle \)  2  58  199  1  21  1  7  8  1972  \(\nicefrac {5}{6}\)  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{3}\)  0  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  
43  \(\langle 24,13\rangle \)  2  56  199  1  21  1  6  8  1972  \(\nicefrac {5}{6}\)  0  \(\nicefrac {2}{3}\)  0  0  \(\nicefrac {1}{6}\)  \(\nicefrac {1}{2}\)  
44  \(\langle 32,7\rangle \)  3  65  206  2  27  2  9  10  1818  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  
45  \(\langle 32,11\rangle \)  6  95  312  3  45  2  12  16  2478  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{8}\)  0  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{8}\)  \(\nicefrac {1}{2}\)  
46  \(\langle 32,11\rangle \)  6  95  318  4  48  2  12  18  2496  0  0  0  0  0  0  0  
47  \(\langle 32,34\rangle \)  6  105  312  3  45  3  15  16  2478  \(\nicefrac {1}{2}\)  0  0  0  0  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{2}\)  
48  \(\langle 32,43\rangle \)  3  65  207  2  27  2  9  10  1822  \(\nicefrac {1}{2}\)  0  0  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{4}\)  \(\nicefrac {1}{2}\)  
49  \(\langle 32,43\rangle \)  3  68  206  2  27  2  10  10  1818  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{8}\)  0  0  0  \(\nicefrac {3}{8}\)  \(\nicefrac {1}{2}\)  
50  \(\langle 32,49\rangle \)  3  68  207  2  27  2  10  10  1822  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{8}\)  0  0  0  \(\nicefrac {3}{8}\)  \(\nicefrac {1}{2}\)  
51  \(\langle 42,1\rangle \)  2  38  122  1  15  1  5  6  1158  \(\nicefrac {5}{6}\)  0  \(\nicefrac {2}{3}\)  0  0  \(\nicefrac {1}{6}\)  \(\nicefrac {1}{2}\)  
52  \(\langle 48,3\rangle \)  4  61  208  2  30  1  7  12  1656  \(\nicefrac {2}{3}\)  0  \(\nicefrac {2}{3}\)  0  0  0  0  
53  \(\langle 48,48\rangle \)  1  33  103  1  12  1  5  5  997  \(\nicefrac {2}{3}\)  \(\nicefrac {1}{8}\)  \(\nicefrac {1}{3}\)  0  0  \(\nicefrac {5}{24}\)  \(\nicefrac {1}{2}\)  
54  \(\langle 64,134\rangle \)  3  56  159  2  24  2  9  9  1248  \(\nicefrac {1}{2}\)  \(\nicefrac {1}{16}\)  0  \(\nicefrac {1}{8}\)  0  \(\nicefrac {5}{16}\)  \(\nicefrac {1}{2}\)  
55  \(\langle 96,64\rangle \)  2  34  104  1  15  1  5  6  828  \(\nicefrac {5}{6}\)  \(\nicefrac {1}{8}\)  \(\nicefrac {1}{3}\)  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{8}\)  \(\nicefrac {1}{2}\)  
56  \(\langle 96,64\rangle \)  2  34  110  2  18  1  5  8  846  \(\nicefrac {1}{3}\)  0  \(\nicefrac {1}{3}\)  0  0  0  0  
57  \(\langle 96,72\rangle \)  2  35  104  1  15  1  5  6  828  \(\nicefrac {5}{6}\)  0  \(\nicefrac {2}{3}\)  0  0  \(\nicefrac {1}{6}\)  \(\nicefrac {1}{2}\)  
58  \(\langle 168,42\rangle \)  2  19  52  2  12  1  4  6  366  \(\nicefrac {1}{3}\)  0  \(\nicefrac {1}{3}\)  0  0  0  0  
59  \(\langle 192,956\rangle \)  1  21  55  1  9  1  4  4  423  \(\nicefrac {2}{3}\)  \(\nicefrac {1}{16}\)  \(\nicefrac {1}{3}\)  \(\nicefrac {1}{8}\)  0  \(\nicefrac {7}{48}\)  \(\nicefrac {1}{2}\)  
60  \(\langle 336,208\rangle \)  1  12  26  1  6  1  3  3  183  \(\nicefrac {2}{3}\)  0  \(\nicefrac {1}{3}\)  \(\nicefrac {1}{4}\)  0  \(\nicefrac {1}{12}\)  \(\nicefrac {1}{2}\) 
Sato–Tate statistics for Fermat twists over degree one primes \(p\le 2^{26}\)
\(\#\)  \(\mathrm {ST}(C_k)\)  \(\mathrm {ST}(C_{kM})\)  

\(\overline{\mathrm {M }}_{101}\)  \(\mathrm {M }_{101}\)  \(\overline{\mathrm {M }}_{030}\)  \(\mathrm {M }_{030}\)  \(\overline{\mathrm {M }}_{202}\)  \(\mathrm {M }_{202}\)  \(\overline{\mathrm {M }}_{101}\)  \(\mathrm {M }_{101}\)  \(\overline{\mathrm {M }}_{030}\)  \(\mathrm {M }_{030}\)  \(\overline{\mathrm {M }}_{202}\)  \(\mathrm {M }_{202}\)  
1  26.99  27  620.72  621  2365.83  2367  53.99  54  1214.69  1215  4732.64  4734 
4  12.98  13  304.55  305  1185.12  1187  25.98  26  610.36  611  2371.23  2374 
5  12.99  13  318.75  319  1185.94  1187  25.99  26  610.61  611  2372.38  2374 
11  8.99  9  215.63  216  787.39  789  17.98  18  404.33  405  1575.11  1578 
12  9.00  9  206.88  207  788.45  789  18.00  18  404.84  405  1577.24  1578 
13  6.98  7  154.56  155  595.23  597  13.97  14  308.25  309  1190.96  1194 
14  6.99  7  160.84  161  596.37  597  13.99  14  308.75  309  1192.99  1194 
17  5.99  6  160.80  161  596.20  597  11.99  12  308.67  309  1192.65  1194 
20  6.99  7  167.74  168  595.89  597  13.98  14  308.54  309  1192.02  1194 
24  11.99  12  234.80  235  806.10  807  23.99  24  442.70  443  1612.54  1614 
27  11.99  12  220.64  221  805.56  807  23.98  24  442.43  443  1611.64  1614 
28  5.99  6  167.77  168  596.03  597  11.98  12  308.60  309  1192.31  1194 
32  4.00  4  111.98  112  397.87  398  8.00  8  205.99  206  795.91  796 
33  6.00  6  112.89  113  406.55  407  12.00  12  224.88  225  813.43  814 
34  3.00  3  85.98  86  301.92  302  6.00  6  157.99  158  603.96  604 
35  5.99  6  125.77  126  405.04  406  11.98  12  224.57  225  810.25  812 
36  5.99  6  118.74  119  404.95  406  11.98  12  224.52  225  810.06  812 
37  4.00  4  85.99  86  301.98  302  8.00  8  158.00  158  604.09  604 
38  2.99  3  88.81  89  301.24  302  5.99  6  157.65  158  602.60  604 
41  5.99  6  125.74  126  405.90  407  11.98  12  224.53  225  811.97  814 
44  6.00  6  118.88  119  406.47  407  11.99  12  224.80  225  813.12  814 
46  3.99  4  88.73  89  300.88  302  7.98  8  157.50  158  601.89  604 
47  2.00  2  57.88  58  198.48  199  3.99  4  102.77  103  397.05  398 
48  1.99  2  55.81  56  198.20  199  3.99  4  102.64  103  396.49  398 
49  2.99  3  64.85  65  205.42  206  5.99  6  115.72  116  410.92  412 
50  6.00  6  94.92  95  311.67  312  12.00  12  182.87  183  623.48  624 
51  5.99  6  104.72  105  310.80  312  11.98  12  182.47  183  621.72  624 
52  3.00  3  64.94  65  206.72  207  6.00  6  115.89  116  413.52  414 
53  2.99  3  67.84  68  205.30  206  5.99  6  115.71  116  410.68  412 
54  3.00  3  67.88  68  206.50  207  5.99  6  115.78  116  413.08  414 
55  1.00  1  32.93  33  102.73  103  1.99  2  54.86  55  205.50  206 
56  2.99  3  55.77  56  158.06  159  5.98  6  94.56  95  316.20  318 
57  1.99  2  33.87  34  103.46  104  3.99  4  60.76  61  206.96  208 
58  2.00  2  34.93  35  103.71  104  4.00  4  60.87  61  207.45  208 
59  1.00  1  20.99  21  54.95  55  2.00  2  33.98  34  109.92  110 
Sato–Tate statistics for Klein twists over degree one primes \(p\le 2^{26}\)
\(\#\)  \(\mathrm {ST}(C_k)\)  \(\mathrm {ST}(C_{kM})\)  

\(\overline{\mathrm {M }}_{101}\)  \(\mathrm {M }_{101}\)  \(\overline{\mathrm {M }}_{030}\)  \(\mathrm {M }_{030}\)  \(\overline{\mathrm {M }}_{202}\)  \(\mathrm {M }_{202}\)  \(\overline{\mathrm {M }}_{101}\)  \(\mathrm {M }_{101}\)  \(\overline{\mathrm {M }}_{030}\)  \(\mathrm {M }_{030}\)  \(\overline{\mathrm {M }}_{202}\)  \(\mathrm {M }_{202}\)  
1  26.99  27  620.78  621  2366.04  2367  53.99  54  1214.67  1215  4732.50  4734 
2  12.99  13  318.73  319  1185.85  1187  25.98  26  610.52  611  2371.92  2374 
3  11.99  12  308.67  309  1192.59  1194  11.99  12  308.67  309  1192.59  1194 
4  8.99  9  215.76  216  787.97  789  17.98  18  404.55  405  1576.08  1578 
5  9.00  9  206.95  207  788.79  789  18.00  18  404.94  405  1577.71  1578 
6  7.00  7  154.90  155  596.58  597  14.00  14  308.88  309  1193.45  1194 
7  6.99  7  167.80  168  596.13  597  13.99  14  308.63  309  1192.36  1194 
8  4.01  4  103.36  103  399.55  398  4.01  4  103.36  103  399.55  398 
9  3.99  4  111.68  112  396.63  398  7.98  8  205.37  206  793.34  796 
10  5.99  6  113.83  114  365.24  366  11.99  12  200.67  201  730.55  732 
11  3.00  3  85.94  86  301.73  302  6.00  6  157.89  158  603.51  604 
12  2.01  2  55.11  55  206.43  206  2.01  2  55.11  55  206.43  206 
13  2.00  2  37.97  38  121.81  122  4.00  4  66.94  67  243.65  244 
14  1.00  1  11.94  12  25.70  26  2.00  2  18.87  19  51.40  52 
Proof
Tables 7 and 8 show Sato–Tate statistics for the Fermat and Klein twists C / k and their base changes \(C_{kM}\). In each row, we list moment statistics \(\overline{\mathrm {M }}_{101}, \overline{\mathrm {M }}_{030}, \overline{\mathrm {M }}_{202}\) for the three moments \(\mathrm {M }_{101}, \mathrm {M }_{030}, \mathrm {M }_{202}\) that uniquely determine the Sato–Tate group \(\mathrm {ST}(C)\), by Proposition 3.13. These were computed by averaging over all good primes of degree one and norm \(p\le 2^{26}\).
For comparison, we also list the actual value of each moment, computed using the method described in Sect. 3.2.2. In every case, the moment statistics agree with the corresponding moments of the Sato–Tate groups to within 1.5 percent, and in almost all cases, to within 0.5 percent.
4 Tables
In this final section, we present tables of characters, curves, Sato–Tate distributions, and moment statistics referred to elsewhere in this article. Let us briefly describe their contents.
Table 3 lists characters of the automorphism groups of the Fermat and Klein quartics specified via conjugacy class representatives expressed using the generators s, t, u defined in (3.2) and (3.3).
Tables 4 and 5 list explicit curve equations for twists C of the Fermat and Klein quartics corresponding to subgroups H of \(G_{C^0}:={\text {Aut}}(C^0_{M})\rtimes \mathrm {Gal}(M/\mathbb Q)\), as described in Remark 3.10. The group \(H_0:=H\cap {\text {Aut}}(C^0_{M})\) is specified in terms of the generators s, t, u listed in (3.2) and (3.3). When \(kM=k\) we have \(H=H_0\), and otherwise H is specified by listing an element \(h\in {\text {Aut}}(C_M^0)\) for which \(H=H_0\cup H_0\cdot (h,\tau )\), where \(\mathrm {Gal}(M/\mathbb Q)=\langle \tau \rangle \); see (3.6). The isomorphism classes of H and \(H_0\) are specified by GAP identifiers \(\mathrm {ID}(H)\) and \(\mathrm {ID}(H_0)\). The minimal field K over which all endomorphisms of \({\text {Jac}}(C_{\overline{\mathbb Q}})\) are defined is given as an explicit extension of \(\mathbb Q\), or as the splitting field \(\mathrm {Gal}(f(x))\) of a monic \(f\in \mathbb Z[x]\). In the last 2 columns of Tables 4 and 5 we identify the Sato–Tate distributions of \(\mathrm {ST}(C)\) and \(\mathrm {ST}(C_{kM})\) by their row numbers in Table 6. Among twists with the same Sato–Tate group \(\mathrm {ST}(C)\) (which is uniquely identified by its distribution), we list curves with isogenous Jacobians, per Remark 3.31.
In Table 6, we list the 60 Sato–Tate distributions that arise among twists of the Fermat and Klein quartics. Each component group is identified by its GAP ID, and we list the joint moments \(\mathrm {M }_{101}\), \(\mathrm {M }_{030}\), \(\mathrm {M }_{202}\) sufficient to uniquely determine the Sato–Tate distribution, along with the first two nontrivial independent coefficient moments for \(a_1,a_2,a_3\). We also list the proportion \(z_{i,j}\) of components on which the coefficient \(a_i\) takes the fixed integer value j; for \(i=1,3\) we list only \(z_1:=z_{1,0}\) and \(z_3:=z_{3,0}\), and for \(i=2\) we list the vector \(z_2:=[z_{2,1},z_{2,0},z_{2,1},z_{2,2},z_{2,3}]\); see Lemma 3.15 for details. There are 6 pairs of Sato–Tate distributions whose independent coefficient measures coincide; these pairs are identified by roman letters that appear in the last column.
Tables 7 and 8 list moment statistics for twists of the Fermat and Klein quartics computed over good primes \(p\le 2^{26}\), along with the corresponding moment values. Twists with isogenous Jacobians necessarily have the same moment statistics, so we list only one twist in each isogeny class.
Acknowledgements
We thank Josep González for his help with Lemma 3.18, and we are grateful to the Banff International Research Station for hosting a May 2017 workshop on Arithmetic Aspects of Explicit Moduli Problems where we worked on this article. Fité is grateful to the University of California at San Diego for hosting his visit in spring 2012, the period in which this project was conceived. Fité received financial support from the German Research Council (CRC 701), the Excellence Program María de Maeztu MDM20140445, and MTM201563829P. Sutherland was supported by NSF Grants DMS1115455 and DMS1522526. This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant Agreement No. 682152), and from the Simons Foundation (Grant #550033).
Using Gassmann triples, one can construct examples (of large dimension) where the converse of the first implication in (1.3) also fails to hold, but we will not pursue this here.
When we need not specify the number field k over which C is defined, we will simply say that C is a twist of \(C^0\). Thus, by saying that C is a twist of \(C^0\), we do not necessarily mean that C is defined over \(\mathbb Q\).
Note that this does not hold for the hyperelliptic curves considered in [12] where \([L:\! K]\) may be 1 or 2.
The objective of this test is not to prove anything, it is simply a mechanism for catching mistakes, of which we found several; most were our own, but some were due to minor errors in the literature, and at least one was caused by a defect in one of the computer algebra systems we used.
Declarations
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Authors’ Affiliations
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