ON PAIRS OF 17-CONGRUENT ELLIPTIC CURVES

T.A. FISHER

Abstract. We compute explicit equations for the surfaces Z(17, 1) and Z(17, 3) parametrising pairs of 17-congruent elliptic curves. We find that each is a double cover of the same elliptic K3-surface. We use these equations to exhibit the first non-trivial example of a pair of symplectically 17-congruent elliptic curves over the rationals. We also compute the corresponding genus 2 curve whose Jacobian has a (17, 17)-splitting.

1. Introduction

Let p be a prime number. Elliptic curves over the rationals are said to be p-congruent if their p-torsion subgroups are isomorphic as Galois modules, and symplectically p-congruent if the isomorphism can be chosen to respect the Weil pairing. For example if ϕ : E → E′ is an isogeny of degree d, and d is coprime to p, then E and E′ are p-congruent, and symplectically p-congruent if d is a quadratic residue mod p. Such congruences, arising from an isogeny, are said to be trivial.

Examples of non-trivial symplectic p-congruences were previously known for all primes p ≤ 13. We exhibit the first such example with p = 17. Specifically, the elliptic curves

E_{1} : y^{2} + xy = x^{3} − x^{2} − 128973503459x + 17827877649739965
E_{2} : y^{2} + xy = x^{3} − x^{2} − 184201215542543714x − 34187608332483214491862380

with conductors

N(E_{1}) = 279809270 = 2 · 5 · 13 · 59 · 191^{2},
N(E_{2}) = 3077901970 = 2 · 5 · 11 · 13 · 59 · 191^{2},

are symplectically 17-congruent. This claim may be verified using either of the techniques we review in Sections 2 and 3.

A pair of anti-symplectically 17-congruent elliptic curves was previously found by Cremona [B, CF, F1]. These are the elliptic curves

E′_{1} :  y^{2} + xy = x^{3} − 8x + 27
E′_{2} :  y^{2} + xy = x^{3} + 8124402x − 11887136703

Date: 3rd June 2021.

with conductors

N(E′_{1}) = 3675 = 3 · 5^{2} · 7^{2},
N(E′_{2}) = 47775 = 3 · 5^{2} · 7^{2} · 13.

In [F3] we completed the proof that for all primes p ≤ 13 there are infinitely many non-trivial pairs of p-congruent elliptic curves (with infinitely many pairs of j-invariants) both symplectic and anti-symplectic. The Frey-Mazur conjecture predicts that for p sufficiently large, there are no such examples. We suggest the following strong form of their conjecture.

Conjecture 1.1. Let p ≥ 17 be a prime. Then any pair of p-congruent elliptic curves over the rationals is either explained by an isogeny, or the elliptic curves are simultaneous quadratic twists of one of the pairs (E_{1}, E_{2}) or (E′_{1}, E′_{2}).

We say that a p-congruence has power k if the isomorphism of p-torsion subgroups raises the Weil pairing to the power k. Let Z(p, k) be the surface parametrising all pairs of elliptic curves that are p-congruent with power k, up to simultaneous quadratic twist. This surface comes with an involution ι whose moduli interpretation is that we swap over the two elliptic curves. We write W (p, k) for the quotient of Z(p, k) by ι. These surfaces only depend (up to isomorphism) on whether k is a quadratic residue or a quadratic non-residue mod p. As above, we call these the symplectic and anti-symplectic cases.

For p ≤ 13 it is known [F2, F3, Kum] that the surfaces W (p, k) are rational (i.e., birational to ℙ^{2}) over ℚ.

Theorem 1.2. The surfaces W (17, 1) and W (17, 3) are birational over ℚ to the elliptic K3-surface with Weierstrass equation

(1)  y^{2} + (T + 1)(T − 2)xy + T^{3}y = x^{3} − x^{2}.

The surfaces Z(17, 1) and Z(17, 3) are birational over ℚ to the double covers z^{2} = F_{1}(T, x, y) and z^{2} = F_{3}(T, x, y) where F_{1} and F_{3} are recorded in Appendix A.

Let j_{1} and j_{2} be the rational functions on Z(p, k) giving the j-invariants of the two elliptic curves. Then j_{1} + j_{2} and j_{1}j_{2} are rational functions on W (p, k). In the cases (p, k) = (17, 1) and (17, 3) we have also computed these rational functions. The formulae are too complicated to record here, but are available electronically from [F4].

We used these formulae to find the pairs of elliptic curves (E_{1}, E_{2}) and (E′_{1}, E′_{2}) specified above. In each case the curves are not isogenous, since for example they do not have the same conductor. The only previous method for finding such examples was to search in tables of elliptic curves with small conductor. (See for example [CF, Section 3] or [BM, Section 4.3].) Accordingly only the second pair was previously known.

Our evidence for Conjecture 1.1 when p = 17 is that we searched for further rational points on Z(17, 1) and Z(17, 3), but none of the points we found give rise to new pairs of 17-congruent elliptic curves. Conjecture 1.1 has also been verified by Cremona and Freitas [CF, Theorem 1.3 and Section 3.7] for all pairs of elliptic curves with conductor less than 500 000.

We have no theoretical explanation for our observation that the surfaces W (17, 1) and W (17, 3) are birational. It would of course be interesting to find one. We note that a wealth of information about the complex geometry of the surfaces Z(n, k) was computed by Kani and Schanz [KS]. In particular the surfaces Z(17, 1) and Z(17, 3) are surfaces of general type with geometric genus 10. We do not expect that these surfaces are birational.

In Section 2 we verify the 17-congruences claimed above by comparing traces of Frobenius mod 17. In Section 3 we compute a genus 2 curve whose Jacobian is isogenous to E_{1} × E_{2}, and note that this gives another proof that E_{1} and E_{2} are 17-congruent. We construct our birational models for Z(17, 1) and Z(17, 3) as quotients of X(17) × X(17). In Section 4 we give explicit equations for X(17), and in Sections 5 and 6 we compute the quotients in the symplectic and antisymplectic cases. In the final two sections we describe some of the interesting curves and points that we have so far found on these surfaces.

2. Verification via modularity

In [Ma, p.133] Mazur asked whether there are any non-trivial symplectic ncongruences for any integer n ≥ 7. The question was answered by Kraus and Oesterlé [KO] who gave the example of the pair of symplectically 7-congruent elliptic curves 152a1 and 7448e1. (We use the subsequent labelling of these curves in Cremona’s tables.) They also established the following results.

Lemma 2.1. [KO, Proposition 2] Let p be a prime number. Let E and E′ by pcongruent elliptic curves over ℚ, with minimal discriminants Δ and Δ′. Suppose that E and E′ have multiplicative reduction at a prime ℓ ≠ p and that the exponent v_{ℓ}(Δ) is coprime to p. Then v_{ℓ}(Δ′) is coprime to p, and the p-congruence is symplectic if and only if the ratio v_{ℓ}(Δ)/v_{ℓ}(Δ′) is a square mod p.

Lemma 2.2. [KO, Proposition 4] Let E and E′ be modular elliptic curves over ℚ with conductors N and N′. Let S be the set of primes for which one of the curves has split multiplicative reduction, and the other has non-split multiplicative reduction. Let M = lcm(N, N′) ∏_{ℓ∈S} ℓ and

μ(M) = [SL_{2}(ℤ) : Γ_{0}(M)] = #ℙ^{1}(ℤ/Mℤ) = M ∏_{ℓ|M}(1 + ℓ^{−1}).

Then the following conditions are equivalent.

(i) The Galois modules E[p] and E′[p] have isomorphic semi-simplifications. (ii) a_{ℓ}(E) ≡ a_{ℓ}(E′) (mod p) for all primes ℓ < μ(M)/6 with v_{ℓ}(NN′) = 0; and

a_{ℓ}(E)a_{ℓ}(E′) ≡ ℓ+1 (mod p) for all primes ℓ < μ(M)/6 with v_{ℓ}(NN′) = 1.

Since X_{0}(17) is a rank 0 elliptic curve, there are only finitely many j-invariants of elliptic curves over ℚ admitting a rational 17-isogeny. As noted in [C, Section 3.8], the exceptional j-invariants are −17^{2} · 101^{3}/2 and −17 · 373^{3}/2^{17}. Ignoring these two j-invariants, the conclusion of Lemma 2.2(i), when p = 17, is that E and E′ are 17-congruent.

Example 2.3. Let E_{1} and E_{2} be the elliptic curves defined in the introduction. For the primes ℓ < 50 the traces of Frobenius are as follows.

\begin{matrix}ℓ & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 & 43 & 47 \\ a_{ℓ}(E_{1}) & −1 & 0 & 1 & −1 & −5 & 1 & 2 & 4 & −1 & −3 & −2 & −11 & 5 & −4 & −9 \\ a_{ℓ}(E_{2}) & −1 & 0 & 1 & −1 & 1 & 1 & 2 & 4 & −1 & −3 & −2 & 6 & −12 & −4 & −9 \end{matrix}

In the notation of Lemma 2.2 we have S = ∅ and M = N(E_{2}). It takes about three hours^{1} to verify that a_{ℓ}(E_{1}) ≡ a_{ℓ}(E_{2}) (mod 17) for all primes ℓ < μ(M)/6 ≈ 1.033 × 2^{30} with ℓ ≠ 11. This shows that E_{1} and E_{2} are 17-congruent. Since

Δ(E_{1}) = 2^{3} · 5^{3} · 13 · 59^{2} · 191^{3},
Δ(E_{2}) = −2^{14} · 5^{11} · 11^{17} · 13 · 59 · 191^{9},

it follows by Lemma 2.1 (with ℓ = 2, 5, 13 or 59) that the congruence is symplectic.

Example 2.4. Let E′_{1} and E′_{2} be the elliptic curves defined in the introduction. For the primes ℓ < 50 the traces of Frobenius are as follows.

\begin{matrix}ℓ & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 & 43 & 47 \\ a_{ℓ}(E′_{1}) & −1 & 1 & 0 & 0 & 0 & −3 & 2 & −1 & −2 & −8 & 8 & −7 & 0 & 8 & −10 \\ a_{ℓ}(E′_{2}) & −1 & 1 & 0 & 0 & 0 & 1 & 2 & −1 & −2 & 9 & −9 & 10 & 0 & 8 & 7 \end{matrix}

In the notation of Lemma 2.2 we have S = ∅ and M = N(E′_{2}). It takes a fraction of a second to verify that a_{ℓ}(E_{1}) ≡ a_{ℓ}(E_{2}) (mod 17) for all primes ℓ < μ(M)/6 = 15680 with ℓ ≠ 13. This shows that E′_{1} and E′_{2} are 17-congruent. Since

Δ(E′_{1}) = −3^{5} · 5^{2} · 7^{2},
Δ(E′_{2}) = −3^{2} · 5^{2} · 7^{2} · 13^{17},

it follows by Lemma 2.1 (with ℓ = 3) that the congruence is anti-symplectic.

1Running Magma on a single core of the author’s desktop.

Remark 2.5. (i) It may be possible to reduce the Sturm bound, and hence the runtime, in Example 2.3 by using a result similar to [St, Theorem 9.21] or by using level lowering. We did not pursue this. (ii) Methods for determining the symplectic type in situations where Lemma 2.1 does not apply have recently been studied in [CF, FKr]. (iii) The existence of a prime ℓ for which Lemma 2.1 applies, together with the Weil pairing and the fact our elliptic curves do not admit a rational 17-isogeny, is already enough (see [Se, Chapter IV, Section 3.2]) to show that in each case the mod 17 Galois representation is surjective.

3. Verification via genus 2 Jacobians

Let E_{1} and E_{2} be n-congruent elliptic curves over ℚ, where the congruence ψ reverses the sign of the Weil pairing. Then the quotient J of E_{1} × E_{2} by the graph of ψ is a principally polarised abelian surface. It is shown in [FKa, Section 1] that if n is odd and E_{1} and E_{2} are not isogenous, then J is the Jacobian of a genus 2 curve C defined over ℚ, and there are degree n morphisms C → E_{1} and C → E_{2}, also defined over ℚ. For further details of this construction of reducible genus 2 Jacobians, see for example [BHLS, BD, FKa, Kum, Kuh, Sh].

Since −1 is a quadratic residue mod 17, the elliptic curves E_{1} and E_{2} defined in the introduction are of the form considered in the last paragraph. In this section we compute the corresponding genus 2 curve, and note that this gives another proof that E_{1} and E_{2} are 17-congruent.

Lemma 3.1. Let E_{1} = ℂ/(ℤ+τ_{1}ℤ) and E_{2} = ℂ/(ℤ+τ_{2}ℤ) with Im(τ_{1}), Im(τ_{2}) > 0. Let ψ : E_{1}[n] → E_{2}[n] be the isomorphism given by \frac{1}{n}(r + sτ_{1}) ↦ \frac{1}{n}(r − sτ_{2}) for r, s = 0, 1, …, n − 1. (Note the minus sign!) Then the quotient J of E_{1} × E_{2} by the graph of ψ is represented in the Siegel upper half-space by

τ = (\begin{matrix}nτ_{1} & τ_{1} \\ τ_{1} & (τ_{1} + τ_{2})/n \end{matrix}) .

Proof. We have J ≅ ℂ^{2}/Λ where Λ is the lattice spanned by the columns b_{1}, …, b_{4} of the matrix

(\begin{matrix}τ_{1} & τ_{1}/n & 1/n & 0 \\ 0 & −τ_{2}/n & 1/n & −1 \end{matrix}) .

The principal polarisation on J is given by the Hermitian Riemann form

H((z_{1}, z_{2}), (w_{1}, w_{2})) = n (\frac{z_{1}\overline{w}_{1}}{Im(τ_{1})} + \frac{z_{2}\overline{w}_{2}}{Im(τ_{2})}) ,

whose imaginary part is given with respect to the basis b_{1}, …, b_{4} for Λ by

(\begin{matrix}0 & I_{2} \\ −I_{2} & 0 \end{matrix}) .

We therefore take

τ = (b_{3}|b_{4})^{−1}(b_{1}|b_{2}) = (\begin{matrix}n & 0 \\ 1 & −1 \end{matrix}) (\begin{matrix}τ_{1} & τ_{1}/n \\ 0 & −τ_{2}/n \end{matrix}) = (\begin{matrix}nτ_{1} & τ_{1} \\ τ_{1} & (τ_{1} + τ_{2})/n \end{matrix}) . □

The elliptic curves E_{1} and E_{2} defined in the introduction are represented in the upper half-plane by τ_{1} ≈ 0.1142862335i and τ_{2} ≈ 0.5000000000+1.415897663i. We computed τ in the Siegel upper half-space corresponding to our genus 2 Jacobian by applying Lemma 3.1 (with n = 17) to τ_{1} and τ′_{2} = (64τ_{2} −15)/(−17τ_{2} +4). The formula for τ′_{2} had to be guessed, but since the congruence must respect complex conjugation there were only 8 possibilities to try. We then used the methods described by van Wamelen [W] to compute the Igusa-Clebsch invariants of our genus 2 curve to 300 decimal digits of precision. Recognising these as rational numbers, we next used the method of Mestre [Me] to find a genus 2 curve over ℚ with these invariants. Up to quadratic twist, this gave the genus 2 curve C with equation y^{2} = f_{1}(x)f_{2}(x) where

f_{1}(x) = 196081931x^{3} + 1143338037x^{2} − 801791940x + 135616700,
f_{2}(x) = −25996x^{3} + 1698260x^{2} − 6845267x + 3822078.

We chose this particular quadratic twist since it satisfies # Jac(C)(𝔽_{p}) = #E_{1}(𝔽_{p})· #E_{2}(𝔽_{p}) for many primes p of good reduction.

To prove that our equation for C is correct (without relying on the numerical approximations in the last paragraph) we also computed the degree 17 morphisms ϕ_{1} : C → E_{1} and ϕ_{2} : C → E_{2}. The x-coordinate of ϕ_{i} is given by ξ_{i}(x) = h_{i}(x)/(f_{i}(x)g_{i}(x)^{2}) where g_{i} and h_{i} are certain polynomials of degrees 7 and 17. Working mod p = 101 we find

g_{1}(x) = 25x^{7} + 56x^{6} + 31x^{5} + 99x^{4} + 100x^{3} + 42x^{2} + 79x + 5,
g_{2}(x) = 3x^{7} + 76x^{6} + 44x^{5} + 97x^{4} + 52x^{3} + 38x^{2} + 75x + 2,
h_{1}(x) = 16x^{17} + 6x^{16} + 57x^{15} + 54x^{14} + 94x^{13} + 79x^{12} + 77x^{11} + 55x^{10}

+ 74x^{9} + 78x^{8} + 97x^{7} + 79x^{6} + 25x^{5} + 96x^{4} + 98x^{3} + 46x^{2} + 4x + 99,

h_{2}(x) = 67x^{17} + 25x^{16} + x^{15} + 22x^{14} + 84x^{13} + 94x^{12} + 93x^{11} + 95x^{10}

+ 34x^{9} + 40x^{8} + 99x^{7} + 84x^{6} + 43x^{5} + 12x^{4} + 59x^{3} + 13x^{2} + 26x + 98.

The full expressions for g_{1}, g_{2}, h_{1}, h_{2} ∈ ℤ[x] may be found in [F4]. We do not record these here, since some of the coefficients have nearly 100 decimal digits.

Our method to compute these polynomials was to compute them mod p for many primes p and then use the Chinese remainder theorem. To compute them mod p we looped over all possibilities for the map C(𝔽_{p}) → E_{i}(𝔽_{p}), compatible with the group laws on the Jacobians, and then solved for the rational function ξ_{i} (with numerator and denominator of degree at most 17) by interpolation.

The y-coordinates of the maps ϕ_{i} : C → E_{i} are of course even more complicated to write down. However, a convenient alternative to recording these directly is to note that the invariant differentials on E_{1} and E_{2} pull back to the following “elliptic differentials” on C:

ϕ_{1}^{∗} (\frac{dx}{2y + x}) = \frac{(273857x − 336364)dx}{y},
ϕ_{2}^{∗} (\frac{dx}{2y + x}) = \frac{(2758x + 1630)dx}{y}.

Our second proof that E_{1} and E_{2} are 17-congruent is completed by the next lemma, which we record for convenience, but is essentially well known. Compared to the proof in Section 2, this proof takes a fraction of the computer time, since we only have to check that our formulae for ϕ_{1} and ϕ_{2} do indeed define morphisms C → E_{1} and C → E_{2}.

Lemma 3.2. Let C be a genus 2 curve and let p be a prime. Let ϕ_{1} : C → E_{1} and ϕ_{2} : C → E_{2} be morphisms of degree p, where E_{1} and E_{2} are non-isogenous elliptic curves. Then E_{1} and E_{2} are p-congruent.

Proof. Since E_{1}, E_{2} and J = Jac C are principally polarised abelian varieties, we identify them with their duals without further comment.

The map ϕ_{1} : C → E_{1} induces by pull back a map E_{1} → J. This map is injective since otherwise, by [BL, Proposition 11.4.3], ϕ_{1} would have to factor via a non-trivial isogeny of elliptic curves, which is not possible by our assumption that ϕ_{1} has prime degree. Since E_{1} and E_{2} are not isogenous, the composite of the maps E_{1} → J and J → E_{2} induced by ϕ_{1} and ϕ_{2} must be the zero map. The same observations apply with the roles of E_{1} and E_{2} swapped over. The pull back and push forward maps associated to ϕ_{1} and ϕ_{2} therefore define dual isogenies

E_{1} × E_{2} \underrightarrow{ϕ̂} J ⟶^{ϕ} E_{1} × E_{2}

whose composite is multiplication-by-p. In particular deg ϕ = deg ϕ̂ = p^{2} and there are isomorphisms of Galois modules E_{1}[p] ≅ J[ϕ] ≅ E_{2}[p]. □

4. The modular curve X(17)

Let ζ = e^{2πi/17} and ξ_{k} = ζ^{k} + ζ^{−k}. Let G ≅ PSL_{2}(ℤ/17ℤ) be the subgroup of SL_{9}(ℂ) generated by M_{2} and M_{17} where

M_{2} = \frac{−1}{\sqrt{17}} (\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & ξ_{3} & ξ_{8} & ξ_{7} & ξ_{4} & ξ_{5} & ξ_{2} & ξ_{6} & ξ_{1} \\ 2 & ξ_{8} & ξ_{7} & ξ_{4} & ξ_{5} & ξ_{2} & ξ_{6} & ξ_{1} & ξ_{3} \\ 2 & ξ_{7} & ξ_{4} & ξ_{5} & ξ_{2} & ξ_{6} & ξ_{1} & ξ_{3} & ξ_{8} \\ 2 & ξ_{4} & ξ_{5} & ξ_{2} & ξ_{6} & ξ_{1} & ξ_{3} & ξ_{8} & ξ_{7} \\ 2 & ξ_{5} & ξ_{2} & ξ_{6} & ξ_{1} & ξ_{3} & ξ_{8} & ξ_{7} & ξ_{4} \\ 2 & ξ_{2} & ξ_{6} & ξ_{1} & ξ_{3} & ξ_{8} & ξ_{7} & ξ_{4} & ξ_{5} \\ 2 & ξ_{6} & ξ_{1} & ξ_{3} & ξ_{8} & ξ_{7} & ξ_{4} & ξ_{5} & ξ_{2} \\ 2 & ξ_{1} & ξ_{3} & ξ_{8} & ξ_{7} & ξ_{4} & ξ_{5} & ξ_{2} & ξ_{6} \end{matrix})

and M_{17} = Diag(1, ζ, ζ^{9}, ζ^{13}, ζ^{15}, ζ^{16}, ζ^{8}, ζ^{4}, ζ^{2}). The pattern of subscripts in the definition of M_{2} is the sequence of powers of 3 in (ℤ/17ℤ)/{±1}.

We write ℂ[x_{0}, …, x_{8}]_{d} for the space of homogeneous polynomials of degree d. An invariant of degree d is a polynomial I ∈ ℂ[x_{0}, …, x_{8}]_{d} satisfying I ∘ g = I for all g ∈ G. In degrees 2 and 3 the only invariants are

Q = x_{0}^{2} + x_{1}x_{5} + x_{2}x_{6} + x_{3}x_{7} + x_{4}x_{8},
D = 2x_{0}(x_{1}x_{5} − x_{2}x_{6} + x_{3}x_{7} − x_{4}x_{8})
− x_{1}^{2}x_{4} + x_{2}^{2}x_{5} − x_{3}^{2}x_{6} + x_{4}^{2}x_{7} − x_{5}^{2}x_{8} + x_{6}^{2}x_{1} − x_{7}^{2}x_{2} + x_{8}^{2}x_{3}.

In degree 4 we have the invariants Q^{2} and

F = x_{0}^{4} + x_{0}(x_{1}^{2}x_{4} + x_{2}^{2}x_{5} + x_{3}^{2}x_{6} + x_{4}^{2}x_{7} + x_{5}^{2}x_{8} + x_{1}x_{6}^{2} + x_{2}x_{7}^{2} + x_{3}x_{8}^{2})

+ x_{1}x_{3}x_{5}x_{7} + x_{2}x_{4}x_{6}x_{8} + x_{1}x_{2}x_{5}x_{6} + x_{2}x_{3}x_{6}x_{7} + x_{3}x_{4}x_{7}x_{8} + x_{1}x_{4}x_{5}x_{8} + x_{1}^{2}x_{3}x_{8} + x_{1}x_{2}^{2}x_{4} + x_{2}x_{3}^{2}x_{5} + x_{3}x_{4}^{2}x_{6} + x_{4}x_{5}^{2}x_{7} + x_{5}x_{6}^{2}x_{8} + x_{1}x_{6}x_{7}^{2} + x_{2}x_{7}x_{8}^{2}.

Proposition 4.1. Let C ⊂ ℙ^{8} be the curve defined by the vanishing of Q and all partial derivatives of F. Then C = C_{1} ∪ C_{2} where C_{1} and C_{2} are curves of degrees 96 and 168, each isomorphic to the modular curve X(17). The 144 cusps on C_{1} are cut out (each with multiplicity 2) by the cubic form D. Moreover D vanishes identically on C_{2}.

Proof. Let p ≥ 5 be a prime. The group PSL_{2}(ℤ/pℤ) acts on X(p) with quotient the j-line, and the group of divisor classes fixed by this group action is an infinite cyclic group, generated by λ of degree (p^{2} − 1)/24. Let m = (p − 1)/2. Klein gave equations for X(p) embedded in ℙ^{m−1} and ℙ^{m} with hyperplane sections (m − 1)λ and mλ. Following [AR] we call these models the z-curve and the A-curve. See [AR, Section 24] or [F3, Section 4] for further details.

We take p = 17. Let z_{1}, …, z_{8} be coordinates on ℙ^{7}. We write z_{0} = 0, z_{−i} = −z_{i} and agree to read all subscripts mod 17. According to [F1, Section 2] the z-curve for X(17) is the curve in ℙ^{7} defined by the 4 by 4 Pfaffians of the 17 by 17 skew symmetric matrix (z_{i−j}z_{i+j}). We define maps ϕ_{i} : X(17) → ℙ^{8} for i = 1, 2 by

ϕ_{1} = (1 : \frac{z_{2}}{z_{1}} : \frac{z_{6}}{z_{3}} : \frac{−z_{1}}{z_{8}} : \frac{−z_{3}}{z_{7}} : \frac{z_{8}}{z_{4}} : \frac{−z_{7}}{z_{5}} : \frac{z_{4}}{z_{2}} : \frac{−z_{5}}{z_{6}} )

and

ϕ_{2} = (z_{1}z_{4} + z_{2}z_{8} + z_{3}z_{5} − z_{6}z_{7} − \frac{2z_{4}z_{7}z_{2}}{z_{1}}

: −z_{8}^{2} − \frac{2z_{5}z_{7}z_{2}}{z_{1}} : z_{7}^{2} − \frac{2z_{2}z_{4}z_{6}}{z_{3}} : −z_{4}^{2} + \frac{2z_{5}z_{6}z_{1}}{z_{8}} : z_{5}^{2} + \frac{2z_{1}z_{2}z_{3}}{z_{7}}

: −z_{2}^{2} + \frac{2z_{3}z_{6}z_{8}}{z_{4}} : z_{6}^{2} − \frac{2z_{1}z_{8}z_{7}}{z_{5}} : −z_{1}^{2} − \frac{2z_{3}z_{7}z_{4}}{z_{2}} : z_{3}^{2} + \frac{2z_{4}z_{8}z_{5}}{z_{6}} ).

Let C_{1} and C_{2} be the images of ϕ_{1} and ϕ_{2}. We find using Magma [BCP] that Q, the partial derivatives of F, and the quartics x_{0}^{4} + x_{1}x_{3}x_{5}x_{7} and x_{0}^{4} + x_{2}x_{4}x_{6}x_{8} vanish on C_{1}. Likewise, Q, D, and the partial derivatives of F vanish on C_{2}. These equations are sufficient to define each curve set-theoretically. In fact, the homogeneous ideal of C_{1} is generated by one quadratic form, 9 cubic forms and 117 quartic forms, and the homogeneous ideal of C_{2} is generated by one quadratic form and 28 cubic forms. To prove the decomposition C = C_{1} ∪ C_{2} we checked that (x_{0}^{4} + x_{1}x_{3}x_{5}x_{7})D^{2} and (x_{0}^{4} + x_{2}x_{4}x_{6}x_{8})D^{2} belong to the ideal generated by Q and the partial derivatives of F.

Since Q and F are invariants, the group G acts on C, and hence on C_{1} and C_{2}. It is shown in [AR, Lemma 20.40] that for p ≥ 7 a prime, the curve X(p) has automorphism group PSL_{2}(ℤ/pℤ). So up to an automorphism of G, the Gactions on C_{1} and C_{2} correspond to the usual action of PSL_{2}(ℤ/17ℤ) on X(17). The points on X(17) above j = 0, 1728, ∞ form G-orbits of sizes 816, 1224, 144. All other G-orbits have size |G| = 2448. The intersection of C_{1} with {D = 0} has 3 × 96 = 288 points counted with multiplicity. Being preserved by the G-action, it must therefore be the set of cusps, each counted with multiplicity 2. □

Remark 4.2. The formula for ϕ_{1} (up to signs and ordering) is that given in [AR, Section 51], and accordingly C_{1} is the A-curve. The formula for ϕ_{2} was found by using the G-actions to compute a complement to the image of S^{2}L(ζ) in L(2ζ), where L(δ) denotes the Riemann Roch space of a divisor δ, and ζ ∼ 7λ is the hyperplane section for the z-curve.

Definition 4.3. A covariant of degree d is a column vector v of polynomials in ℂ[x_{0}, …, x_{8}]_{d} satisfying v ∘ g = gv for all g ∈ G.

Starting from an invariant I of degree d we may construct a covariant of degree d − 1 as

(2)  ∇_{Q}I = H(Q)^{−1} (∂I/∂x_{0} ; ⋮ ; ∂I/∂x_{8})

where H(Q) is the 9 by 9 matrix of second partial derivatives of Q. Going in the other direction, if v and w are covariants of degrees d and e then

(3)  v · w := v^{T} H(Q)w = coeff(Q(v + tw), t)

is an invariant of degree d + e. If we think of a covariant as a G-equivariant polynomial map ℂ^{9} → ℂ^{9} then the composition of covariants v and w of degrees d and e is a covariant v ∘ w of degree de.

We put v_{1} = (x_{0}, …, x_{8})^{T}, v_{2} = ∇_{Q}D, v_{3} = ∇_{Q}F, v_{4} = v_{2}∘v_{2} and v_{6} = v_{3}∘v_{2}. Then c_{4} = v_{4} · v_{6} is an invariant of degree 10.

Lemma 4.4. Let X = X(17) be the curve denoted C_{1} in Proposition 4.1. Then the j-map X → ℙ^{1} is given by j = −2^{7}c_{4}^{3}/D^{10}.

Proof. The calculation at the end of this proof shows that c_{4} does not vanish identically on X. So the intersection of X with {c_{4} = 0} is a set of 10 × 96 = 816 + 144 points. Arguing as in the proof of Proposition 4.1, this set is the union of the points above j = 0 and j = ∞. Our formula for the j-invariant therefore has the correct divisor, and so is correct up to scaling.

Let u = (θ^{3} − θ^{2} − θ + 2i − 1)/4 where θ = \sqrt[4]{1 − 4i} and i = \sqrt{−1}. Let σ be the generator for Gal(ℚ(θ)/ℚ(i)) given by σ(θ) = iθ. The point

(1 : u : σ(u) : σ^{2}(u) : σ^{3}(u) : u : σ(u) : σ^{2}(u) : σ^{3}(u)) ∈ X

is fixed by a permutation matrix of order 2 in G, and so lies above j = 1728. The function c_{4}^{3}/D^{10} takes the value −1728/2^{7} at this point. □

5. Computations in the symplectic case

Let X = X(17) ⊂ ℙ^{8} be the curve denoted C_{1} in Proposition 4.1. By [F3, Lemma 3.2] the surface Z(17, 1) is birational to the quotient of X × X ⊂ ℙ^{8} × ℙ^{8} by the diagonal action of G ≅ PSL_{2}(ℤ/17ℤ). We write x_{0}, …, x_{8} and y_{0}, …, y_{8} for our coordinates on the first and second copies of ℙ^{8}.

Definition 5.1. A bi-invariant of degree (m, n) is a polynomial in x_{0}, …, x_{8} and y_{0}, …, y_{8}, that is homogeneous of degrees m and n in the two sets of variables, and is invariant under the diagonal action of G.

In principle we may find equations for Z(17, 1) by computing generators and relations for the ring of bi-invariants mod I(X × X). In practice we find it is sufficient to compute only some of the generators and some of the relations.

The calculations that follow rely on showing that certain bi-invariants vanish identically on X ×X. Initially we only checked that they vanish at many 𝔽_{p}-points for some moderately large prime p. For a full proof in characteristic 0 we used the G-action and numerical approximations to verify the conditions in the following lemma. This is sufficient since if the absolute value of the norm of an algebraic integer is less than one, then it must be zero.

Lemma 5.2. Let I be a bihomogeneous form of degree (m, n) with m, n ≤ 22. If I vanishes at all points (P, Q) ∈ X × X with j(P), j(Q) ∈ {0, 1728, ∞} then I vanishes on X × X.

Proof. This follows from Bezout’s theorem, using that 144 + 816 + 1224 > 22 × 96. The argument is then identical to that used in the proof of [F3, Lemma 6.2]. □

It is easy to compute the dimension of the space of bi-invariants of any given degree from the character table of G. However we need to work with explicit bases for these spaces. Let Q, D and F be the invariants of degrees 2, 3 and 4 defined in Section 4. We define bi-invariants Q_{ij}, D_{ij} and F_{ij} by the rules

Q(λx_{0} + μy_{0}, …, λx_{8} + μy_{8}) = λ^{2}Q_{20} + λμQ_{11} + μ^{2}Q_{02},
D(λx_{0} + μy_{0}, …, λx_{8} + μy_{8}) = λ^{3}D_{30} + λ^{2}μD_{21} + … + μ^{3}D_{03},
F (λx_{0} + μy_{0}, …, λx_{8} + μy_{8}) = λ^{4}F_{40} + λ^{3}μF_{31} + … + μ^{4}F_{04}.

Then, writing H for the 9 by 9 matrix of second partial derivatives with respect to x_{0}, …, x_{8}, we put

D_{x} = H(Q)^{−1}H(D_{30}), D_{y} = H(Q)^{−1}H(D_{21}), F_{xy} = H(Q)^{−1}H(F_{31}).

The space of bi-invariants of degree (2, 2) has dimension 4, with basis A_{1}, …, A_{4} where A_{1} = Q_{20}Q_{02}, A_{2} = Q_{11}^{2}, A_{3} = F_{22}, and

tr(D_{x}D_{y}F_{xy}) = −16A_{1} + 8A_{2} − 8A_{4}.

Each of these bi-invariants is symmetric (under interchanging the x’s and y’s), but only the first vanishes identically on X × X.

The space of bi-invariants of degree (3, 3) has dimensional 16. Under interchanging the x’s and y’s, this breaks up as the direct sum of symmetric and skew-symmetric subspaces of dimensions 14 and 2. The subspace of symmetric bi-invariants has basis B′_{1}, …, B′_{14} given by

Q_{11}A_{2}, Q_{11}A_{3}, Q_{11}A_{4}, D_{30}D_{03}, D_{21}D_{12}, tr(D_{x}^{3}D_{y}^{3}), tr(D_{x}^{2}D_{y}D_{x}D_{y}^{2}),
tr(D_{x}^{2}D_{y}^{2}F_{xy}), tr(D_{x}D_{y}D_{x}D_{y}F_{xy}), tr(D_{x}D_{y}F_{xy}^{2}), tr(F_{xy}^{3}),
Q_{11}A_{1}, Q_{20}F_{13} + Q_{02}F_{31}, 2∇_{Q}F (x_{0}, …, x_{8}) · ∇_{Q}F (y_{0}, …, y_{8}),

where in the final expression (for B′_{14}) we use the notation (2) and (3).

We changed our choice of basis for this space of bi-invariants first so that the bi-invariants themselves have small integer coefficients, then so that the relations considered below have small integer coefficients, and finally to facilitate writing down an elliptic fibration. To simplify the calculations that follow, we therefore (with the benefit of hindsight) switch to the basis B_{1}, …, B_{14} that is related to the basis B′_{1}, …, B′_{14} (as specified in the last paragraph) by the first change of basis matrix recorded in Appendix A. In fact we keep the first and last three basis elements the same, i.e. B_{i} = B′_{i} for i = 1, 2, 3, 12, 13, 14.

The subspace of bi-invariants vanishing on X × X is spanned by B_{12}, B_{13}, B_{14}. We write I_{1} ⊂ ℚ[z_{1}, …, z_{11}] for the ideal generated by all quadratic and cubic forms vanishing on the image of the map

X × X → ℙ^{10} ; (x_{0}, …, x_{8}; y_{0}, …, y_{8}) ↦ (B_{1} : …: B_{11}).

We find that I_{1} is minimally generated by 13 quadratic forms and 21 cubic forms. Moreover the subvariety Σ_{1} ⊂ ℙ^{10} defined by I_{1} is a surface of degree 29.

Proposition 5.3. The surface Σ_{1} is birational over ℚ to the elliptic surface Σ defined by the Weierstrass equation (1) in the statement of Theorem 1.2.

Proof. The rational map Σ_{1} → ℙ^{3} × 𝔸^{1} given by

(z_{1} : …: z_{11}) ↦ ((z_{1} : z_{2} : z_{3} : z_{4}), T) = ((z_{1} : z_{2} : z_{3} : z_{4}), z_{5}/z_{6})

has image satisfying

2T z_{1}^{2} + 3T z_{1}z_{2} − (5T − 2)z_{1}z_{3} − 3T^{2}z_{1}z_{4} + T z_{2}^{2} − 4T z_{2}z_{3} − 2T^{2}z_{2}z_{4}
+ 2(2T − 1)z_{3}^{2} + T (4T − 1)z_{3}z_{4} + T^{3}z_{4}^{2} = 0,
(T + 1)^{2}z_{1}^{2} + T (2T + 1)z_{1}z_{2} − (T + 1)^{2}z_{1}z_{3} − T (T + 1)^{2}z_{1}z_{4} + T^{2}z_{2}^{2}
− 2T^{2}z_{2}z_{3} − T^{3}z_{2}z_{4} = 0.

These same equations define a genus one curve in ℙ^{3} over the function field ℚ(T). Making the linear change of coordinates

u_{1} = T (T + 1)z_{1},  u_{3} = (T + 1)z_{3},
u_{2} = T (z_{1} + z_{2} − 2z_{3} − T z_{4}),  u_{4} = T (−z_{1} + 2z_{3} + T z_{4}),

gives the simplified quadric intersection

u_{1}u_{2} + u_{1}u_{3} + (T + 1)u_{2}^{2} − u_{3}u_{4} = 0,
u_{1}u_{4} + T u_{2}^{2} − T^{2}u_{2}u_{4} − T u_{3}u_{4} = 0,

which in turn is isomorphic to the elliptic curve (1) via

x = \frac{−T u_{1}}{u_{4}} , y = \frac{T u_{1}(u_{1} − u_{2} − u_{4})}{u_{2}u_{4}}.

Composing these maps gives a birational map Σ_{1} → Σ. The inverse map, represented as an explicit 11-tuple of elements in the function field ℚ(Σ), is recorded in the accompanying computer file [F4]. □

Since the rational map X × X → Σ is defined by symmetric bi-invariants, it factors via a rational map π : W (17, 1) → Σ. We will see below that π is birational, thereby proving the first part of Theorem 1.2.

One way to compute equations for the double cover Z(17, 1) → W (17, 1) is to find a skew-symmetric bi-invariant of degree (3, 3) that does not vanish identically on X × X, and then write its square, modulo I(X × X), as a quadratic form in B_{1}, …, B_{11}. We omit the details, since this calculation is superceded by the calculation of the j-maps, which we do next.

We consider the symmetric bi-invariants α_{1} = D_{30}D_{03}, α_{2} = D_{12}D_{21}, α_{3} = D_{21}^{3}D_{03} + D_{12}^{3}D_{30}, α_{4} = c_{4}(x_{0}, …, x_{8})c_{4}(y_{0}, …, y_{8}) and

α_{5} = c_{4}(x_{0}, …, x_{8})D_{03}^{3}D_{12} + c_{4}(y_{0}, …, y_{8})D_{30}^{3}D_{21},

of degrees (m, m) for m = 3, 3, 6, 10, 11. Let S = ℚ[u, v, w, z_{4}, …, z_{11}] be the graded polynomial ring where the variables have weights 1, 2, 2, 3, …, 3. In the accompanying Magma file [F4] we record g_{1}, …, g_{5} ∈ S of weighted degrees 3, 3, 9, 15, 15 and h_{1}, …, h_{5} ∈ S of weighted degrees 0, 0, 3, 5, 4, such that each of the bi-invariants

g_{i}(Q_{11}, A_{3}, A_{4}, B_{4}, …, B_{11}) − h_{i}(Q_{11}, A_{3}, A_{4}, B_{4}, …, B_{11})α_{i}

vanishes on X × X. We use these expressions to solve for the α_{i} as elements of the function field ℚ(Σ). Then the polynomials f_{1}(Y) = Y^{2} − α_{3}Y + α_{1}α_{2}^{3} and f_{2}(Y) = Y^{2} − α_{5}Y + α_{1}^{3}α_{2}α_{4} have roots defined over the same quadratic extension of ℚ(Σ). Let these roots be r_{1}, s_{1} and r_{2}, s_{2}. If we order these roots appropriately then by Lemma 4.4, and the definition of the α_{i}, we have

j_{1} + j_{2} = \frac{−2^{7}}{α_{1}^{9}} (\frac{r_{2}^{3}}{r_{1}} + \frac{s_{2}^{3}}{s_{1}} )  and  j_{1}j_{2} = \frac{2^{14}α_{4}^{3}}{α_{1}^{10}} .

Let Σ̃ → Σ be the double cover defined by the requirement that disc f_{1}, disc f_{2}, or (j_{1} + j_{2})^{2} − 4j_{1}j_{2} is a square. Then the product of j-maps X → ℙ^{1} factors as

(4)  X × X ⟶ Z(17, 1) ⟶ Σ̃ ⟶ ℙ^{1} × ℙ^{1}.

The composite corresponds to a Galois extension of function fields, with Galois group G × G. Since G ≅ PSL_{2}(ℤ/17ℤ) is a simple group, the diagonal subgroup Δ_{G} ⊂ G × G is a maximal subgroup. Therefore one of the last two maps in (4) is birational. However if the last map were birational, then this would mean that in attempting to quotient out by Δ_{G}, we had in fact quotiented out by G × G. To exclude this possibility we may check, for example, that the rational function F_{22}/Q_{11}^{2} on X × X is not G × G-invariant.

In conclusion, Z(17, 1) is birational to Σ̃, and W (17, 1) is birational to Σ. This completes the proof of Theorem 1.2 in the symplectic case.

Remark 5.4. We initially hoped that we might compute W (17, 1) using the biinvariants of degree (2, 2), without needing those of degree (3, 3). In hindsight we see that this is not possible, since the map W (17, 1) → ℙ^{2} given by (A_{2} : A_{3} : A_{4}) is generically 5-to-1. This may be seen by eliminating z_{4} from the first quadric intersection in the proof of Proposition 5.3, and noting that the resulting quartic in z_{1}, z_{2}, z_{3} has degree 5 in T.

6. Computations in the anti-symplectic case

In Section 4 we defined G as the subgroup of SL_{9}(ℂ) generated by certain matrices with entries in ℚ(ζ), where ζ is a primitive 17th root of unity. Replacing each matrix entry by its image under the automorphism ζ ↦ ζ^{3} of ℚ(ζ) defines an outer automorphism g ↦ g̃ of G.

Definition 6.1. A skew bi-invariant of degree (m, n) is a polynomial in x_{0}, …, x_{8} and y_{0}, …, y_{8}, that is homogeneous of degrees m and n in the two sets of variables, and is invariant under the action of G via g : (x, y) ↦ (gx, g̃y).

Since the map g ↦ ˜ g̃ is an inner automorphism of G we see that if f is a skew bi-invariant, then so too is

f^{†}(x_{0}, …, x_{8}; y_{0}, …, y_{8}) := f(y_{0}, …, y_{8}; −x_{0}, −x_{2}, −x_{3}, …, −x_{8}, −x_{1}).

We have f^{††} = f.

The space of skew bi-invariants of degree (2, 2) has basis A_{1}, A_{2}, A_{3} where A_{1} = Q_{20}Q_{02} and

A_{2} = 4x_{0}^{2}y_{0}^{2} + ∑(2x_{0}x_{1}y_{3}y_{4} + 2x_{4}x_{5}y_{0}y_{1} + x_{1}^{2}y_{1}y_{7} + x_{2}x_{8}y_{1}^{2} + x_{1}x_{2}y_{2}y_{5}
+ x_{3}x_{6}y_{1}y_{2} + x_{1}x_{3}y_{5}y_{8} + x_{1}x_{6}y_{1}y_{3} + \frac{1}{2}(x_{1}x_{5}y_{2}y_{6} + x_{3}x_{7}y_{1}y_{5})),
A_{3} = 4x_{0}^{2}y_{0}^{2} + ∑(2x_{0}x_{1}y_{2}y_{8} + 2x_{1}x_{3}y_{0}y_{1} + x_{1}^{2}y_{2}y_{3} + x_{3}x_{4}y_{1}^{2} + x_{1}x_{2}y_{2}y_{5}
+ x_{3}x_{6}y_{1}y_{2} + x_{1}x_{3}y_{5}y_{8} + x_{1}x_{6}y_{1}y_{3} + \frac{1}{2}(x_{1}x_{5}y_{1}y_{5} + x_{2}x_{6}y_{1}y_{5})).

Here ∑ denotes the sum over all simultaneous cyclic permutations of x_{1}, …, x_{8} and y_{1}, …, y_{8} (fixing x_{0} and y_{0}). We have A_{i}^{†} = A_{i} for i = 1, 2, 3.

The space of skew bi-invariants of degree (3, 1) is 1-dimensional, spanned by

S_{31} = −16x_{0}^{3}y_{0} + ∑((3x_{0}x_{1}x_{5} + 3x_{1}^{2}x_{4})y_{0} + (6x_{0}x_{1}x_{3} + 6x_{0}x_{4}x_{5} + 3x_{2}^{2}x_{3}
+ 3x_{1}x_{4}^{2} + x_{5}^{3} + 3x_{4}x_{6}^{2} + 6x_{1}x_{2}x_{7} + 6x_{3}x_{5}x_{8} + 6x_{6}x_{7}x_{8})y_{1}).

We write S_{13} = S_{31}^{†} for the corresponding skew bi-invariant of degree (1, 3).

Earlier we wrote H(f) for the 9 by 9 matrix of second partial derivatives of f with respect to x_{0}, …, x_{8}. We now write H = H_{xx} and define H_{xy}, H_{yx}, H_{yy} in the analogous way. We further put H_{xx}(f) = H(Q)^{−1}H_{xx}(f), H_{xy}(f) = H(Q)^{−1}H_{xy}(f) and so on. The following are skew bi-invariants of degree (3, 3).

P = 2(Q_{20}S_{13} + Q_{02}S_{31}),
Θ_{ij} = tr(H_{xx}(D_{30})H_{xy}(A_{i})H_{yy}(D_{03})H_{yx}(A_{j})),
Ψ_{ij} = tr(H_{xx}(S_{31})H_{xy}(A_{i})H_{yx}(A_{j})),
U_{i} = tr(H_{xy}(P)H_{yx}(A_{i})),
V = tr(H_{xy}(Θ_{12})H_{yx}(A_{3})).

The space of skew bi-invariants of degree (3, 3) has dimension 15. The subspace fixed by the involution f ↦ f^{†} has basis B′_{1}, …, B′_{12} given by

D_{30}D_{03}, P, Θ_{12}, Θ_{13}, Θ_{22}, Θ_{23}, Θ_{33}, U_{2}, U_{3}, Ψ_{22} + Ψ_{22}^{†}, Ψ_{23} + Ψ_{23}^{†}, V.

For the calculations that follow we switch (with the benefit of hindsight) to the basis B_{1}, …, B_{12} that is related to the basis B′_{1}, …, B′_{12} by the second change of basis matrix recorded in Appendix A.

The subspace of bi-invariants vanishing on X × X has basis B_{11}, B_{12}. We write I_{3} ⊂ ℚ[z_{1}, …, z_{10}] for the ideal generated by all quadratic and cubic forms vanishing on the image of the map

X × X → ℙ^{9} ; (x_{0}, …, x_{8}; y_{0}, …, y_{8}) ↦ (B_{1} : …: B_{10}).

We find that I_{3} is minimally generated by 14 quadratic forms and 2 cubic forms. Moreover the variety Σ_{3} ⊂ ℙ^{9} defined by I_{3} is a surface of degree 24.

Proposition 6.2. The surface Σ_{3} is birational over ℚ to the elliptic surface Σ defined by the Weierstrass equation (1) in the statement of Theorem 1.2.

Proof. The rational map Σ_{3} → ℙ^{3} × 𝔸^{1} given by

(z_{1} : …: z_{10}) ↦ ((z_{1} : z_{2} : z_{3} : z_{4}), T) = ((z_{1} : z_{2} : z_{3} : z_{4}), z_{5}/z_{6})

has image satisfying

z_{1}^{2} − T z_{1}z_{2} + T z_{1}z_{3} + T z_{2}z_{3} − T z_{3}^{2} + T z_{3}z_{4} = 0,
z_{1}z_{3} − T z_{2}z_{3} + 2(T + 1)z_{1}z_{4} − (T^{2} − 1)z_{2}z_{4} + (T + 1)z_{4}^{2} = 0.

These same equations define a genus one curve in ℙ^{3} over ℚ(T). Making the linear change of coordinates

u_{1} = z_{1} − T z_{2},  u_{3} = (T + 1)z_{3},
u_{2} = (T + 2)z_{1} + z_{2} − (T + 1)(z_{3} − z_{4}),  u_{4} = (T + 1)z_{4}

gives the simplified quadric intersection

u_{1}^{2} + T u_{1}u_{2} + T u_{2}u_{3} − T u_{1}u_{4} = 0,
u_{1}u_{3} + T u_{1}u_{4} + u_{2}u_{4} + u_{3}u_{4} = 0,

which in turn is isomorphic to the elliptic curve (1) via

x = \frac{T (u_{1} + T u_{2})}{u_{4}}, y = \frac{T (u_{1} + T u_{2})^{2}}{u_{1}u_{4}} .

Composing these maps gives a birational map Σ_{3} → Σ. The inverse map, represented as an explicit 10-tuple of elements in the function field ℚ(Σ), is recorded in the accompanying computer file [F4]. □

Since the B_{i} are skew bi-invariants satisfying B_{i}^{†} = B_{i}, the rational map X × X → Σ factors via a rational map π : W (17, 3) → Σ. We will see below that π is birational, thereby proving the first part of Theorem 1.2.

For the purpose of computing the j-maps we decided to work with the skew bi-invariants of degree (2, 2), alongside those of degree (3, 3). We consider the map from X × X to the weighted projective space ℙ(2, 2, 3, …, 3) given by

(v, w, z_{1}, …, z_{10}) = (A_{2}, A_{3}, B_{1}, …, B_{10}).

Among the equations defining the image of this map we found the relations

vz_{5}z_{10} − w(z_{1}z_{5} − z_{3}z_{5} + z_{1}z_{6} − z_{5}z_{8} + z_{5}z_{9}) = 0,
vw^{2}z_{7} − (z_{2} − z_{3} − z_{5})(z_{6} − z_{7})z_{10} + z_{5}z_{8}z_{10} = 0.

We used these relations to extend our map Σ → ℙ^{9} to a map Σ → ℙ(2, 2, 3, …, 3).

The space of skew bi-invariants of degree (3, 2) has dimension 5. We picked one of these skew bi-invariants, not vanishing on X ×X, and called it T_{32}. We also put T_{23} = T_{32}^{†}. We define skew bi-invariants α_{1} = D_{30}D_{03}, α_{2} = S_{31}S_{13}, α_{3} = T_{32}T_{23}, α_{4} = D_{03}S_{31}T_{32} + D_{30}S_{13}T_{23}, α_{5} = S_{31}T_{23}^{2} + S_{13}T_{32}^{2},

α_{6} = c_{4}(x_{0}, …, x_{8})c_{4}(y_{0}, …, y_{8}),
α_{7} = D_{03}^{3}T_{23}c_{4}(x_{0}, …, x_{8}) + D_{30}^{3}T_{32}c_{4}(y_{0}, …, y_{8}),

of degrees (m, m) for m = 3, 4, 5, 6, 7, 10, 12. Let S = ℚ[v, w, z_{1}, …, z_{10}] be the coordinate ring of ℙ(2, 2, 3, …, 3), i.e., the graded polynomial ring where the variables have these weights. In the accompanying Magma file [F4], we record g_{1}, …, g_{7} ∈ S of weighted degrees 3, 6, 8, 8, 12, 18, 18, and h_{1}, …, h_{7} ∈ S of weighted degrees 0, 2, 3, 2, 5, 8, 6, such that each of the skew bi-invariants

g_{i}(A_{2}, A_{3}, B_{1}, …, B_{10}) − h_{i}(A_{2}, A_{3}, B_{1}, …, B_{10})α_{i}

vanishes on X × X. We use these expressions to solve for the α_{i} as elements of the function field ℚ(Σ). Then the polynomials

f_{1}(Y) = Y^{2} − α_{4}Y + α_{1}α_{2}α_{3},
f_{2}(Y) = Y^{2} − α_{5}Y + α_{2}α_{3}^{2},
f_{3}(Y) = Y^{2} − α_{7}Y + α_{1}^{3}α_{3}α_{6},

have roots defined over the same quadratic extension of ℚ(Σ). Let f_{i} have roots r_{i}, s_{i}. If we order these roots appropriately then by Lemma 4.4, and the definition of the α_{i}, we have

j_{1} + j_{2} = \frac{−2^{7}α_{2}}{α_{1}^{9}} (\frac{r_{3}^{3}}{r_{1}r_{2}} + \frac{s_{3}^{3}}{s_{1}s_{2}} ) and j_{1}j_{2} = \frac{2^{14}α_{6}^{3}}{α_{1}^{10}} .

Let Σ̃ → Σ be the double cover defined by the requirement that disc f_{1}, disc f_{2}, disc f_{3} or (j_{1} + j_{2})^{2} − 4j_{1}j_{2} is a square. Then the product of j-maps X → ℙ^{1} factors as

X × X ⟶ Z(17, 3) ⟶ Σ̃ ⟶ ℙ^{1} × ℙ^{1}.

Exactly as in Section 5 it follows that Z(17, 3) is birational to Σ̃ and W (17, 3) is birational to Σ. This completes the proof of Theorem 1.2 in the anti-symplectic case.

Remark 6.3. The elliptic K3-surface (1) admits many different elliptic fibrations. The fibrations we initially found on W (17, 1) and W (17, 3) were different, and it was only after we discovered that these surfaces are birational that we adjusted the calculations in this section so as to find the same elliptic fibration.

7. Some modular curves

Let m ≥ 2 be an integer coprime to 17. Then any pair of m-isogenous elliptic curves are 17-congruent with power k, where k = 1 if m is a quadratic residue mod 17, and k = 3 otherwise. There is therefore a copy of the modular curve X_{0}(m) on the surface Z(17, k). In Table 1 we explicitly identify these curves in all cases where X_{0}(m) has genus 0 or 1. The polynomials F_{1} and F_{3} are those appearing in the statement of Theorem 1.2, and explicitly recorded in Appendix A.

Table 1. Copies of X_{0}(m) on Z(17, 1) and Z(17, 3)
m  Formula specifying a curve on (a blow up of) z^{2} = F_{k}(T, x, y)
2 F_{1}(−2 + ε, −4 + 8ε − 5ε^{2} + ε^{3} + tε^{4}, 4 + O(ε)) = 2^{18}(8t + 1)ε^{4} + O(ε^{5})
3 F_{3}(−1/2 + ε, 1/2 − ε + tε^{3}, 1/4 + O(ε^{2})) = −2^{−20}(27t − 16)ε^{4} + O(ε^{5})
4 F_{1}(1 + ε, −tε^{2}, −1 + O(ε)) = 2^{4}(32t + 1)ε^{2} + O(ε^{3})
5 F_{3}(−2 + ε, 2 − ε + tε^{2}, 2 + O(ε)) = −2^{12}3^{4}(t^{2} − 11t − 1)ε^{4} + O(ε^{5})
6 F_{3}(−1 + ε, 1 − 2ε + tε^{2}, 2ε + O(ε^{2})) = (t^{2} − 36t + 36)ε^{10} + O(ε^{11})
7 F_{3}(tε^{−2}, tε^{−3}, ε^{−2} + O(ε^{−1})) = t^{22}(t + 1)(t − 27)ε^{−48} + O(ε^{−47})
8 F_{1}(ε, ε^{2} + ε^{3} + 8tε^{4}, ε^{2} + 2ε^{3} + O(ε^{4})) = 2^{4}(t^{2} + 6t + 1)ε^{18} + O(ε^{19})
9 F_{1}(ε^{−1}, ε^{−3} + tε^{−1}, O(ε^{−4})) = (t^{2} + 20t − 8)ε^{−24} + O(ε^{−23})
10 F_{3}((t + 1)ε^{2}, t^{−1}(t + 1)^{3}ε^{4}, t^{−1}(t + 1)^{3}(ε^{4} + ε^{5}) + O(ε^{6}))
= t^{−4}(t + 1)^{16}(t^{2} + 18t + 1)ε^{32} + O(ε^{33})
11 F_{3}(ε^{2} − ε^{3}, ε^{3} + ε^{4} + (t − 1)ε^{5}, ε^{3} + 2ε^{4} + O(ε^{5}))
= t(t^{3} + 20t^{2} + 56t + 44)ε^{36} + O(ε^{37})
12 F_{3}(ε^{−1}, −ε^{−1} − (t + 1), tε^{−2} + O(ε^{−1})) = (t^{2} − 14t + 1)ε^{−24} + O(ε^{−23})
13 F_{1}(−tε^{−1}, (t + 1)ε^{−1}, t^{−2}(t + 1)^{3} + O(ε)) = t^{16}(t^{2} + 12t − 16)ε^{−20} + O(ε^{−19})
14 F_{3}(ε^{−1}, −tε^{−3}, tε^{−5} + O(ε^{−4})) = t^{2}(t + 1)^{4}(t^{4} − 14t^{3} + 19t^{2} − 14t + 1)ε^{−30} + O(ε^{−29})
15 F_{1}(ε^{−1}, tε^{−2}, −tε^{−4} + O(ε^{−3})) = t^{2}(t − 1)^{2}(t^{2} − t − 1)(t^{2} + 11t − 1)ε^{−24} + O(ε^{−23})
16 F_{1}(−1 + ε, tε, 1 + O(ε)) = (t^{2} − 12t + 4)ε^{4} + O(ε^{5})
18 F_{1}(t, −t, −t(t + 1)) = t^{16}(t + 1)^{2}(t^{2} + 10t + 1)
19 F_{1}(−1 + 2ε, (t + 4)ε, (t + 4)^{2}ε^{2} + O(ε^{3})) = −8(t + 3)(t^{3} − 2t + 2)ε^{4} + O(ε^{5})
20 F_{3}(−ε + t^{2}ε^{2}, ε − tε^{2}, ε − ε^{2} + O(ε^{3})) = (t^{4} + 8t^{3} − 2t^{2} + 8t + 1)ε^{12} + O(ε^{13})
21 F_{1}(−ε, t(t + 1)ε^{3}, (t + 1)^{2}ε^{3} + O(ε^{4})) = (t^{4} + 6t^{3} − 17t^{2} + 6t + 1)ε^{16} + O(ε^{17})
24 F_{3}(t, 1, 0) = (t + 1)^{8}(t^{3} + t^{2} − 1)^{4}(t^{4} − 8t^{3} + 2t^{2} + 8t + 1)
25 F_{1}(t, t^{2}, t^{2}) = −t^{18}(t + 1)^{4}(16t^{2} + 4t − 1)
27 F_{3}(t, t^{2}(t + 1), t^{2}(t + 1)^{2}) = −t^{18}(t^{2} + 2t + 2)^{4}(t − 1)(11t^{3} + 15t^{2} + 9t + 1)
32 F_{1}(t, t^{2}(t + 1), −t^{2}(t^{3} + t^{2} − 1))
= t^{16}(t + 1)^{4}(t^{2} + t + 1)^{4}(t^{4} + 8t^{3} + 12t^{2} + 16t + 4)
36 F_{1}(t, (t + 1)(t^{2} + t + 1), (t + 1)^{2}(t^{2} + 2t + 2))

= (t + 1)^{4}(t^{3} + t^{2} + 2t + 1)^{2}(t^{3} + 2t^{2} + 3t + 1)^{4}(4t^{4} + 8t^{3} + 12t^{2} + 8t + 1)

49 F_{1}(t, t(t^{2} − 1), −t(t^{2} − 1)^{2}) = t^{20}(t + 1)^{4}(t^{2} − t − 1)^{2}(t^{4} + 6t^{3} + 3t^{2} − 18t − 19)

In compiling Table 1 we used the SmallModularCurve database in Magma [BCP] to check the moduli interpretations. For example, the entry with m = 18 shows that Z(17, 1) contains a curve isomorphic to y^{2} = t^{2} + 10t + 1. We parametrise this curve by putting t = −T/((T + 2)(T + 3)), and find, using our expressions for j_{1} + j_{2} and j_{1}j_{2} as rational functions on W (17, 1), that

X^{2} − (j_{1} + j_{2})X + j_{1}j_{2} = (X − j_{18}(T))(X − j_{18}(6/T))

where

j_{18}(T) = \frac{((T + 2)^{12} − 8(T + 2)^{9} + 16(T + 2)^{3} + 16)^{3}}{(T + 2)^{9}((T + 2)^{3} − 8)((T + 2)^{3} + 1)^{2}}

is the j-map on X_{0}(18).

To find most of these curves it was necessary to blow up the surfaces in Theorem 1.2. In such cases we specify the arguments T, x, y of F_{k} as power series in ε, given to sufficient precision to determine a unique solution of (1). For example, the entry with m = 20 shows that blowing up our model for Z(17, 3) above (T, x, y) = (0, 0, 0) we found a curve isomorphic to y^{2} = t^{4} + 8t^{3} − 2t^{2} + 8t + 1. Putting this elliptic curve in Weierstrass form we find it has Cremona label 20a1, and in particular is isomorphic to X_{0}(20).

8. Examples and further questions

We restate Conjecture 1.1 in the case p = 17. As usual we say a p-congruence is trivial if it is explained by an isogeny of degree coprime to p.

Conjecture 8.1. (i) The only non-trivial pairs of symplectically 17-congruent elliptic curves over ℚ are the simultaneous quadratic twists of the elliptic curves E_{1} and E_{2} (with conductors 279809270 and 3077901970) as defined in the introduction.

(ii) The only non-trivial pairs of anti-symplectically 17-congruent elliptic curves over ℚ are the simultaneous quadratic twists of the elliptic curves E′_{1} and E′_{2} (with conductors 3675 and 47775) as defined in the introduction.

We make a related conjecture.

Conjecture 8.2. Let Z̃(17, k) be the surface in 𝔸^{4} with equations

y^{2} + (T + 1)(T − 2)xy + T^{3}y = x^{3} − x^{2}

and z^{2} = F_{k}(T, x, y) where F_{k} is as recorded in Appendix A.

(i) The only ℚ-points on Z̃(17, 1) lie above one of the curves

(x, y) = (0, −T^{3}), (−T, −T^{2} − T), (−T, −T), (T^{2}, T^{2}), (T^{2}, −T^{4} + T^{2}),
(T^{3} + T^{2}, T^{4} + 2T^{3} + T^{2}), (T^{3} − T, T^{4} − T^{2} − T),

or the curve T = 0, or one of the points in Table 2.

(ii) The only ℚ-points on Z̃(17, 3) lie above one of the curves

(x, y) = (−T, −T), (−T, −T^{2} − T), (T^{2}, −T^{4} + T^{2}),
(T^{3} + T^{2}, −T^{5} − T^{4} + T^{2}),

or the curve T = 0, or one of the points in Table 2.

The points on Z̃(17, k) lying above one of the curves listed in Conjecture 8.2 do not correspond to non-trivial pairs of 17-congruent elliptic curves, either because there is an m-isogeny (with m = 18 or 25), or the j-maps have a pole, or the point is spurious since F_{k} vanishes to even multiplicity.

It should be possible to prove that Conjectures 8.1 and 8.2 are equivalent by computing biregular (not just birational) models for Z(17, 1) and Z(17, 3). This belief is based on the fact that in compiling Table 1 we carried out some of the necessary blow ups, but did not find any new examples of 17-congruences. Nonetheless we leave the full verification to future work.

The height of a rational number x = a/b (where a, b are coprime integers) is H(x) = max(|a|, |b|). Our evidence for Conjecture 8.2 is that we found no further points with H(T) ≤ 3000 and H(x) ≤ 10000. We see little hope of proving this conjecture using existing methods. A possibly more tractable problem, the answer to which would still be interesting, would be to determine all curves of genus 0 or 1 on these surfaces (either over ℚ or over \overline{ℚ}).

In Table 2 we list elliptic curves via their Cremona labels [C], writing instead the conductor followed by a star for curves beyond the range of Cremona’s tables. The latter convention is only needed for the first entry, where the relevant elliptic curves are the ones defined in the introduction. The first column records whether the congruence is symplectic (k = 1) or anti-symplectic (k = 3). The final column records the degree of the isogeny when the curves are isogenous. Each pair of elliptic curves we list is only determined up to simultaneous quadratic twist.

Table 2. Some rational points on Z(17, 1) and Z(17, 3)

k T  x  y  17-congruent ell. curves

degree

1 1/3  −2/75  −11/125  279809270∗, 3077901970∗ −

   1 −3  −27  108  1849a1, 1849a2  43

1 −5/6 −5/24  5/16  4489a1, 4489a2  67

   3 1  2  4  27a2, 27a4  27

3 9/7  27/49  −54/49  3675b1, 47775b1  −
3 −5/14 125/392  375/1568  1225h1, 1225h2  37
3 11/39 1771/6591 116380/257049  26569a1, 26569a2  163

ON PAIRS OF 17-CONGRUENT ELLIPTIC CURVES 21

Appendix A. Formulae

We record the polynomials F_{1}(T, x, y) and F_{3}(T, x, y) in Theorem 1.2. These define the double covers Z(17, 1) → W (17, 1) and Z(17, 3) → W (17, 3).

F_{1}(T, x, y) = x^{10} − 2T (T − 1)x^{8}y + T (T^{3} − 2T^{2} − 11T + 4)x^{9}

− T^{2}(T^{4} − 3T^{3} − 3T^{2} + 3T − 10)x^{7}y + T^{2}(8T^{4} + 58T^{3} + 15T^{2} − 64T + 5)x^{8} − T^{3}(8T^{5} + 51T^{4} + 80T^{3} + 51T^{2} − 2T − 20)x^{6}y − 2T^{4}(16T^{4} − 41T^{3} − 205T^{2} − 97T + 71)x^{7} + 2T^{4}(16T^{6} + 49T^{5} − 13T^{4} − 142T^{3} − 113T^{2} − T + 10)x^{5}y − T^{4}(149T^{6} + 576T^{5} + 180T^{4} − 956T^{3} − 579T^{2} + 148T + 5)x^{6} − 2T^{5}(13T^{7} − 39T^{6} − 229T^{5} − 202T^{4} + 126T^{3} + 169T^{2} + 9T − 5)x^{4}y + T^{5}(80T^{8} + 318T^{7} − 192T^{6} − 1800T^{5} − 1376T^{4} + 819T^{3} + 750T^{2} − 67T − 4)x^{5} − T^{6}(T + 1)(24T^{7} + 156T^{6} − 24T^{5} − 558T^{4} − 285T^{3} + 192T^{2} + 21T − 2)x^{3}y − T^{6}(16T^{10} + 72T^{9} − 239T^{8} − 1300T^{7} − 870T^{6} + 1952T^{5} + 2295T^{4} + 18T^{3} − 449T^{2} + 4T + 1)x^{4} + T^{8}(T + 1)^{2}(12T^{6} − 50T^{5} − 226T^{4} + 36T^{3} + 292T^{2} − 31T − 6)x^{2}y − T^{8}(T + 1)(76T^{8} + 273T^{7} − 275T^{6} − 1505T^{5} − 631T^{4} + 1016T^{3} + 472T^{2} − 94T − 4)x^{3} − T^{10}(T + 1)^{3}(T^{5} − 14T^{4} + 55T^{3} + 118T^{2} − 68T − 4)xy − T^{10}(T + 1)^{2}(131T^{6} + 328T^{5} − 234T^{4} − 700T^{3} − 18T^{2} + 138T + 3)x^{2} − T^{13}(T + 1)^{4}(T^{2} − 2T + 28)y − 2T^{13}(T + 1)^{3}(49T^{3} + 63T^{2} − 63T − 27)x − 27T^{16}(T + 1)^{4}.

F_{3}(T, x, y) = x^{10} − 14T x^{8}y − T (4T^{2} − 71T − 16)x^{9} + T^{2}(17T^{2} − 89T − 18)x^{7}y

− T^{2}(14T^{4} + 288T^{3} + 165T^{2} − 220T − 19)x^{8} + T^{3}(76T^{4} + 480T^{3} + 545T^{2} − 176T − 4)x^{6}y + T^{3}(94T^{6} + 513T^{5} + 234T^{4} − 1412T^{3} − 732T^{2} + 242T + 4)x^{7} − T^{5}(163T^{5} + 837T^{4} + 1320T^{3} − 72T^{2} − 898T + 106)x^{5}y − T^{5}(159T^{7} + 590T^{6} − 103T^{5} − 3276T^{4} − 3150T^{3} + 1326T^{2} + 820T − 112)x^{6} + T^{6}(80T^{7} + 418T^{6} + 501T^{5} − 936T^{4} − 2496T^{3} − 948T^{2} + 390T − 4)x^{4}y + T^{6}(98T^{9} + 386T^{8} − 350T^{7} − 3439T^{6} − 3894T^{5} + 3010T^{4} + 4872T^{3} − 306T^{2} − 284T + 4)x^{5} + T^{8}(4T^{8} + 130T^{7} + 917T^{6} + 2787T^{5} + 4078T^{4} + 2292T^{3} − 482T^{2} − 480T − 60)x^{3}y − T^{8}(27T^{10} + 122T^{9} − 177T^{8} − 1496T^{7} − 987T^{6} + 5032T^{5} + 8446T^{4} + 1124T^{3} − 2621T^{2} + 54T − 61)x^{4} − T^{10}(10T^{9} + 132T^{8} + 738T^{7} + 2126T^{6} + 3179T^{5} + 1902T^{4} − 718T^{3} − 1376T^{2} − 482T − 256)x^{2}y + T^{10}(T^{10} − 48T^{9}

22 T.A. FISHER

− 122T^{8} + 882T^{7} + 4304T^{6} + 6244T^{5} + 973T^{4} − 4506T^{3} − 1714T^{2} + 428T − 242)x^{3} − T^{12}(T^{10} − T^{9} − 70T^{8} − 368T^{7} − 814T^{6} − 714T^{5} + 237T^{4} + 963T^{3} + 800T^{2} + 522T + 312)xy − T^{12}(26T^{9} + 283T^{8} + 1018T^{7} + 1256T^{6} − 810T^{5} − 3237T^{4} − 1848T^{3} + 648T^{2} + 108T − 265)x^{2} − T^{14}(T + 2)(T^{8} + 12T^{7} + 44T^{6} + 74T^{5} + 64T^{4} + 20T^{3} − 43T^{2} − 92T − 60)y − 2T^{14}(10T^{8} + 94T^{7} + 323T^{6} + 471T^{5} + 129T^{4} − 367T^{3} − 263T^{2} + 69T + 36)x + T^{16}(T^{8} + 4T^{7} − 8T^{6} − 66T^{5} − 120T^{4} − 56T^{3} + 53T^{2} + 36T − 16).

We also record the change of basis matrices we used in Sections 5 and 6. In each case the rows of the matrix give the B′_{i} in terms of the B_{i}.

( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ −24 & −12 & 28 & 8 & −4 & 0 & 4 & 4 & 0 & −4 & 4 & 0 & 0 & 0 \\ −34 & −15 & 34 & 7 & −8 & −3 & 0 & 5 & −2 & −9 & 7 & 0 & −1 & 0 \\ −480 & −288 & 320 & 160 & −64 & −32 & 192 & 160 & 0 & −96 & 96 & 512 & −96 & 0 \\ 248 & 144 & −96 & −144 & 0 & −16 & −192 & −144 & −32 & 48 & 16 & 448 & −48 & 16 \\ −24 & −56 & −128 & −40 & 0 & −24 & 32 & 56 & −16 & −24 & −8 & −32 & −40 & 8 \\ −384 & −144 & 688 & 96 & −64 & 0 & 32 & −48 & −16 & 0 & 128 & −64 & 0 & 8 \\ 328 & 204 & −240 & −76 & 32 & 4 & −144 & −92 & 8 & 36 & −20 & 24 & 20 & −12 \\ −366 & −180 & 96 & 72 & −48 & 12 & 192 & 168 & −12 & −48 & −84 & 36 & 0 & 36 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

( \begin{matrix}−4 & −12 & 8 & −4 & −8 & −8 & 0 & −4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 \\ −56 & −176 & 148 & −68 & −160 & −140 & 36 & −52 & 24 & 24 & −8 & 4 \\ −104 & −240 & 156 & −76 & −128 & −164 & 12 & −92 & 8 & 8 & 8 & 4 \\ −16 & −128 & 124 & −36 & −144 & −92 & 12 & −36 & 8 & 8 & 8 & 12 \\ −40 & −48 & 48 & −24 & −32 & −40 & 48 & −24 & 32 & 32 & 0 & 16 \\ −64 & −176 & 132 & −60 & −128 & −148 & 4 & −76 & −8 & −8 & 24 & −12 \\ −36 & −72 & 96 & −96 & 12 & 24 & −36 & −12 & 0 & 24 & 24 & −12 \\ −60 & −48 & 48 & −24 & −12 & −48 & 60 & −60 & 48 & 48 & 72 & 24 \\ 36 & 72 & −72 & 24 & 60 & 24 & −60 & 12 & −72 & −72 & 24 & −48 \\ −72 & −84 & 108 & −48 & 0 & 24 & −12 & −24 & 12 & 24 & 0 & 12 \\ −208 & 204 & −72 & 28 & 364 & 116 & 288 & −44 & 208 & 224 & 32 & 104 \end{matrix} )

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University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

Email address: T.A.Fisher@dpmms.cam.ac.uk