ON FAMILIES OF n-CONGRUENT ELLIPTIC CURVES

T.A. FISHER

Abstract. We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n = 7, 9, 11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose n-torsion subgroups are isomorphic as Galois modules. We also show by giving explicit non-trivial examples over ℚ(T) that there are infinitely many examples over ℚ in the cases n = 9 and n = 11.

1. Introduction

Elliptic curves E_{1} and E_{2} over a field K are n-congruent if their n-torsion subgroups E_{1}[n] and E_{2}[n] are isomorphic as Galois modules. They are directly n-congruent if the isomorphism ϕ : E_{1}[n] ≅ E_{2}[n] respects the Weil pairing and reverse n-congruent if e_{n}(ϕP, ϕQ) = e_{n}(P, Q)^{−1} for all P, Q ∈ E_{1}[n]. The elliptic curves directly n-congruent to a given elliptic curve E are parametrised by the modular curve Y_{E}(n) = X_{E}(n) \ {cusps}.

For n ≤ 5 we have X_{E}(n) ≅ ℙ^{1} and the corresponding families of elliptic curves were computed by Rubin and Silverberg [RS1], [RS2], [S]. However for n ≥ 7 the genus is greater than 1. This prompted Mazur [M] to ask whether there are any pairs of non-isogenous elliptic curves over ℚ that are directly n-congruent for any n ≥ 7. This was answered by Kraus and Oesterlé [KO] who gave the example of the directly 7-congruent elliptic curves 152a1 and 7448e1. The labels here are those in Cremona’s tables [C]. Nowadays it is easy to find further examples by searching in Cremona’s tables, for example

\begin{matrix}n = 11 & 190b1 & + & 2470a1, \\ n = 13 & 52a2 & + & 988b1, \\ n = 17 & 3675b1 & − & 47775b1. \end{matrix}

In each case the n-congruence is proved by computing sufficiently many traces of Frobenius; see [KO, Proposition 4]. Then [KO, Proposition 2] shows that the congruences are direct or reverse as indicated by the ±.

Motivated by Mazur’s question, Kani and Schanz [KS] studied the geometry of the surfaces that parametrise pairs of n-congruent of elliptic curves. This prompted them to conjecture that for any n ≤ 12 there are infinitely many pairs of n-congruent non-isogenous elliptic curves over ℚ. It is understood that we are looking for examples with distinct j-invariants, since otherwise from any single example we could construct infinitely many by taking quadratic twists. The conjecture was proved in the case n = 7 by Halberstadt and Kraus [HK1], who subsequently [HK2] gave an explicit formula for X_{E}(7) and used it to show that there are infinitely many 6-tuples of directly 7-congruent non-isogenous elliptic curves over ℚ. We find the corresponding formulae for X_{E}(9) and X_{E}(11) and use them to construct explicit infinite families of pairs of elliptic curves that prove the conjecture for n = 9 and n = 11. In contrast the proof of the conjecture for n = 11 in [KR] does not yield a single explicit example.

Date: 9th May 2011.

We briefly mention three further motivations for studying n-congruence of elliptic curves.

• The modular approach to solving Diophantine equations sometimes requires us to find all elliptic curves n-congruent to a given elliptic curve. For example the paper of Poonen, Schaefer and Stoll [PSS] makes essential use of the formula for X_{E}(7) due to Halberstadt and Kraus.

• There is a correspondence between pairs of reverse n-congruent elliptic curves and curves of genus 2 that admit a degree n morphism to an elliptic curve. See for example [Fr].

• It was observed by Cremona and Mazur [CM] that if elliptic curves E and F are n-congruent then the Mordell-Weil group of F can sometimes be used to explain elements of the Tate-Shafarevich group of E.

As each of these motivations makes clear, we should also be interested in congruences that do not respect the Weil pairing. The elliptic curves reverse n-congruent to E are parametrised by the modular curve Y_{E}^{−}(n) = X_{E}^{−}(n) \ {cusps}. The families of elliptic curves parametrised by Y_{E}^{−}(3) and Y_{E}^{−}(4) were computed in [F2], and the case of Y_{E}^{−}(5) will be treated in [F3]. An equation for X_{E}^{−}(7) was given in [PSS, Section 7.2]. We find corresponding formulae for X_{E}^{−}(9) and X_{E}^{−}(11). In the cases n = 7 and n = 9 we use these formulae to construct explicit infinite families of pairs of reverse n-congruent non-isogenous elliptic curves over ℚ. We do not know if any such families exist in the case n = 11.

In Section 1.1 we recall the definitions of X(n) and its twists. We then state the formulae for X_{E}(n) and X_{E}^{−}(n) for n = 3, 7, 9, 11 in Section 1.2. We include the cases n = 3 and n = 7 since our methods are quite different from those in [F2] and [HK2]. The results in the case n = 3 are also needed in Section 6 to treat the case n = 9. In the next two subsections we explain our basic strategy for computing twists and illustrate how it works in the case n = 3.

In Section 2 we recall Klein’s equations for X(n) for n ≥ 5 an odd integer. The original approach of Klein was via theta functions, but our treatment is purely algebraic. We also recall explicit formulae for the action of SL_{2}(ℤ/nℤ) on X(n).

Then in Section 3 we use invariant theory for SL_{2}(ℤ/nℤ) to compute the twists X_{E}(n) and X_{E}^{−}(n) for n = 7, 9, 11.

In Section 4 we show that in the special case of an elliptic curve E whose n-torsion contains a copy of the Galois module μ_{n}, there are particularly simple formulae for X_{E}(n) and X_{E}^{−}(n). This section may be read independently of Section 3. Again the result for X_{E}(7) is already given in [HK2].

Our formulae reduce the problem of finding elliptic curves n-congruent to E to that of finding rational points on X_{E}(n) and X_{E}^{−}(n). However before searching for rational points it helps to simplify the equations by making a change of coordinates. In Section 5 we describe how to do this in the case K = ℚ. The problem naturally splits into two parts called minimisation and reduction.

In Section 6 we give formulae for the families of elliptic curves parametrised by Y_{E}(n) and Y_{E}^{−}(n) for n = 7, 9, 11. This is easiest in the case n = 9 since the natural maps X_{E}(9) → X_{E}(3) and X_{E}^{−}(9) → X_{E}^{−}(3) have a simple geometric description. In the cases n = 7 and n = 11 it is easy to compute the maps j : X_{E}(n) → ℙ^{1} and j : X_{E}^{−}(n) → ℙ^{1}. However determination of the right quadratic twists takes considerably more work. (In specific numerical examples one can always fall back on the method in [KO], [HK1].) In the case of Y_{E}(7) a formula is given in [HK2], although this formula does not quite cover all cases. We give a new proof leading to formulae that work in all cases. We then generalise to the families of elliptic curves parametrised by Y_{E}^{−}(7), Y_{E}(11) and Y_{E}^{−}(11).

Finally in Section 7 we give examples of two different sorts. First we have written a program in Magma [BCP] that given an elliptic curve E/ℚ and n ∈ {7, 9, 11} searches for rational points (up to a specified height bound) on minimised and reduced models for X_{E}(n) and X_{E}^{−}(n) and returns the corresponding list of elliptic curves n-congruent to E. For n = 9 and n = 11 we have run this program on every elliptic curve in the Cremona database. In particular we found three triples of directly 9-congruent non-isogenous elliptic curves over ℚ.

Secondly we have found some non-trivial pairs of n-congruent elliptic curves over ℚ(T) for n = 7, 9, 11. The infinite families mentioned above are obtained by specialising T.

All computer calculations in support of this work were preformed using Magma [BCP]. A Magma file checking all our formulae, together with extended versions of the tables in Section 7, is available from the author’s website [F4].

1.1. Some modular curves. We work over a field K of characteristic 0 and write \overline{K} for the algebraic closure. Let n ≥ 3 be an integer, and M a Galois module isomorphic to (ℤ/nℤ)^{2} as an abelian group and equipped with a nondegenerate alternating Galois equivariant pairing M × M → μ_{n}. We temporarily write Y_{M} for the algebraic curve defined over K whose L-rational points (L a field extension of K) parametrise the isomorphism classes of pairs (E, ϕ), where E is an elliptic curve defined over L and ϕ : E[n] ≅ M is a symplectic isomorphism (i.e. one that matches up the given pairing on M with the Weil pairing on E[n]) commuting with the action of Gal(\overline{L}/L). Two such pairs (E_{1}, ϕ_{1}) and (E_{2}, ϕ_{2}) are isomorphic if there is an L-isomorphism α : E_{1} → E_{2} such that ϕ_{1} = ϕ_{2} ∘ (α|_{E_{1}[n]}).

Let X_{M} be the smooth projective model of Y_{M}. We define X(n) to be X_{M} in the case M = μ_{n} × ℤ/nℤ with pairing

⟨(ζ, a), (ξ, b)⟩ = ζ^{b}ξ^{−a}.

Given an elliptic curve E over K we define X_{E}(n) to be X_{M} in the case M is E[n] equipped with the Weil pairing. More generally we may take M to be E[n] equipped with the rth power of the Weil pairing for any r ∈ (ℤ/nℤ)^{×}. However the curve obtained only depends on the class of r mod squares. Since we will later be specialising to n ∈ {3, 7, 9, 11} it suffices to take r = ±1. Taking r = 1 gives the curve X_{E}(n) defined above. Taking r = −1 gives the curve X_{E}^{−}(n).

We identify SL_{2}(ℤ/nℤ) with the group of symplectic automorphisms of μ_{n} × ℤ/nℤ. There is then a natural action of PSL_{2}(ℤ/nℤ) := SL_{2}(ℤ/nℤ)/{±I_{2}} on X(n) with quotient map j : X(n) → ℙ^{1}. From the analytic theory we know that the j-map is ramified above 0, 1728 and ∞ with ramification indexes 3, 2 and n. Hence by the Riemann-Hurwitz formula the genus of X(n) is

g(n) = \frac{n − 6}{12n} #PSL_{2}(ℤ/nℤ) + 1

where for n ≥ 3 we have #PSL_{2}(ℤ/nℤ) = (n^{3}/2) ∏_{p|n}(1 − 1/p^{2}). For some small values of n the genus is as follows.

\begin{matrix}n & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 \\ g(n) & 0 & 0 & 0 & 0 & 1 & 3 & 5 & 10 & 13 & 26 & 25 & 50 & 49 & 73 & 81 & 133 \end{matrix}

1.2. Statement of results. The family of elliptic curves parametrised by Y_{E}(n) for n = 2, 3, 4, 5 is given by formulae of Rubin and Silverberg [RS1], [RS2], [S]. In [F2] we developed an alternative invariant-theoretic approach that also gives formulae for Y_{E}^{−}(n). In the case n = 3 we have

Theorem 1.1. Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x−54c_{6}. Then the families of elliptic curves parametrised by Y_{E}(3) and Y_{E}^{−}(3) are

y^{2} = x^{3} − 27c_{4}(λ, μ)x − 54c_{6}(λ, μ)

and

y^{2} = x^{3} − 27c_{4}^{∗}(λ, μ)x − 54c_{6}^{∗}(λ, μ)

where

c_{4}(λ, μ) = c_{4}λ^{4} + 4c_{6}λ^{3}μ + 6c_{4}^{2}λ^{2}μ^{2} + 4c_{4}c_{6}λμ^{3} − (3c_{4}^{3} − 4c_{6}^{2})μ^{4},
c_{6}(λ, μ) = c_{6}λ^{6} + 6c_{4}^{2}λ^{5}μ + 15c_{4}c_{6}λ^{4}μ^{2} + 20c_{6}^{2}λ^{3}μ^{3}
+ 15c_{4}^{2}c_{6}λ^{2}μ^{4} + 6(3c_{4}^{4} − 2c_{4}c_{6}^{2})λμ^{5} + (9c_{4}^{3}c_{6} − 8c_{6}^{3})μ^{6},

and

c_{4}^{∗}(λ, μ) = −4(λ^{4} − 6c_{4}λ^{2}μ^{2} − 8c_{6}λμ^{3} − 3c_{4}^{2}μ^{4})/(c_{4}^{3} − c_{6}^{2}),
c_{6}^{∗}(λ, μ) = −8c_{6}(λ, μ)/(c_{4}^{3} − c_{6}^{2})^{2}.

Proof: These formulae (and their relation to those in [RS1]) may be found in [F2]. We recall the idea of the proof. Let F be the ternary cubic obtained by homogenising the Weierstrass equation for E. Then the family of curves parametrised by X_{E}(3) is the pencil of plane cubics spanned by F and its Hessian, each with base point the same as that for E. It only remains to put these curves in Weierstrass form. One way of carrying out this final calculation is by using the formula for the Jacobian (due to Weil) coming from classical invariant theory. For this reason the polynomials c_{4}(λ, μ) and c_{6}(λ, μ) may already be found in [Sa, Art. 230], although not with the interpretation given here.

The corresponding formula for Y_{E}^{−}(3) is obtained by replacing the Hessian (which is a covariant) by suitable contravariants. □

A formula for X_{E}(7) was obtained by Halberstadt and Kraus [HK2]. Their method relies on studying the points on the Klein quartic

X(7) = {x^{3}y + y^{3}z + z^{3}x = 0} ⊂ ℙ^{2}

corresponding to an elliptic curve E and the elliptic curves E_{a}, E_{b}, E_{c} that are 2-isogenous to E. By combining this result with some classical invariant theory Poonen, Schaefer and Stoll [PSS, Section 7.2] then gave a formula for X_{E}^{−}(7).

Theorem 1.2 (Halberstadt, Kraus, Poonen, Schaefer, Stoll). Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} + ax + b. Then X_{E}(7) ⊂ ℙ^{2} has equation F = 0 where

F = ax^{4} + 7bx^{3}z + 3x^{2}y^{2} − 3a^{2}x^{2}z^{2} − 6bxyz^{2}
− 5abxz^{3} + 2y^{3}z + 3ay^{2}z^{2} + 2a^{2}yz^{3} − 4b^{2}z^{4},

and X_{E}^{−}(7) ⊂ ℙ^{2} has equation G = 0 where

G = −a^{2}x^{4} + 2abx^{3}y − 12bx^{3}z − (6a^{3} + 36b^{2})x^{2}y^{2} + 6ax^{2}z^{2}
+ 2a^{2}bxy^{3} − 12abxy^{2}z + 18bxyz^{2} + (3a^{4} + 19ab^{2})y^{4}
− (8a^{3} + 42b^{2})y^{3}z + 6a^{2}y^{2}z^{2} − 8ayz^{3} + 3z^{4}.

We give new proofs of Theorems 1.1 and 1.2. We then extend to the cases n = 9 and n = 11. Although we believe that the formulae in Theorems 1.3 and 1.4 below are correct for all elliptic curves E we assume for simplicity that j(E) ≠ 0, 1728. In Section 2 we recall that X(9) may be embedded in ℙ^{3} as the complete intersection of two cubics:

X(9) = \{ \begin{matrix}x^{2}y + y^{2}z + z^{2}x & = 0 \\ xy^{2} + yz^{2} + zx^{2} & = t^{3} \end{matrix}\} ⊂ ℙ^{3}.

Theorem 1.3. Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} + ax + b. If j(E) ≠ 0, 1728 then X_{E}(9) ⊂ ℙ^{3} has equations F_{1} = F_{2} = 0 where

F_{1} = x^{2}t + 6xyz + 6bxt^{2} + 6y^{3} − 9ay^{2}t + 6a^{2}yt^{2} − 3bz^{3} + 3a^{2}z^{2}t
+ 9abzt^{2} − (a^{3} − 12b^{2})t^{3},
F_{2} = x^{2}z + 6xy^{2} − 6axyt + 2a^{2}xt^{2} − 9ay^{2}z − 18byz^{2} + 12a^{2}yzt + a^{2}z^{3}
+ 9abz^{2}t − 3a^{3}zt^{2} + a^{2}bt^{3},

and X_{E}^{−}(9) ⊂ ℙ^{3} has equations G_{1} = G_{2} = 0 where

G_{1} = 9x^{2}y + 3x^{2}z − 6axyt + 6bxt^{2} − 6ay^{3} + 27by^{2}t + 3ayz^{2}
+ 18byzt + 3a^{2}yt^{2} + az^{3} + 3bz^{2}t + a^{2}zt^{2} − abt^{3},
G_{2} = x^{3} + 6axyz + 18bxyt + 3axz^{2} + 6bxzt + a^{2}xt^{2} + 9by^{3} + 6a^{2}y^{2}t
− 9byz^{2} + 6a^{2}yzt − 3abyt^{2} − 4bz^{3} + 2a^{2}z^{2}t + 2b^{2}t^{3}.

It was observed by Klein that X(11) may be embedded in ℙ^{4} as the singular locus of the Hessian of the cubic threefold

{v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v = 0} ⊂ ℙ^{4}.

Theorem 1.4. Let E be an elliptic curve with Weierstrass equation y^{2} = x^{3} + ax + b. If j(E) ≠ 0, 1728 then X_{E}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

F = v^{3} + av^{2}z − 2avx^{2} + 4avxy − 6bvxz + avy^{2} + 6bvyz + a^{2}vz^{2} − w^{3}
+ aw^{2}z − 4awx^{2} − 12bwxz + a^{2}wz^{2} − 2bx^{3} + 3bx^{2}y + 2a^{2}x^{2}z + 6bxy^{2}
+ 4abxz^{2} + by^{3} − a^{2}y^{2}z + abyz^{2} + 2b^{2}z^{3},

and X_{E}^{−}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

G = v^{2}z + 2vwy + 4vxy + 2w^{2}x − aw^{2}z + 2wx^{2} − 2awy^{2} − 6bwyz
+ 6x^{3} − 6ax^{2}z + 2a^{2}xz^{2} + by^{3} − 2a^{2}y^{2}z − 5abyz^{2} − b^{2}z^{3}.

1.3. Preliminaries on twisting. We identify SL_{2}(ℤ/nℤ) with the group of symplectic automorphisms of μ_{n} × ℤ/nℤ via ( \begin{matrix}a & b \\ c & d \end{matrix}) : (ζ_{n}^{x}, y) ↦ (ζ_{n}^{ax+by}, cx + dy), where ζ_{n} is a fixed primitive nth root of unity. The Galois action on SL_{2}(ℤ/nℤ) will be that arising from this identification. There is then a Galois equivariant group homomorphism

(1)  \overline{ρ} : SL_{2}(ℤ/nℤ) → Aut(X(n)).

where the action of \overline{ρ}(γ) on Y (n) is given by (E, ϕ) ↦ (E, γ ∘ ϕ).

Lemma 1.5. Let E/K be an elliptic curve and ϕ : E[n] ≅ μ_{n}×ℤ/nℤ a symplectic, respectively anti-symplectic, isomorphism over \overline{K}. Then there is an isomorphism α : X_{E}(n) → X(n), respectively α : X_{E}^{−}(n) → X(n), over \overline{K} such that

σ(α)α^{−1} = \overline{ρ}(σ(ϕ)ϕ^{−1})

for all σ ∈ Gal(\overline{K}/K).

Proof: The points on Y_{E}(n), respectively Y_{E}^{−}(n), correspond to pairs (F, ψ) where F is an elliptic curve and ψ : F [n] ≅ E[n] is a symplectic, respectively anti-symplectic, isomorphism. The modular interpretation of α is that it sends (F, ψ) to (F, ϕ ∘ ψ). □

We are interested in the case X(n) is embedded in ℙ^{m−1} for some m, and \overline{ρ} is realised as a projective representation (also denoted \overline{ρ} by abuse of notation):

\overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{m}(\overline{K}).

Writing ∝ for equality in PGL_{m}(\overline{K}), we further suppose there exists^{1} ε ∈ GL_{m}(K) such that

(2)  \overline{ρ}(ιγι) ∝ ε\overline{ρ}(γ)^{−T} ε^{−1}

for all γ ∈ SL_{2}(ℤ/nℤ), where ι = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Our strategy for computing X_{E}(n) and X_{E}^{−}(n) as twists of X(n) is explained by the following lemma.

Lemma 1.6. Let E/K be an elliptic curve and ϕ : E[n] ≅ μ_{n} ×ℤ/nℤ a symplectic isomorphism over \overline{K}. Suppose h_{1}, h_{2} ∈ GL_{m}(\overline{K}) satisfy

σ(h_{1})h_{1}^{−1} ∝ \overline{ρ}(σ(ϕ)ϕ^{−1}) σ(h_{2})h_{2}^{−1} ∝ \overline{ρ}(σ(ϕ)ϕ^{−1})^{−T}

for all σ ∈ Gal(\overline{K}/K). Then X_{E}(n) ⊂ ℙ^{m−1} and X_{E}^{−}(n) ⊂ ℙ^{m−1} are the twists of X(n) ⊂ ℙ^{m−1} given by X_{E}(n) ≅ X(n); x ↦ h_{1}x and X_{E}^{−}(n) ≅ X(n); x ↦ εh_{2}x.

Proof: We apply Lemma 1.5 to the pairs (E, ϕ) and (E, ι ∘ ϕ). □

1By Remark 2.4(iii) below we can usually take ε = I_{m}.

Remark 1.7. If the projective representation \overline{ρ} lifts to a representation

ρ : SL_{2}(ℤ/nℤ) → GL_{m}(\overline{K})

then the existence of the matrix h_{1} in Lemma 1.6 is clear from the generalised form of Hilbert’s Theorem 90 which states that H^{1}(Gal(\overline{K}/K), GL_{m}(\overline{K})) = 0. We could then take h_{2} = h_{1}^{−T}. We will instead use invariant theory for the group SL_{2}(ℤ/nℤ) to give explicit formulae for h_{1} and h_{2}.

1.4. Introductory example: the case n = 3. The family of curves parametrised by X(3) ≅ ℙ^{1} is the Hesse pencil of plane cubics

{A(x^{3} + y^{3} + z^{3}) + Bxyz = 0} ⊂ ℙ^{2}

with base point (x : y : z) = (0 : 1 : −1). An equation for this family in Weierstrass form is y^{2} = x^{3}−27c_{4}(A, B)x−54c_{6}(A, B) with discriminant Δ = (c_{4}^{3}−c_{6}^{2})/1728 = D^{3} where

D(A, B) = −27A^{4} − AB^{3}
c_{4}(A, B) = −216A^{3}B + B^{4}
c_{6}(A, B) = 5832A^{6} − 540A^{3}B^{3} − B^{6}.

The action of SL_{2}(ℤ/3ℤ) on X(3) ≅ ℙ^{1} lifts to a representation

ρ : SL_{2}(ℤ/3ℤ) → SL_{2}(\overline{K})

given on the generators S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) by

ρ(S) = \frac{−1}{ζ_{3} − ζ_{3}^{2}} (\begin{matrix}1 & 1/3 \\ 6 & −1 \end{matrix}) , ρ(T) = (\begin{matrix}ζ_{3} & 0 \\ 0 & ζ_{3}^{2} \end{matrix}) .

The hypothesis (2) is satisfied with ε = ( \begin{matrix}1 & 0 \\ 0 & 18 \end{matrix} ).

Lemma 1.8. Let E/K be an elliptic curve and ϕ : E[3] ≅ μ_{3} × ℤ/3ℤ a symplectic isomorphism over \overline{K}. Let (A : B) be the corresponding \overline{K}-point on X(3) ≅ ℙ^{1} with co-ordinates (A, B) scaled so that

(3)  c_{4}(A, B) = c_{4}(E) and c_{6}(A, B) = c_{6}(E)

where E has Weierstrass equation y^{2} = x^{3}−27c_{4}(E)x−54c_{6}(E). If j(E) ≠ 0, 1728 then

h_{1} = (\begin{matrix}A & −\frac{∂D}{∂B} \\ B & \frac{∂D}{∂A} \end{matrix} )

satisfies the hypotheses of Lemma 1.6.

Proof: Since (A : B) ∈ X(3) is not a cusp we have det h_{1} = 4D ≠ 0 and so h_{1} ∈ GL_{2}(\overline{K}). Let G = Im(ρ). There is a unique character χ : G → μ_{3} such that

(4)  \begin{aligned}D ∘ g & = χ(g)D \\ c_{4} ∘ g & = χ(g)^{2}c_{4} \\ c_{6} ∘ g & = χ(g)^{3}c_{6} \end{aligned}

for all g ∈ G. It follows by the chain rule that

(5)  (−\frac{∂D}{∂B}_{\frac{∂D}{∂A}} ) ∘ g = χ(g)g (−\frac{∂D}{∂B}_{\frac{∂D}{∂A}} )

for all g ∈ G.

Let ξ_{σ} = σ(ϕ)ϕ^{−1} ∈ SL_{2}(ℤ/3ℤ). Since ρ describes the action of SL_{2}(ℤ/3ℤ) on X(3) ≅ ℙ^{1} we have

(6)  σ (\begin{matrix}A \\ B \end{matrix}) = λ_{σ}ρ(ξ_{σ}) (\begin{matrix}A \\ B \end{matrix})

for some λ_{σ} ∈ \overline{K}^{×}. Then by (4) we have

σ(c_{4}(A, B)) = λ_{σ}^{4}χ_{σ}^{2}c_{4}(A, B) and σ(c_{6}(A, B)) = λ_{σ}^{6}χ_{σ}^{3}c_{6}(A, B)

where χ_{σ} = χ(ρ(ξ_{σ})). Since c_{4}(E), c_{6}(E) ∈ K it follows by (3) and our assumption j(E) ≠ 0, 1728 that λ_{σ}^{2}χ_{σ} = 1. Using (5) and (6) we compute

σ(h_{1}) = h_{1} ∘ (λ_{σ}ρ(ξ_{σ})) = ρ(ξ_{σ})h_{1} (\begin{matrix}λ_{σ} & 0 \\ 0 & λ_{σ}^{3}χ_{σ} \end{matrix}) ∝ ρ(ξ_{σ})h_{1}.

Hence σ(h_{1})h_{1}^{−1} ∝ ρ(ξ_{σ}) as required. □

Lemma 1.6 shows that X_{E}(3) and X_{E}^{−}(3) are the twists of X(3) ≅ ℙ^{1} by h_{1} and εh_{1}^{−T}. To compute the families of curves parametrised by X_{E}(3) and X_{E}^{−}(3) we put

c_{4}(λ, μ) = c_{4}(λA − μ\frac{∂D}{∂B}, λB + μ\frac{∂D}{∂A}),
c_{6}(λ, μ) = c_{6}(λA − μ\frac{∂D}{∂B}, λB + μ\frac{∂D}{∂A}),

and

c_{4}^{∗}(λ, μ) = \frac{1}{(36D)^{4}} c_{4}(λB + μ\frac{∂D}{∂A}, −18(λA − μ\frac{∂D}{∂B})),
c_{6}^{∗}(λ, μ) = \frac{1}{(36D)^{6}} c_{6}(λB + μ\frac{∂D}{∂A}, −18(λA − μ\frac{∂D}{∂B})).

The coefficients of c_{4}(λ, μ), c_{6}(λ, μ), c_{4}^{∗}(λ, μ), c_{6}^{∗}(λ, μ) may be written as rational functions in c_{4}(A, B) and c_{6}(A, B), thereby giving the formulae in Theorem 1.1. The following alternative proof of Theorem 1.1 has the advantage over that in [F2] of not requiring any knowledge of the Hessian or the contravariants. We include it to illustrate the approach we take for n = 7, 9, 11.

Proof of Theorem 1.1: We assume j(E) ≠ 0, 1728 so that Lemma 1.8 applies. By [RS1, Proposition 2.1] the formulae in the statement of the theorem define a family of elliptic curves with constant mod 3 Galois representation. In the case of Y_{E}(3) the proof is completed by specialising to (λ : μ) = (1 : 0). In general, that is, to give an argument that also applies to Y_{E}^{−}(3), we note that since we have specified a family of elliptic curves with the right j-invariant, it must be correct up to quadratic twist, say by δ ∈ K^{×}. It remains to show that δ is a square. As noted in [HK2, Section 7.1] it suffices to prove this in the case ϕ : E[3] ≅ μ_{3} × ℤ/3ℤ is defined over K. But in that case the families of curves parametrised by X(3), X_{E}(3) and X_{E}^{−}(3) are the same, and this is born out by our construction. □

2. Equations for X(n)

We recall equations of Klein [K1], [K2], [K3] for the modular curves X(n). Our treatment follows the survey in [F0, Chapter 4], but see also [AR], [V2].

2.1. Klein’s equations. Let E be an elliptic curve and P, Q a basis for E[n] with e_{n}(P, Q) = ζ_{n}. If we embed E ⊂ ℙ^{n−1} by a complete linear system |D| of degree n then the translation maps τ_{P} and τ_{Q} extend to automorphisms of ℙ^{n−1}. The following lemma is recalled from [F1].

Lemma 2.1. (i) We may change co-ordinates on ℙ^{n−1} (over \overline{K}) so that τ_{P} and τ_{Q} are given by

M_{P} = (\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & ζ_{n} & 0 & ⋯ & 0 \\ 0 & 0 & ζ_{n}^{2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ζ_{n}^{n−1} \end{matrix}) and M_{Q} = (\begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}).

(ii) If n is odd and [−1]^{∗}D ∼ D then there is a unique choice of co-ordinates (over \overline{K}) such that τ_{P}, τ_{Q} and multiplication by −1 are given by M_{P}, M_{Q} and

[−1] = (\begin{matrix}1 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \end{matrix}).

We restrict to n ≥ 5 an odd integer. Given a symplectic isomorphism ϕ : E[n] ≅ μ_{n} × ℤ/nℤ we let P, Q be the basis for E[n] with ϕ(iP + jQ) = (ζ_{n}^{i}, j) for all i, j ∈ ℤ/nℤ. If we embed E ⊂ ℙ^{n−1} via the complete linear system |n.0_{E}| and choose co-ordinates as in Lemma 2.1 then sending (E, ϕ) to the image of 0_{E} defines an embedding Y (n) ⊂ ℙ^{n−1}. Indeed if we know the co-ordinates of 0_{E} ∈ ℙ^{n−1} then M_{P} and M_{Q} allow us to write down n^{2} points on E. Since E is defined by quadrics, it follows by Bézout’s theorem that these n^{2} points suffice to determine E. The embedding Y (n) ⊂ ℙ^{n−1} is defined over K since ⟨M_{P} , M_{Q}⟩ ⊂ PGL_{n}(\overline{K}) is isomorphic to μ_{n} × ℤ/nℤ as a Galois module.

We write (x_{0} : x_{1} : …: x_{n−1}) for our co-ordinates on ℙ^{n−1} and agree to read all subscripts mod n. Since n is odd we have

n.0_{E} ∼ 0_{E} + P + 2P + … + (n − 1)P.

Therefore 0_{E} belongs to exactly one of the hyperplanes fixed by M_{P}. But 0_{E} is fixed by [−1] so we have either

\begin{aligned}& 0 & = & (0 : a_{1} : a_{2} : …: a_{2} : a_{1}) & (+) \\ or & 0 & = & (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) & (−) \end{aligned}

where a_{1}, a_{2}, … are non-zero.

Let W be the vector space of quadrics on ℙ^{n−1} and V the subspace of quadrics vanishing on E. Then dim W = n(n + 1)/2 and dim V = n(n − 3)/2. The action of M_{P} allows us to write these as direct sums V = ⊕V_{i} and W = ⊕W_{i} with

V_{i} ⊂ W_{i} = ⟨x_{i}^{2}, x_{i−1}x_{i+1}, …⟩.

Since n is odd it follows by the action of M_{Q} that dim V_{i} = (n−3)/2 and dim W_{i} = (n + 1)/2. The requirement that the quadrics in V_{0} vanish at 0_{E} and its translates under M_{Q} imposes some linear conditions on the coefficients of these quadrics. This leads us to rule out the case (+) and to make the following definition.

Definition 2.2. For n ≥ 5 an odd integer let Z(n) ⊂ ℙ^{n−1} be the subvariety defined by a_{0} = 0, a_{n−i} = −a_{i} and

(7)  rank(a_{i−j}a_{i+j})_{i,j=0}^{n−1} ≤ 2.

We note that (7) is equivalent to the vanishing of the 4 × 4 Pfaffians of this skew-symmetric matrix. The above construction shows that Y (n) ⊂ Z(n). It is natural to ask whether Z(n) = X(n). Vélu [V2] proved this in the case n = p is a prime. However when n is composite Z(n) has extra components.

When n = 7 we put 0_{E} = (0 : a : b : −c : c : −b : −a). Then Z(7) is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} & −c^{2} \\ a^{2} & 0 & ac & −bc \\ b^{2} & −ac & 0 & ab \\ c^{2} & bc & −ab & 0 \end{matrix}) ≤ 2.

Thus X(7) = Z(7) is the Klein quartic {a^{3}b + b^{3}c + c^{3}a = 0} ⊂ ℙ^{2}.

When n = 9 we put 0_{E} = (0 : a : −b : d : c : −c : −d : b : −a). Then Z(9) is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} & −d^{2} & −c^{2} \\ a^{2} & 0 & −ad & bc & cd \\ b^{2} & ad & 0 & ac & −bd \\ d^{2} & −bc & −ac & 0 & −ab \\ c^{2} & −cd & bd & ab & 0 \end{matrix}) ≤ 2,

equivalently

(a^{2}b + b^{2}c + c^{2}a)d = 0  bc^{3} − ba^{3} − cd^{3} = 0
ab^{3} − ac^{3} − bd^{3} = 0  ca^{3} − cb^{3} − ad^{3} = 0.

Adding together the last three equations and factoring shows that Z(9) is the union of the curve

X(9) = \{ \begin{matrix}a^{2}b + b^{2}c + c^{2}a & = 0 \\ ab^{2} + bc^{2} + ca^{2} & = d^{3} \end{matrix}\} ⊂ ℙ^{3}

and four isolated points

(0 : 0 : 0 : 1), (1 : 1 : 1 : 0), (1 : ζ_{3} : ζ_{3}^{2} : 0), (1 : ζ_{3}^{2} : ζ_{3} : 0).

When n = 11 we put 0_{E} = (0 : a : −c : b : e : d : −d : −e : −b : c : −a). Then X(11) ⊂ ℙ^{4} is the singular locus of the Hessian of the cubic threefold

{a^{2}b + b^{2}c + c^{2}d + d^{2}e + e^{2}a = 0} ⊂ ℙ^{4}.

We refer to [AR] for further details. We checked using Magma that the homogeneous ideal of X(11) is generated by the 4 × 4 minors of the Hessian matrix of the cubic form.

2.2. The action of SL_{2}(ℤ/nℤ). We continue to take n ≥ 5 odd. In Section 2.1 we defined an embedding X(n) ⊂ ℙ^{m−1} where m = (n − 1)/2. The action of SL_{2}(ℤ/nℤ) on X(n) extends to a projective representation \overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{m}(\overline{K}). We show that it lifts to a representation. See [AR, Appendix I] for a discussion of how this relates to work of Weil.

Proposition 2.3. The projective representation \overline{ρ} : SL_{2}(ℤ/nℤ) → PGL_{m}(\overline{K}) lifts to a representation ρ : SL_{2}(ℤ/nℤ) → GL_{m}(\overline{K}).

Proof: The action of γ = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ/nℤ) on Y (n) is given by

(8)  γ : (E, P, Q) ↦ (E, dP − cQ, −bP + aQ).

If we embed X(n) ⊂ ℙ^{n−1} as described in Section 2.1 then the action (8) extends to a projective representation \overline{π} : SL_{2}(ℤ/nℤ) → PGL_{n}(\overline{K}) where the image of γ = ( \begin{matrix}a & b \\ c & d \end{matrix}) is uniquely determined by the properties that

(9)  \overline{π}(γ)^{−1}M_{P}^{u}M_{Q}^{v}\overline{π}(γ) ∝ M_{P}^{du−bv}M_{Q}^{−cu+av}

for all u, v ∈ ℤ/nℤ, and \overline{π}(γ) commutes with [−1]. We regard \overline{π} as describing an action on ℙ^{n−1} = ℙ(W) where W is an n-dimensional vector space. The action of [−1] gives an eigenspace decomposition W = W_{+} ⊕ W_{−} with dim W_{±} = (n ± 1)/2. We may then identify \overline{ρ} with the restriction of \overline{π} to ℙ(W_{−}) = ℙ^{m−1}. To prove the proposition we show more generally that \overline{π} lifts to a representation π : SL_{2}(ℤ/nℤ) → GL_{n}(\overline{K}).

Let S = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) be the usual generators of SL_{2}(ℤ/nℤ). In view of the relations (ST)^{3} = S^{4} = T^{n} = I_{2}, the only 1-dimensional characters of SL_{2}(ℤ/nℤ) are the ones in the case n is a multiple of 3 that factor via PSL_{2}(ℤ/3ℤ) ≅ A_{4}. Using (8) and (9) we compute

(10)  \overline{π}(S) ∝ (ζ_{n}^{ij})_{i,j=0}^{n−1}  \overline{π}(T) ∝ Diag(ζ_{n}^{i^{2}/2})_{i=0}^{n−1}

where the exponents are read as elements of ℤ/nℤ.

If M ∈ GL_{n}(\overline{K}) acts on W_{±} then we write M_{±} for its restriction to W_{±}. Since

3(1^{2} + 2^{2} + … + m^{2}) ≡ 0 (mod n)

it is clear by (10) that there is a lift π(T) of \overline{π}(T) and 1-dimensional characters χ_{±} of SL_{2}(ℤ/nℤ) such that det(π(T)_{±}) = χ_{±}(T). Next we lift \overline{π}(S) to a matrix π(S) such that

(11)  π(S)π(T)^{−1}π(S) = π(T)π(S)π(T).

Restricting to W_{±} and taking determinants it follows that det(π(S)_{±}) = 1 = χ_{±}(S). For each γ ∈ SL_{2}(ℤ/nℤ) we now let π(γ) be the unique lift of \overline{π}(γ) such that det(π(γ)_{±}) = χ_{±}(γ). These lifts exist since S and T generate SL_{2}(ℤ/nℤ) and are unique since dim W_{−} and dim W_{+} are coprime. It is evident that the map π so defined is a group homomorphism. □

Remark 2.4. (i) A calculation using (11) shows that π(S) = g_{n}^{−1}(ζ^{ij})_{i,j=0}^{n−1} where the Gauss sum g_{n} = ∑_{i=0}^{n−1} ζ^{−i^{2}/2} satisfies g_{n}^{2} = (−1)^{(n−1)/2}n.

(ii) If we take 0_{E} = (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) then with respect to co-ordinates (a_{1} : …: a_{m}) we may take

ρ(S) = g_{n}^{−1}(ζ^{ij} − ζ^{−ij})_{i,j=1}^{m} ρ(T) = Diag(ζ^{i^{2}/2})_{i=1}^{m}.

In particular ρ(−I_{2}) = (−1)^{(n+1)/2}I_{m}. (iii) Since ρ(S) and ρ(T) are symmetric we see that (2) holds with ε = I_{m}. (iv) If 3 ∤ n then SL_{2}(ℤ/nℤ) has no 1-dimensional characters and so the lifts we have constructed are unique. If 3 | n then 3 ∤ m and we can make ρ unique by demanding that det ρ(T) = 1, equivalently ρ takes values in SL_{m}(\overline{K}).

3. Equations for X_{E}(n) and X_{E}^{−}(n)

We derive our equations for X_{E}(n) and X_{E}^{−}(n) by using invariant theory for the group SL_{2}(ℤ/nℤ) to twist the equations for X(n) in Section 2.1. We split into the cases n = 7, 9, 11.

3.1. Formulae in the case n = 7. We recall that X(7) is the Klein quartic {F = 0} ⊂ ℙ^{2} where F = a^{3}b + b^{3}c + c^{3}a. Let G ≅ PSL_{2}(ℤ/7ℤ) be the image of ρ : SL_{2}(ℤ/7ℤ) → GL_{3}(\overline{K}). It is generated by

\frac{1}{g_{7}} (\begin{matrix}ζ_{7} − ζ_{7}^{6} & ζ_{7}^{2} − ζ_{7}^{5} & ζ_{7}^{4} − ζ_{7}^{3} \\ ζ_{7}^{2} − ζ_{7}^{5} & ζ_{7}^{4} − ζ_{7}^{3} & ζ_{7} − ζ_{7}^{6} \\ ζ_{7}^{4} − ζ_{7}^{3} & ζ_{7} − ζ_{7}^{6} & ζ_{7}^{2} − ζ_{7}^{5} \end{matrix}) and (\begin{matrix}ζ_{7} & 0 & 0 \\ 0 & ζ_{7}^{4} & 0 \\ 0 & 0 & ζ_{7}^{2} \end{matrix})

where g_{7} = 1 + 2(ζ_{7}^{3} + ζ_{7}^{5} + ζ_{7}^{6}).

Definition 3.1. An invariant of degree m is a homogeneous polynomial I = I(a, b, c) of degree m such that I ∘ g = I for all g ∈ G.

Recalling formulae of Klein we put

H = (−1/54) × | \begin{matrix}\frac{∂^{2}F}{∂a^{2}} & \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂a∂c} \\ \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂b^{2}} & \frac{∂^{2}F}{∂b∂c} \\ \frac{∂^{2}F}{∂a∂c} & \frac{∂^{2}F}{∂b∂c} & \frac{∂^{2}F}{∂c^{2}} \end{matrix} | ,

c_{4} = (1/9) × | \begin{matrix}\frac{∂^{2}F}{∂a^{2}} & \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂a∂c} & \frac{∂H}{∂a} \\ \frac{∂^{2}F}{∂a∂b} & \frac{∂^{2}F}{∂b^{2}} & \frac{∂^{2}F}{∂b∂c} & \frac{∂H}{∂b} \\ \frac{∂^{2}F}{∂a∂c} & \frac{∂^{2}F}{∂b∂c} & \frac{∂^{2}F}{∂c^{2}} & \frac{∂H}{∂c} \\ \frac{∂H}{∂a} & \frac{∂H}{∂b} & \frac{∂H}{∂c} & 0 \end{matrix} | , c_{6} = (1/14) × | \begin{matrix}\frac{∂F}{∂a} & \frac{∂F}{∂b} & \frac{∂F}{∂c} \\ \frac{∂H}{∂a} & \frac{∂H}{∂b} & \frac{∂H}{∂c} \\ \frac{∂c_{4}}{∂a} & \frac{∂c_{4}}{∂b} & \frac{∂c_{4}}{∂c} \end{matrix} | .

The ring of invariants K[a, b, c]^{G} is generated by F, H, c_{4} and c_{6} subject to a single relation which reduces when we set F = 0 to

c_{4}^{3} − c_{6}^{2} ≡ 1728H^{7} (mod F).

Since F, H, c_{4} and c_{6} have degrees 4, 6, 14 and 21 it is clear that every invariant of odd degree is divisible by c_{6}.

Lemma 3.2. The j-invariant X(7) → ℙ^{1} is given by j = c_{4}^{3}/H^{7}.

Proof: Both j and j_{0} = c_{4}^{3}/H^{7} define maps X(7) → ℙ^{1} that quotient out by the action of G ≅ PSL_{2}(ℤ/7ℤ). So they can differ by at most a Möbius map. We recall that j is ramified above 0, 1728 and ∞ with ramification indices 3, 2 and 7. Since

#\{F = c_{4} = 0\} ≤ 4 deg(c_{4}) = \frac{1}{3}|G|
#\{F = c_{6} = 0\} ≤ 4 deg(c_{6}) = \frac{1}{2}|G|
#\{F = H = 0\} ≤ 4 deg(H) = \frac{1}{7}|G|

and j_{0} − 1728 = c_{6}^{2}/H^{7} it follows that j = j_{0} as required. □

Definition 3.3. A covariant column, respectively contravariant column, of degree m is a column vector v = (v_{1}, v_{2}, v_{3})^{T} of homogeneous polynomials of degree m in variables a, b, c such that v ∘ g = gv, respectively v ∘ g = g^{−T} v, for all g ∈ G.

We note that x = (a, b, c)^{T} is a covariant column of degree 1, whereas if I is an invariant of degree m then ∇I = (\frac{∂I}{∂a}, \frac{∂I}{∂b}, \frac{∂I}{∂c})^{T} is a contravariant column of degree m − 1.

Lemma 3.4. Let E/K be an elliptic curve and ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ a symplectic isomorphism over \overline{K}. Let (a : b : c) be the corresponding \overline{K}-point on X(7) ⊂ ℙ^{2} with co-ordinates (a, b, c) scaled so that

(12)  c_{4}(a, b, c) = c_{4}(E) and c_{6}(a, b, c) = c_{6}(E)

where E has Weierstrass equation y^{2} = x^{3}−27c_{4}(E)x−54c_{6}(E). If j(E) ≠ 0, 1728 and h ∈ GL_{3}(\overline{K}) is a matrix whose columns are covariant columns, respectively contravariant columns, of the same degree mod 7 evaluated at (a, b, c) then

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1}),

respectively

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1})^{−T} ,

for all σ ∈ Gal(\overline{K}/K).

Proof: Let ξ_{σ} = σ(ϕ)ϕ^{−1} ∈ SL_{2}(ℤ/7ℤ). Recalling that ρ describes the action of SL_{2}(ℤ/7ℤ) on X(7) ⊂ ℙ^{2} we have

(13)  σ((a, b, c)^{T}) = λ_{σ}ρ(ξ_{σ})(a, b, c)^{T}

for some λ_{σ} ∈ \overline{K}^{×}. Since c_{4} and c_{6} are invariants of degrees 14 and 21 we deduce

σ(c_{4}(a, b, c)) = λ_{σ}^{14}c_{4}(a, b, c) and σ(c_{6}(a, b, c)) = λ_{σ}^{21}c_{6}(a, b, c)

for all σ ∈ Gal(\overline{K}/K). Since c_{4}(E), c_{6}(E) ∈ K it follows by (12) and our assumption j(E) ≠ 0, 1728 that λ_{σ}^{14} = λ_{σ}^{21} = 1. Hence λ_{σ} is a 7th root of unity. Now suppose the columns of h are obtain by specialising polynomials whose degrees are all congruent to r mod 7. Then by (13) and Definition 3.3 we have

σ(h) = h ∘ (λ_{σ}ρ(ξ_{σ})) = λ_{σ}^{r}ρ(ξ_{σ})h,

respectively

σ(h) = h ∘ (λ_{σ}ρ(ξ_{σ})) = λ_{σ}^{r}ρ(ξ_{σ})^{−T} h.

Hence σ(h)h^{−1} ∝ ρ(ξ_{σ}), respectively σ(h)h^{−1} ∝ ρ(ξ_{σ})^{−T}, as required. □

We use Lemmas 1.6 and 3.4 to compute equations for X_{E}(7) and X_{E}^{−}(7). First we classify the covariant and contravariant columns. It is evident that

• The dot product of a covariant column and a contravariant column is an invariant.

• The cross product of two covariant columns is a contravariant column.

• The cross product of two contravariant columns is a covariant column. We also write [v_{1}, v_{2}, v_{3}] = (v_{1} × v_{2}) · v_{3} for the scalar triple product. It is easy to solve for the covariant and contravariant columns of any given degree by linear algebra. Let e and f be the covariant columns of degrees 9 and 11 given by

e = \frac{I_{22}(∇F × ∇H) − c_{4}(∇F × ∇c_{4}) + 12H^{2}(∇H × ∇c_{4})}{14c_{6}}
f = \frac{I_{24}(∇F × ∇H) − (16F^{4} − 104F H^{2})(∇F × ∇c_{4}) + c_{4}(∇H × ∇c_{4})}{14c_{6}}

where

I_{22} = 448F^{4}H − 48F^{2}c_{4} − 2048F H^{3}
I_{24} = 128F^{6} − 160F^{3}H^{2} − 236F Hc_{4} − 336H^{4}.

Lemma 3.5. (i) The covariant columns of odd degree, respectively even degree, form a free k[F, H, c_{4}]-module of rank 3 generated by x, e, f, respectively ∇F × ∇H, ∇F × ∇c_{4}, ∇H × ∇c_{4}.

(ii) The contravariant columns of odd degree, respectively even degree, form a free k[F, H, c_{4}]-module of rank 3 generated by ∇F, ∇H, ∇c_{4}, respectively x × e, x × f, e × f.

Proof: By direct calculation we have [x, e, f] = −c_{6}, whereas the definition of c_{6} may be rewritten as [∇F, ∇H, ∇c_{4}] = 14c_{6}.

Let v be a covariant column of odd degree. We write v = I_{1}x + I_{2}e + I_{3}f where I_{1}, I_{2}, I_{3} are rational functions in a, b, c. Taking the dot product with e × f shows that [v, e, f] = I_{1}[x, e, f]. But [v, e, f] is an invariant of odd degree and therefore divisible by c_{6}. It follows that I_{1} is an invariant and likewise for I_{2} and I_{3}.

The other cases are similar. □

Theorem 3.6. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. If j(E) ≠ 0, 1728 then X_{E}(7) ⊂ ℙ^{2} has equation F = 0 where

F = 12x^{3}z + 108x^{2}y^{2} + 3c_{4}x^{2}z^{2} + 72c_{4}xy^{2}z − 108c_{4}y^{4} − 12c_{6}xyz^{2}
+ 84c_{6}y^{3}z + c_{4}^{2}xz^{3} − 15c_{4}^{2}y^{2}z^{2} + c_{4}c_{6}yz^{3} + 768Δz^{4},

and X_{E}^{−}(7) ⊂ ℙ^{2} has equation G = 0 where

G = 3x^{4} + c_{4}x^{3}z − 18c_{4}x^{2}y^{2} − 3c_{6}x^{2}yz + 24c_{6}xy^{3} + 3c_{4}^{2}xy^{2}z
− 9c_{4}^{2}y^{4} − c_{4}c_{6}y^{3}z + 168Δxz^{3} + 1728Δy^{2}z^{2} + 5c_{4}Δz^{4}.

Proof: The covariant columns x, ∇F × ∇H, He have degrees 1, 8, 15, and the contravariant columns ∇F, x×e, H^{2}∇H have degrees 3, 10, 17. The determinants of the matrices formed from these columns are

(14)

det(x, (∇F × ∇H), He) = 72H^{4} − 4c_{4}F H
det(∇F, (x × e), H^{2}∇H) = 72H^{5} − 4c_{4}F H^{2}.

The coefficients of the quartic F̃(x, y, z) = F (xx + y(∇F × ∇H) + zHe) are invariants. Using linear algebra to rewrite these invariants as polynomials in F, H, c_{4} and c_{6} we find

F̃(x, y, z) = F x^{4} + 12H^{3}x^{3}z + (108H^{3} − 6c_{4}F)x^{2}y^{2} − 8c_{6}F xy^{3}
+ 3c_{4}H^{3}x^{2}z^{2} + (72c_{4}H^{3} + 4128F^{2}H^{4} + 48c_{4}F^{3}H − 768F^{5}H^{2})xy^{2}z
+ (−108c_{4}H^{3} − 3c_{4}^{2}F − 11376F^{2}H^{4} + 32c_{4}F^{3}H + 3392F^{5}H^{2} − 256F^{8})y^{4}
− 12c_{6}H^{3}xyz^{2} + (84c_{6}H^{3} − 16c_{6}F^{3}H)y^{3}z + (c_{4}^{2}H^{3} + 688F H^{7}
+ 8c_{4}F^{2}H^{4} − 128F^{4}H^{5})xz^{3} + (−15c_{4}^{2}H^{3} − 10512F H^{7} − 384c_{4}F^{2}H^{4}
+ 6144F^{4}H^{5} + 96c_{4}F^{5}H^{2} − 768F^{7}H^{3})y^{2}z^{2} + (c_{4}c_{6}H^{3} − 8c_{6}F^{2}H^{4})yz^{3}
+ (768H^{10} − 36c_{4}F H^{7} − c_{4}^{2}F^{2}H^{4} + 176F^{3}H^{8} + 16c_{4}F^{4}H^{5} − 64F^{6}H^{6})z^{4}.

Likewise G̃(x, y, z) = F (x∇F + y(x × e) + zH^{2}∇H) becomes

G̃(x, y, z) = (3H^{2} + 28F^{3})x^{4} + (c_{4}H^{2} + 168F^{2}H^{3})x^{3}z + (−18c_{4}H^{2}
− 816F^{2}H^{3} − 24c_{4}F^{3} + 192F^{5}H)x^{2}y^{2} − 3c_{6}H^{2}x^{2}yz + 24c_{6}H^{2}xy^{3}
+ (222F H^{6} + 24F^{4}H^{4})x^{2}z^{2} + (3c_{4}^{2}H^{2} + 3744F H^{6} − 576F^{4}H^{4})xy^{2}z
+ (−9c_{4}^{2}H^{2} − 5184F H^{6} − 240c_{4}F^{2}H^{3} − 4c_{4}^{2}F^{3} + 2240F^{4}H^{4} + 64c_{4}F^{5}H
− 256F^{7}H^{2})y^{4} + (−c_{4}c_{6}H^{2} + 8c_{6}F^{2}H^{3})y^{3}z + (168H^{9} + 3c_{4}F H^{6}
+ 24F^{3}H^{7})xz^{3} + (1728H^{9} − 78c_{4}F H^{6} + 816F^{3}H^{7} + 24c_{4}F^{4}H^{4}
− 192F^{6}H^{5})y^{2}z^{2} + c_{6}F H^{6}yz^{3} + (5c_{4}H^{9} + 35F^{2}H^{10} − 4F^{5}H^{8})z^{4}.

Let (a : b : c) be the \overline{K}-point on X(7) corresponding to (E, ϕ) for some choice of symplectic isomorphism ϕ : E[7] ≅ μ_{7}×ℤ/7ℤ. By Lemma 3.2 we may scale (a, b, c) to satisfy (12). Then by Lemmas 1.6 and 3.4 a formula for X_{E}(7), respectively X_{E}^{−}(7), is given by specialising the coefficients of F̃, respectively G̃, to this choice of (a, b, c). Explicitly we set F = 0, divide through by H^{3}, respectively H^{2}, and replace H^{7} by Δ. □

Remark 3.7. The equations for X_{E}(7) and X_{E}^{−}(7) given in Theorems 1.2 and 3.6 are related by F(x, y, z) = \frac{1}{4}F(6c_{4}z − \frac{1}{3}y, x, −18z) and G(x, y, z) = G(9c_{4}y + z, 3x, 108y) where a = −27c_{4} and b = −54c_{6}.

3.2. Formulae in the case n = 9. We recall that X(9) = {F_{1} = F_{2} = 0} ⊂ ℙ^{3} where F_{1} = a^{2}b + b^{2}c + c^{2}a and F_{2} = ab^{2} + bc^{2} + ca^{2} − d^{3}. Let G ≅ SL_{2}(ℤ/9ℤ) be the image of ρ : SL_{2}(ℤ/9ℤ) → SL_{4}(\overline{K}). It is generated by

\frac{1}{3} (\begin{matrix}ζ_{9} − ζ_{9}^{8} & ζ_{9}^{7} − ζ_{9}^{2} & ζ_{9}^{4} − ζ_{9}^{5} & ζ_{9}^{3} − ζ_{9}^{6} \\ ζ_{9}^{7} − ζ_{9}^{2} & ζ_{9}^{4} − ζ_{9}^{5} & ζ_{9} − ζ_{9}^{8} & ζ_{9}^{3} − ζ_{9}^{6} \\ ζ_{9}^{4} − ζ_{9}^{5} & ζ_{9} − ζ_{9}^{8} & ζ_{9}^{7} − ζ_{9}^{2} & ζ_{9}^{3} − ζ_{9}^{6} \\ ζ_{9}^{3} − ζ_{9}^{6} & ζ_{9}^{3} − ζ_{9}^{6} & ζ_{9}^{3} − ζ_{9}^{6} & 0 \end{matrix}) and (\begin{matrix}ζ_{9} & 0 & 0 & 0 \\ 0 & ζ_{9}^{4} & 0 & 0 \\ 0 & 0 & ζ_{9}^{7} & 0 \\ 0 & 0 & 0 & ζ_{9}^{6} \end{matrix})

Let χ : G → μ_{3} be the character given by Diag(ζ_{9}, ζ_{9}^{4}, ζ_{9}^{7}, ζ_{9}^{6}) ↦ ζ_{9}^{3}.

Definition 3.8. An invariant of type r ∈ {0, 1, 2} and degree m is a homogeneous polynomial I = I(a, b, c, d) of degree m such that I ∘ g = χ(g)^{r}I for all g ∈ G.

The type 0 invariants of smallest degree are D = −(a^{3} + b^{3} + c^{3} − 3abc)d and I_{6} = 2(a^{5}b+b^{5}c+c^{5}a)+5(a^{4}c^{2} +b^{4}a^{2} +c^{4}b^{2})+20abc(a^{2}b+b^{2}c+c^{2}a)−3d^{6}. Writing A and B for their matrices of second partial derivatives we find

det(A + tB) = 81D^{2} − 1620DI_{6}t + 2700I_{12}t^{2} + 90000D^{4}t^{4}.

where I_{12} is a type 0 invariant of degree 12.

Definition 3.9. A covariant column, respectively contravariant column, of type r ∈ {0, 1, 2} and degree m is a column vector v = (v_{1}, v_{2}, v_{3}, v_{4})^{T} of homogeneous polynomials of degree m in variables a, b, c, d such that v∘g = χ(g)^{r}gv, respectively v ∘ g = χ(g)^{r}g^{−T} v, for all g ∈ G.

We note that x_{1} = (a, b, c, d)^{T} is a covariant column of type 0 and degree 1, whereas if I is an invariant of type r and degree m then ∇I = (\frac{∂I}{∂a}, \frac{∂I}{∂b}, \frac{∂I}{∂c}, \frac{∂I}{∂d})^{T} is a contravariant column of type r and degree m − 1. To construct some further examples we define 4 × 4 alternating matrices

Λ_{i} = (\begin{matrix}0 & 𝔮_{i}(a, b, c)d^{2} & −𝔮_{i}(c, a, b)d^{2} & 𝔯_{i}(a, b, c) \\ −𝔮_{i}(a, b, c)d^{2} & 0 & 𝔮_{i}(b, c, a)d^{2} & 𝔯_{i}(b, c, a) \\ 𝔮_{i}(c, a, b)d^{2} & −𝔮_{i}(b, c, a)d^{2} & 0 & 𝔯_{i}(c, a, b) \\ −𝔯_{i}(a, b, c) & −𝔯_{i}(b, c, a) & −𝔯_{i}(c, a, b) & 0 \end{matrix})

for i = 0, 1, 2 where

𝔮_{0}(a, b, c) = 3(a^{2} + 2bc)  𝔯_{0}(a, b, c) = 2(a^{3}b + c^{3}b) − 3a^{2}c^{2} − b^{4}
𝔮_{1}(a, b, c) = 3(c^{2} + 2ab)  𝔯_{1}(a, b, c) = 2(b^{3}c + a^{3}c) − 3b^{2}a^{2} − c^{4}
𝔮_{2}(a, b, c) = 3(b^{2} + 2ac)  𝔯_{2}(a, b, c) = 2(c^{3}a + b^{3}a) − 3c^{2}b^{2} − a^{4}.

For M a 4 × 4 alternating matrix and v a column vector we define a column vector M ⋆ v by the rule (M ⋆ v)_{l} = ∑ M_{ij}v_{k} where the sum is over all (i, j, k) for which (i, j, k, l) is an even permutation of (1, 2, 3, 4). We define covariant and contravariant columns

x_{7} = Λ_{0}∇D  e_{9} = Λ_{1}∇I_{6}  u_{5} = Λ_{1} ⋆ x_{1}
e_{7} = Λ_{1}∇D  f_{7} = Λ_{2}∇D  v_{11} = Λ_{2} ⋆ x_{7}.

Then c_{4} = x_{7} · u_{5} is an invariant of type 1 and degree 12, and c_{6} = \frac{1}{2}(x_{7} · ∇I_{12}) is an invariant of type 0 and degree 18. Routine Gröbner basis calculations show that c_{4}, c_{6} ∉ (F_{1}, F_{2}) yet

c_{4}^{3} − c_{6}^{2} ≡ 1728D^{9} mod (F_{1}, F_{2}).

Lemma 3.10. The j-invariant X(9) → ℙ^{1} is given by j = c_{4}^{3}/D^{9}.

Proof: Both j and j_{0} = c_{4}^{3}/D^{9} define maps X(9) → ℙ^{1} that quotient out by the action of G/{±I} ≅ PSL_{2}(ℤ/9ℤ). So they can differ by at most a Möbius map.

We recall that j is ramified above 0, 1728 and ∞ with ramification indices 3, 2 and 9. Since

#\{F_{1} = F_{2} = c_{4} = 0\} ≤ 9 deg(c_{4}) = \frac{1}{3}|G/\{±I\}|
#\{F_{1} = F_{2} = c_{6} = 0\} ≤ 9 deg(c_{6}) = \frac{1}{2}|G/\{±I\}|
#\{F_{1} = F_{2} = D = 0\} ≤ 9 deg(D) = \frac{1}{9}|G/\{±I\}|

and j_{0} − 1728 = c_{6}^{2}/D^{9} it follows that j = j_{0} as required. □

Lemma 3.11. Let E/K be an elliptic curve and ϕ : E[9] ≅ μ_{9}×ℤ/9ℤ a symplectic isomorphism over \overline{K}. Let (a : b : c : d) be the corresponding \overline{K}-point on X(9) ⊂ ℙ^{3} with co-ordinates (a, b, c, d) scaled so that

(15)  c_{4}(a, b, c, d) = c_{4}(E) and c_{6}(a, b, c, d) = c_{6}(E)

where E has Weierstrass equation y^{2} = x^{3}−27c_{4}(E)x−54c_{6}(E). If j(E) ≠ 0, 1728 and h ∈ GL_{4}(\overline{K}) is a matrix whose columns are covariant columns, respectively contravariant columns, of types r_{1}, …, r_{4} and degrees m_{1}, …, m_{4} with m_{i} + 6r_{i} constant mod 18, evaluated at (a, b, c, d) then

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1}),

respectively

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1})^{−T} ,

for all σ ∈ Gal(\overline{K}/K).

Proof: Let ξ_{σ} = σ(ϕ)ϕ^{−1} ∈ SL_{2}(ℤ/9ℤ). Recalling that ρ describes the action of SL_{2}(ℤ/9ℤ) on X(9) ⊂ ℙ^{3} we have

(16)  σ((a, b, c, d)^{T}) = λ_{σ}ρ(ξ_{σ})(a, b, c, d)^{T}

for some λ_{σ} ∈ \overline{K}^{×}. Since c_{4} and c_{6} are invariants of types 1 and 0 and degrees 12 and 18 we deduce

σ(c_{4}(a, b, c, d)) = λ_{σ}^{12}χ_{σ}c_{4}(a, b, c, d) and σ(c_{6}(a, b, c, d)) = λ_{σ}^{18}c_{6}(a, b, c, d)

where χ_{σ} = χ(ρ(ξ_{σ})). Since c_{4}(E), c_{6}(E) ∈ K it follows by (15) and our assumption j(E) ≠ 0, 1728 that λ_{σ}^{12}χ_{σ} = λ_{σ}^{18} = 1. Hence λ_{σ} is an 18th root of unity and χ_{σ} = λ_{σ}^{6}. Now suppose the columns of h are obtained by specialising covariant columns, respectively contravariant columns, of types r_{1}, …, r_{4} and degrees m_{1}, …, m_{4} with m_{i} + 6r_{i} ≡ r (mod 18). Then by (16) and Definition 3.9 we have

σ(h) = h ∘ (λ_{σ}ρ(ξ_{σ})) = λ_{σ}^{r}ρ(ξ_{σ})h,

respectively

σ(h) = h ∘ (λ_{σ}ρ(ξ_{σ})) = λ_{σ}^{r}ρ(ξ_{σ})^{−T} h.

□

We use Lemmas 1.6 and 3.11 to compute equations for X_{E}(9) and X_{E}^{−}(9). First we construct some more covariant and contravariant columns. Let v_{9} be the contravariant column of type 2 and degree 9 given by

v_{9} = (f_{1}d, f_{2}d, f_{3}d, −af_{1} − bf_{2} − cf_{3})^{T}

where

f_{1}(a, b, c, d) = a^{8} + 35a^{6}bc + 42a^{5}b^{3} + 7a^{5}c^{3} + 105a^{4}b^{2}c^{2} + 28a^{4}bd^{3}
+ 231a^{3}b^{4}c − 196a^{3}bc^{4} + 14a^{3}c^{2}d^{3} + 14a^{2}b^{6} − 70a^{2}b^{3}c^{3} − 84a^{2}b^{2}cd^{3}
− 21a^{2}c^{6} + 28ab^{4}d^{3} − 105ab^{2}c^{5} − 14abc^{3}d^{3} − 27acd^{6} + 19b^{7}c
− 35b^{4}c^{4} + 14b^{3}c^{2}d^{3} + 27b^{2}d^{6} − 27bc^{7} + 14c^{5}d^{3},

f_{2}(a, b, c, d) = f_{1}(b, c, a, d) and f_{3}(a, b, c, d) = f_{1}(c, a, b, d). We further put

x_{13} = Λ_{1}v_{9}  x_{15} = Λ_{1}v_{11}  f_{13} = Λ_{0}v_{9}.

In the following lemma the entries of the covariant and contravariant columns are viewed as elements of the co-ordinate ring K[a, b, c, d]/(F_{1}, F_{2}). The lemma is included to show that we have been systematic, rather than because it is needed in what follows. We therefore omit the proof.

Lemma 3.12. The covariant columns, respectively contravariant columns, of type r ∈ {0, 1, 2}, mod (F_{1}, F_{2}), form a free K[D, c_{6}]-module of rank 4 with basis as indicated in the following table.

Covariants  Contravariants
Type 0 Type 1 Type 2  Type 0 Type 1 Type 2
x_{1}  e_{7}  f_{7}  ∇D  u_{5}  v_{9}
x_{7}  e_{9}  f_{13}  ∇I_{6}  ∇c_{4}  v_{11}
x_{13}  c_{4}x_{1} c_{4}e_{7}  ∇I_{12} c_{4}∇D c_{4}u_{5}
x_{15}  c_{4}x_{7} c_{4}e_{9}  ∇c_{6} c_{4}∇I_{6} c_{4}∇c_{4}

Theorem 3.13. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6}. If j(E) ≠ 0, 1728 then X_{E}(9) ⊂ ℙ^{3} has equations F_{1} = F_{2} = 0 where

F_{1} = 24x^{2}t − 96xyz + 64y^{3} + 24c_{4}y^{2}t + 48c_{4}yz^{2} − 48c_{6}yzt + 12c_{4}^{2}yt^{2}
− 8c_{6}z^{3} + 12c_{4}^{2}z^{2}t − 6c_{4}c_{6}zt^{2} − (c_{4}^{3} − 2c_{6}^{2})t^{3},
F_{2} = 16x^{2}z − 64xy^{2} − 16c_{4}xyt − 16c_{4}xz^{2} + 16c_{6}xzt − 4c_{4}^{2}xt^{2} + 80c_{4}y^{2}z
− 32c_{6}y^{2}t − 32c_{6}yz^{2} + 32c_{4}^{2}yzt − 8c_{4}c_{6}yt^{2} + 8c_{4}^{2}z^{3} − 12c_{4}c_{6}z^{2}t
+ (2c_{4}^{3} + 4c_{6}^{2})zt^{2} − c_{4}^{2}c_{6}t^{3}.

and X_{E}^{−}(9) ⊂ ℙ^{3} has equations G_{1} = G_{2} = 0 where

G_{1} = −72x^{2}y − 144x^{2}z + 24c_{4}xyt − 48c_{4}xzt − 8c_{6}xt^{2} − c_{4}y^{3} + 18c_{4}y^{2}z
+ 3c_{6}y^{2}t + 180c_{4}yz^{2} + 12c_{6}yzt − 3c_{4}^{2}yt^{2} + 792c_{4}z^{3} + 12c_{6}z^{2}t
+ 2c_{4}^{2}zt^{2} + c_{4}c_{6}t^{3},
G_{2} = −864x^{3} + 216c_{4}x^{2}t + 2592c_{4}xyz + 216c_{6}xyt + 15552c_{4}xz^{2}
+ 432c_{6}xzt − 72c_{4}^{2}xt^{2} − 9c_{6}y^{3} + 162c_{6}y^{2}z + 27c_{4}^{2}y^{2}t + 4212c_{6}yz^{2}
+ 108c_{4}^{2}yzt − 27c_{4}c_{6}yt^{2} + 26136c_{6}z^{3} + 972c_{4}^{2}z^{2}t + 18c_{4}c_{6}zt^{2}
+ (5c_{4}^{3} + 4c_{6}^{2})t^{3}.

Proof: The covariant columns x_{1}, f_{7}, De_{9}, Dx_{15} have types 0, 2, 1, 0 and degrees 1, 7, 13, 19. The contravariant columns u_{5}, D^{2}∇D, ∇I_{12}, D^{2}v_{9} have types 1, 0, 0, 2 and degrees 5, 11, 11, 17. The determinants of the matrices formed from these columns satisfy

det(x_{1}, f_{7}, De_{9}, Dx_{15}) ≡ 3456D^{10} mod (F_{1}, F_{2})
det(u_{5}, D^{2}∇D, ∇I_{12}, D^{2}v_{9}) ≡ 1152D^{11} mod (F_{1}, F_{2}).

For certain 2 × 2 matrices α = (α_{ij}) and β = (β_{ij}), specified in Remark 3.14 below, we put

F_{i}(x, y, z, t) = (α_{i1}F_{1} + α_{i2}F_{2})(xx_{1} + yf_{7} + zDe_{9} + tDx_{15})
G_{i}(x, y, z, t) = (β_{i1}F_{1} + β_{i2}F_{2})(xu_{5} + yD^{2}∇D + z∇I_{12} + tD^{2}v_{9})

for i = 1, 2. Using the Gröbner basis machinery in Magma to write each coefficient mod (F_{1}, F_{2}) as a polynomial in c_{4} and c_{6} we obtained the equations in the statement of the theorem. By Lemmas 1.6, 3.10 and 3.11 these are equations for X_{E}(9) and X_{E}^{−}(9). □

Remark 3.14. (i) The 2 × 2 matrices used in the proof of Theorem 3.13 were

α = \frac{1}{9D^{6}} (\begin{matrix}3B & 9A \\ −(108A^{3} + B^{3}) & 9AB^{2} \end{matrix}) and β = \frac{1}{D^{6}} (\begin{matrix}6A & −B \\ 162AB^{2} & 9(108A^{3} + B^{3}) \end{matrix})

where A = d^{3} and B = a^{3} + b^{3} + c^{3} + 6abc. (ii) The equations for X_{E}(9) and X_{E}^{−}(9) in Theorems 1.3 and 3.13 are related by

F_{i}(x, y, z, t) ∝ F_{i}(x − 27c_{4}z − 162c_{6}t, 9y + 81c_{4}t, −54z, 324t)
G_{i}(x, y, z, t) ∝ G_{i}(−x + 9c_{4}t, 36y + 36z, 6z, 108t)

for i = 1, 2 where a = −27c_{4} and b = −54c_{6}.

3.3. Formulae in the case n = 11. We recall that X(11) is the singular locus of the Hessian of the cubic threefold {F = 0} ⊂ ℙ^{4} where

F = a^{2}b + b^{2}c + c^{2}d + d^{2}e + e^{2}a.

Let G ≅ PSL_{2}(ℤ/11ℤ) be the image of ρ : SL_{2}(ℤ/11ℤ) → GL_{5}(\overline{K}). It is generated by

\frac{1}{g_{11}} (\begin{matrix}ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} \\ ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} \\ ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} \\ ζ_{11}^{5} − ζ_{11}^{−5} & ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} \\ ζ_{11}^{4} − ζ_{11}^{−4} & ζ_{11} − ζ_{11}^{−1} & ζ_{11}^{3} − ζ_{11}^{−3} & ζ_{11}^{9} − ζ_{11}^{−9} & ζ_{11}^{5} − ζ_{11}^{−5} \end{matrix})

and Diag(ζ_{11}, ζ_{11}^{9}, ζ_{11}^{4}, ζ_{11}^{3}, ζ_{11}^{5}) where g_{11} = 1 + 2(ζ_{11} + ζ_{11}^{3} + ζ_{11}^{9} + ζ_{11}^{5} + ζ_{11}^{4}). We define the invariants, covariant columns and contravariant columns exactly as in Section 3.1. Let ∑ denote a sum over all cyclic permutations, so that for example F = ∑ a^{2}b. Other examples of invariants of small degree include

H = 3abcde + ∑(a^{3}c^{2} − a^{3}de),
I_{7} = ∑(a^{6}e + 3a^{5}d^{2} − 15a^{4}bce + 5a^{3}b^{3}d + 15a^{3}bcd^{2}),
I_{8} = ∑(a^{7}c − 7a^{4}bd^{3} − 7a^{4}de^{3} + 7a^{3}b^{2}c^{3} + 21a^{3}c^{2}d^{2}e).

Writing A and B for the matrices of second partial derivatives of F and H we find

det(A + tB) = 32H − 32I_{7}t − 24I_{9}t^{2} − 8c_{4}t^{3} + …

where I_{9} and c_{4} are invariants of degrees 9 and 11. We will not need a complete set of generators for the ring of invariants, but note that this is given in [A] and may also be computed using Magma. Let I be the homogeneous ideal of X(11), i.e. the ideal generated by the 4×4 minors of the Hessian matrix of F. The degree 19 polynomial

c̃_{6} = a^{9}b^{10} − 509b^{18}d − 14107b^{14}d^{4}e + 510b^{9}c^{10} + 42326b^{7}d^{12} + 20669b^{3}d^{15}e
− 14107b^{2}d^{2}e^{15} − 277419bc^{2}d^{10}e^{6} − 248909bcd^{16}e − 209926bcd^{5}e^{12}
+ 762409bd^{11}e^{7} + be^{18} − 1018c^{18}e − 14107c^{16}de^{2} − 586835c^{12}d^{3}e^{4}

+ 197780c^{10}d^{4}e^{5} + 1019c^{9}d^{10} − 787130c^{8}d^{5}e^{6} + 15634c^{7}d^{11}e + 42326c^{7}e^{12}

+ 2007576c^{6}d^{6}e^{7} + 247382c^{5}d^{12}e^{2} − 528424c^{5}de^{13} − 616653c^{4}d^{7}e^{8}
+ 376744c^{3}d^{13}e^{3} + 1067732c^{3}d^{2}e^{14} − 225004c^{2}d^{8}e^{9} + 463659cd^{14}e^{4}
− 582142cd^{3}e^{15} + 70511d^{9}e^{10}

is not an invariant but satisfies

c̃_{6}^{2} ≡ abcde(c_{4}^{3} − 1728F^{11}) (mod I).

Lemma 3.15. The j-invariant X(11) → ℙ^{1} is given by j = c_{4}^{3}/F^{11}.

Proof: Both j and j_{0} = c_{4}^{3}/F^{11} define maps X(11) → ℙ^{1} that quotient out by the action of G ≅ PSL_{2}(ℤ/11ℤ). So they can differ by at most a Möbius map. We recall that j is ramified above 0, 1728 and ∞ with ramification indices 3, 2 and 11. It is shown in [AR, Corollary 23.28] that X(11) ⊂ ℙ^{4} has degree 20. Since

#X(11) ∩ \{c_{4} = 0\} ≤ 20 deg(c_{4}) = \frac{1}{3}|G|
#X(11) ∩ {c̃_{6} = 0} ≤ 20 deg(c̃_{6}) < |G|
#X(11) ∩ \{F = 0\} ≤ 20 deg(F) = \frac{1}{11}|G|

and j_{0} − 1728 = c̃_{6}^{2}/((abcde)F^{11}) it follows that j = j_{0} as required. □

Lemma 3.16. Let E/K be an elliptic curve and ϕ : E[11] ≅ μ_{11} × ℤ/11ℤ a symplectic isomorphism over \overline{K}. Let (a : b : c : d : e) be the corresponding \overline{K}-point on X(11) ⊂ ℙ^{4} with co-ordinates (a, b, c, d, e) scaled so that

(17)  c_{4}(a, b, c, d, e) = c_{4}(E)

where E has Weierstrass equation y^{2} = x^{3} − 27c_{4}(E)x − 54c_{6}(E). If j(E) ≠ 0 and h ∈ GL_{5}(\overline{K}) is a matrix whose columns are covariant columns, respectively contravariant columns, of the same degree mod 11 evaluated at (a, b, c, d, e) then

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1}),

respectively

σ(h)h^{−1} ∝ ρ(σ(ϕ)ϕ^{−1})^{−T} ,

for all σ ∈ Gal(\overline{K}/K).

Proof: The proof is similar to that of Lemma 3.4. Recall that c_{4} is a homogeneous polynomial of degree 11 and so (17) determines the scaling of (a, b, c, d, e) up to an 11th root of unity. □

We use Lemmas 1.6 and 3.16 to compute equations for X_{E}(11) and X_{E}^{−}(11). First we construct some covariant columns. Let x_{1} = (a, b, c, d, e)^{T}. If γ ∈ SL_{2}(ℤ/11ℤ) is diagonal then ρ(γ) cyclically permutes the co-ordinates a, b, c, d, e. A covariant column is therefore uniquely determined by its first entry. By averaging over the group we found covariant columns x_{4}, x_{5}, x_{9} with first entries

f_{4} = 2a^{2}e^{2} + 4ab^{2}c − 4ac^{2}d + 4bce^{2} + d^{4},
f_{5} = −5a^{3}ce + 5a^{2}b^{2}d + 5a^{2}cd^{2} + 5abc^{2}e − 10abde^{2} + b^{5} − 5b^{3}cd + 5bd^{3}e + 5c^{2}e^{3},
f_{9} = −14a^{6}bde − 8a^{5}bd^{3} + 9a^{5}c^{2}e^{2} + 2a^{5}de^{3} + 8a^{4}b^{4}e + 5a^{4}b^{2}c^{3} + 63a^{4}b^{2}cde

+ 6a^{4}c^{4}d − 18a^{4}c^{2}d^{2}e + 8a^{4}d^{3}e^{2} + 31a^{3}b^{4}d^{2} − 21a^{3}b^{3}e^{3} + 47a^{3}b^{2}cd^{3} + 35a^{3}bc^{3}e^{2}

+ 14a^{3}bcde^{3} − 12a^{3}c^{2}d^{4} + 10a^{3}d^{5}e + 3a^{2}b^{5}ce − 26a^{2}b^{3}c^{4} − 42a^{2}b^{3}c^{2}de
− 75a^{2}b^{3}d^{2}e^{2} + 3a^{2}b^{2}e^{5} + 18a^{2}bc^{5}d − 30a^{2}bc^{3}d^{2}e − 36a^{2}bcd^{3}e^{2} + 2a^{2}c^{3}e^{4}
− 9a^{2}cde^{5} + a^{2}d^{7} − 2ab^{7}d − 6ab^{5}cd^{2} + 50ab^{4}ce^{3} − 7ab^{3}c^{2}d^{3} − 6ab^{3}d^{4}e
− 54ab^{2}c^{4}e^{2} − 3ab^{2}c^{2}de^{3} − 9ab^{2}d^{2}e^{4} − 29abc^{3}d^{4} + 21abcd^{5}e + abe^{7} + 9ac^{5}de^{2}
+ 25ac^{3}d^{2}e^{3} − 7acd^{3}e^{4} − 10b^{6}c^{2}e − 2b^{6}de^{2} + 4b^{4}c^{5} + 40b^{4}c^{3}de − 6b^{4}cd^{2}e^{2}
+ 13b^{3}ce^{5} − 3b^{3}d^{6} − 15b^{2}c^{4}d^{2}e − 54b^{2}c^{2}d^{3}e^{2} + 31b^{2}d^{4}e^{3} − 11bc^{4}e^{4} + 3bc^{2}de^{5}
− 2bcd^{7} − 7bd^{2}e^{6} − c^{7}d^{2} + 5c^{5}d^{3}e − 9c^{3}d^{4}e^{2} + 8cd^{5}e^{3} − e^{9}.

We temporarily write a_{1}, …, a_{5} for a, b, c, d, e and let Ξ be the 5 × 5 alternating matrix with entries

Ξ_{ij} = \frac{∂F}{∂a_{r}} \frac{∂I_{7}}{∂a_{s}} − \frac{∂F}{∂a_{s}} \frac{∂I_{7}}{∂a_{r}}

where r ≡ (i − j − 2)^{3} + i + 3 (mod 5) and s ≡ (j − i − 2)^{3} + j + 3 (mod 5). Then x_{14} = Ξ∇I_{7} is a covariant column of degree 14.

Theorem 3.17. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. If j(E) ≠ 0, 1728 then X_{E}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

F = v^{3} + 3v^{2}w + c_{4}v^{2}y + 3vw^{2} + 2c_{4}vwy − c_{4}Δvx^{2} + 48Δvxy
+ 9w^{3} + 5c_{4}w^{2}y − c_{4}^{2}w^{2}z + c_{4}^{2}wy^{2} − 576Δwyz + 72c_{4}Δwz^{2}
− 4Δ^{2}x^{3} − 72Δ^{2}x^{2}z + 4c_{4}Δxy^{2} + 2c_{4}^{2}Δxyz − (c_{4}^{3}Δ − 1728Δ^{2})xz^{2}
+ 64Δy^{3} − 72c_{4}Δy^{2}z + 12c_{4}^{2}Δyz^{2} + (c_{4}^{3}Δ − 3456Δ^{2})z^{3},

and X_{E}^{−}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

G = 5v^{3} − c_{4}v^{2}x − 60v^{2}y + 28c_{4}v^{2}z − 2c_{4}Δvw^{2} − 48Δvwx
− 240Δvwz − 16c_{4}vxy + 1680vy^{2} − 872c_{4}vyz + 121c_{4}^{2}vz^{2}
+ 8Δ^{2}w^{3} + 44c_{4}Δw^{2}y − 11c_{4}^{2}Δw^{2}z + c_{4}Δwx^{2} + 336Δwxy
− 122c_{4}Δwxz + 25c_{4}^{2}wy^{2} − 14160Δwyz + 817c_{4}Δwz^{2} − 20Δx^{3}
+ 5c_{4}^{2}x^{2}y − 1884Δx^{2}z − 364c_{4}xy^{2} + 160c_{4}^{2}xyz − 34764Δxz^{2}
+ 19840y^{3} − 10268c_{4}y^{2}z + 1643c_{4}^{2}yz^{2} − 129220Δz^{3}.

Proof: The covariant columns x_{1}, x_{4}, x_{5}, x_{9}, x_{14} have degrees 1, 4, 5, 9, 14 and the contravariant columns ∇F, ∇I_{7}, ∇I_{8}, ∇I_{9}, ∇c_{4} have degrees 2, 6, 7, 8, 10. The determinants of the matrices formed from these columns satisfy

(18)

det(x_{1}, x_{4}, x_{5}, x_{9}, x_{14}) = c_{4}^{3} − 1728F^{11} (mod I)
det(∇F, ∇I_{7}, ∇I_{8}, ∇I_{9}, ∇c_{4}) = 55(c_{4}^{3} − 1728F^{11}) (mod I).

The coefficients of the cubic F̃(v, w, x, y, z) = F (vx_{1}+wx_{4}+xx_{5}+yx_{9}+zx_{14}) are invariants. Using the Gröbner basis machinery in Magma to rewrite the coefficients mod I as polynomials in c_{4} and F we find

F̃ = F v^{3} + 3F^{2}v^{2}w + c_{4}v^{2}y + 3F^{3}vw^{2} + 2F c_{4}vwy − c_{4}vx^{2} + 48F^{5}vxy
+ 9F^{4}w^{3} + 5F^{2}c_{4}w^{2}y − c_{4}^{2}w^{2}z + c_{4}^{2}wy^{2} − 576F^{9}wyz + 72F^{7}c_{4}wz^{2}
− 4F^{5}x^{3} − 72F^{8}x^{2}z + 4F^{4}c_{4}xy^{2} + 2F^{2}c_{4}^{2}xyz − (c_{4}^{3} − 1728F^{11})xz^{2}
+ 64F^{9}y^{3} − 72F^{7}c_{4}y^{2}z + 12F^{5}c_{4}^{2}yz^{2} + (F^{3}c_{4}^{3} − 3456F^{14})z^{3}.

Likewise G̃(v, w, x, y, z) = F (v∇F + w∇I_{7} + x∇I_{8} + y∇I_{9} + z∇c_{4}) becomes

G̃ = 5F^{2}v^{3} − c_{4}v^{2}x − 60F^{4}v^{2}y + 28F c_{4}v^{2}z − 2F c_{4}vw^{2} − 48F^{5}vwx
− 240F^{6}vwz − 16F^{2}c_{4}vxy + 1680F^{6}vy^{2} − 872F^{3}c_{4}vyz + 121c_{4}^{2}vz^{2}
+ 8F^{6}w^{3} + 44F^{3}c_{4}w^{2}y − 11c_{4}^{2}w^{2}z + F^{3}c_{4}wx^{2} + 336F^{7}wxy
− 122F^{4}c_{4}wxz + 25c_{4}^{2}wy^{2} − 14160F^{8}wyz + 817F^{5}c_{4}wz^{2} − 20F^{7}x^{3}
+ 5c_{4}^{2}x^{2}y − 1884F^{8}x^{2}z − 364F^{4}c_{4}xy^{2} + 160F c_{4}^{2}xyz − 34764F^{9}xz^{2}
+ 19840F^{8}y^{3} − 10268F^{5}c_{4}y^{2}z + 1643F^{2}c_{4}^{2}yz^{2} − 129220F^{10}z^{3}.

Let (a : b : c : d : e) be the \overline{K}-point on X(11) corresponding to (E, ϕ) for some choice of symplectic isomorphism ϕ : E[11] ≅ μ_{11} × ℤ/11ℤ. By Lemma 3.15 we may scale (a, b, c, d, e) to satisfy (17) and F (a, b, c, d, e)^{11} = Δ. Moreover the determinants (18) are non-zero by our assumption j(E) ≠ 1728. Then by Lemmas 1.6 and 3.16 cubics describing X_{E}(11) and X_{E}^{−}(11) are obtained by putting

(19)  \begin{aligned}F(v, w, x, y, z) & = \frac{1}{F^{4}} F̃(F v, w, F^{7}x, F^{2}y, F^{4}z) \\ G(v, w, x, y, z) & = \frac{1}{F^{8}} G̃(F^{2}v, F^{8}w, F^{4}x, y, F^{3}z) \end{aligned}

and replacing F^{11} by Δ. □

Remark 3.18. The cubic forms describing X_{E}(11) and X_{E}^{−}(11) in Theorems 1.4 and 3.17 are related by

F(v, w, x, y, z) = \frac{1}{2^{3}c_{6}^{3}} F(−v′, w′, −864x, −36c_{4}x − 108c_{6}z, 72y)
G(v, w, x, y, z) = \frac{1}{2^{5}3^{6}(55c_{6})^{3}} G(v′′, −427680y, x′′, −y′′, −z′′)

where a = −27c_{4}, b = −54c_{6} and

v′ = c_{6}v + 2c_{6}w − 6c_{4}^{2}x + 3c_{4}^{2}y − 9c_{4}c_{6}z  w′ = c_{6}v + 6c_{4}^{2}x + 3c_{4}^{2}y + 9c_{4}c_{6}z
v′′ = 44(2c_{4}v − 6c_{6}w + 33c_{6}x + 135c_{4}^{2}y + 810c_{4}c_{6}z) x′′ = 60(5v + 729c_{4}y + 2187c_{6}z)
y′′ = 11(c_{4}v − 3c_{6}w − 6c_{6}x)  z′′ = 60(v + 27c_{4}y + 81c_{6}z).

4. Diagonal twists

We give an alternative construction of X_{E}(n) and X_{E}^{−}(n) in the case E is an elliptic curve whose n-torsion contains a copy of the Galois module μ_{n}.

Let C → D be an isogeny of elliptic curves with kernel a labelled copy of μ_{n}. Then the dual isogeny has kernel a labelled copy of ℤ/nℤ. The pairs of such curves are parametrised by the modular curve Y_{1}(n). In the cases n = 7, 9 we choose a coordinate λ on X_{1}(n) ≅ ℙ^{1}. In the case n = 11 we recall that X_{1}(11) is the elliptic curve ν^{2} + ν = λ^{3} − λ^{2}. We write λ to indicate λ in the cases n = 7, 9 and the pair λ, ν in the case n = 11. Let C_{λ} and D_{λ} be the corresponding pairs of n-isogenous curves. By [Si, Exercise 8.13] D_{λ} has Weierstrass equation

n = 7 y^{2} − (λ^{2} − λ − 1)xy − (λ^{3} − λ^{2})y = x^{3} − (λ^{3} − λ^{2})x^{2},
n = 9  y^{2} + (λ^{3} − 3λ^{2} + 4λ − 1)xy + λ(λ − 1)^{4}(λ^{2} − λ + 1)y

= x^{3} + λ(λ − 1)(λ^{2} − λ + 1)x^{2},

n = 11  y^{2} + (λν + 2λ − (ν + 1)^{2})xy − λ^{2}ν(ν + 1)(λ − ν − 1)y

= x^{3} − λν(ν + 1)(λ − ν − 1)x^{2}.

On each of these curves P = (0, 0) is a point of order n. If we write the Weierstrass equation for D_{λ} as y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} then by Vélu’s formulae [V1] the n-isogenous curve C_{λ} has Weierstrass equation

(20)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} − 5tx − (a_{1}^{2} + 4a_{2})t − 7w

where t = 6s_{2} + (a_{1}^{2} + 4a_{2})s_{1} + a_{1}a_{3}s_{0}, w = 10s_{3} + 2(a_{1}^{2} + 4a_{2})s_{2} + 3a_{1}a_{3}s_{1} + a_{3}^{2}s_{0} and s_{k} = ∑_{j=1}^{(n−1)/2} x(jP)^{k}. The Weierstrass equations (20) have discriminant

(21) \begin{aligned}n & = 7 & Δ(C_{λ}) & = λ(λ − 1)(λ^{3} − 8λ^{2} + 5λ + 1)^{7} \\ n & = 9 & Δ(C_{λ}) & = λ(λ − 1)(λ^{2} − λ + 1)^{3}(λ^{3} − 6λ^{2} + 3λ + 1)^{9} \\ n & = 11 & Δ(C_{λ,ν}) & = λ(λ − 1)(λν + 2λ^{2} − 2λ + 1)(ν + 1)^{6}f(λ, ν)^{11} \end{aligned}

where f(λ, ν) = (−3λν + 2ν − λ^{3} + 5λ^{2} − 5λ + 1)/(λ − 1).

Lemma 4.1. Let E/K be an elliptic curve and ι : μ_{n} ↪ E[n] an inclusion of Galois modules. Then there exists Q ∈ E(\overline{K})[n] and q ∈ K^{×}/(K^{×})^{n} such that

(i) e_{n}(ι(ζ), Q) = ζ^{−1} for all ζ ∈ μ_{n}, (ii) σ(Q) − Q = ι(\frac{σ \sqrt[n]{q}}{\sqrt[n]{q}}) for all σ ∈ Gal(\overline{K}/K), and (iii) K(E[n]) = K(μ_{n}, \sqrt[n]{q}).

Proof: This follows by standard properties of the Weil pairing, together with Hilbert’s Theorem 90. □

Taking E = C_{λ} we compute q = q(λ) as described in [F1, Section 1.2].

(22)  q(λ) = \{ \begin{matrix}λ^{4}(λ − 1) & if n = 7 \\ λ(λ − 1)^{7}(λ^{2} − λ + 1)^{3} & if n = 9 \\ λν^{2}(λ − 1)(λ − ν − 1)^{3} & if n = 11. \end{matrix}

We consider the following diagonal twists of X(n) for n = 7, 9, 11. Recall in the case n = 11 we take the singular locus of the Hessian of the cubic form.

X[ξ_{1}, ξ_{2}, ξ_{3}] = {ξ_{1}x^{3}y + ξ_{2}y^{3}z + ξ_{3}z^{3}x = 0} ⊂ ℙ^{2}
X[ξ_{1}, ξ_{2}, ξ_{3}; η] = \{ \begin{matrix}ξ_{1}x^{2}y + ξ_{2}y^{2}z + ξ_{3}z^{2}x & = 0 \\ ξ_{1}ξ_{2}xy^{2} + ξ_{2}ξ_{3}yz^{2} + ξ_{1}ξ_{3}zx^{2} & = ηt^{3} \end{matrix}\} ⊂ ℙ^{3}
X[ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}] ∼ {ξ_{1}v^{2}w + ξ_{2}w^{2}x + ξ_{3}x^{2}y + ξ_{4}y^{2}z + ξ_{5}z^{2}v = 0} ⊂ ℙ^{4}.

Lemma 4.2. The diagonal twists are determined up to K-isomorphism by θ ∈ K^{×}/(K^{×})^{n} where

θ = \{ \begin{matrix}ξ_{1}ξ_{2}^{2}ξ_{3}^{4} & if n = 7 \\ ξ_{1}ξ_{2}^{7}ξ_{3}^{4}η^{3} & if n = 9 \\ (ξ_{1}ξ_{2}^{5}ξ_{3}^{3}ξ_{4}^{4}ξ_{5}^{9})^{2} & if n = 11. \end{matrix}

Moreover if we replace θ by θ^{k} where k is a square in (ℤ/nℤ)^{×} then we obtain isomorphic twists.

Proof: The first part is proved by rescaling the co-ordinates and the second part by cyclically permuting them. (The expression for θ in the case n = 11 has been squared to simplify the statement of the next theorem.) □

Theorem 4.3. Let E/K be an elliptic curve and ι : μ_{n} ↪ E[n] an inclusion of Galois modules. If q ∈ K^{×}/(K^{×})^{n} is as specified in Lemma 4.1 then

(i) X_{E}(n) is the diagonal twist with θ = q, and (ii) X_{E}^{−}(n) is the diagonal twist with θ = q^{−1}.

Proof: Let χ : Gal(\overline{K}/K) → ℤ/nℤ be defined by σ(\sqrt[n]{q})/\sqrt[n]{q} = ζ_{n}^{χ(σ)}. Then the symplectic isomorphism ϕ : E[n] ≅ μ_{n} × ℤ/nℤ; ι(ζ_{n}^{x}) + yQ ↦ (ζ_{n}^{−x}, y) satisfies

σ(ϕ)ϕ^{−1} : (ζ_{n}^{x}, y) ↦ (ζ_{n}^{x+χ(σ)y}, y).

Under our identification Aut(μ_{n} × ℤ/nℤ) ≅ SL_{2}(ℤ/nℤ) this corresponds to T^{χ(σ)}. If ρ(T) ∝ Diag(ζ_{n}^{r_{1}}, …, ζ_{n}^{r_{m}}) for some integers r_{1}, …, r_{m} then

h_{1} = Diag((\sqrt[n]{q})^{r_{1}}, …, (\sqrt[n]{q})^{r_{m}})

and h_{2} = h_{1}^{−1} satisfy the hypotheses of Lemma 1.6.

If n = 7 then (r_{1}, r_{2}, r_{3}) = (4, 2, 1) and we deduce

X_{E}(7) ≅ X[q, 1, 1] X_{E}^{−}(7) ≅ X[q^{−1}, 1, 1]

If n = 9 then (r_{1}, r_{2}, r_{3}, r_{4}) = (5, 2, 8, 0) and we deduce

X_{E}(9) ≅ X[1, 1, q; q^{−1}] X_{E}^{−}(9) ≅ X[1, 1, q^{−1}; q]

If n = 11 then (r_{1}, r_{2}, r_{3}, r_{4}, r_{5}) = (6, 10, 2, 7, 8) and we deduce

X_{E}(11) ≅ X[1, 1, q^{−1}, 1, 1] X_{E}^{−}(11) ≅ X[1, 1, q, 1, 1]

The descriptions of X_{E}(n) and X_{E}^{−}(n) now follow by Lemma 4.2. □

It is sometimes convenient to rewrite X[\frac{1}{ξ_{2}} , \frac{1}{ξ_{3}} , \frac{1}{ξ_{1}} ; \frac{η}{ξ_{1}ξ_{2}ξ_{3}} ] as

X^{−}[ξ_{1}, ξ_{2}, ξ_{3}; η] = \{ \begin{matrix}ξ_{1}ξ_{3}x^{2}y + ξ_{1}ξ_{2}y^{2}z + ξ_{2}ξ_{3}z^{2}x & = 0 \\ ξ_{1}xy^{2} + ξ_{2}yz^{2} + ξ_{3}zx^{2} & = ηt^{3} \end{matrix}\} ⊂ ℙ^{3}

For this twist we have θ = ξ_{1}^{2}ξ_{2}^{5}ξ_{3}^{8}η^{3} mod (K^{×})^{9}.

Corollary 4.4. Let E/K be an elliptic curve whose n-torsion contains a copy of the Galois module μ_{n}. Then writing E = C_{λ} we have

X_{E}(7) = X[λ, λ − 1, 1]  X_{E}(9) = X[λ, λ − 1, 1; λ^{2} − λ + 1]
X_{E}^{−}(7) = X[λ − 1, λ, λ(λ − 1)] X_{E}^{−}(9) = X^{−}[λ, λ − 1, 1; 1/(λ^{2} − λ + 1)]

and

X_{E}(11) = X[λ^{2}(λ − 1)^{2}, 1, (λ − ν − 1)^{2}, ν, 1]
X_{E}^{−}(11) = X[λ(λ − 1), 1, λ − ν − 1, ν, ν]

Proof: This follows from Theorem 4.3 and the formula (22) for q(λ). □

Remark 4.5. The formula for X_{E}(7) in Corollary 4.4 was found by Halberstadt and Kraus [HK2, Theorem 7.1] by specialising the formula in Theorem 1.2 and then making a (rather complicated) change of co-ordinates. We worked out the analogue of this in the case n = 9 before discovering the simpler proof presented here.

5. Minimisation and reduction

Let n = 7, 9, 11 and m = (n − 1)/2. Given an elliptic curve E/ℚ the formulae in Section 1.2 give equations for X_{E}(n) ⊂ ℙ^{m−1} and X_{E}^{−}(n) ⊂ ℙ^{m−1}. We search for elliptic curves n-congruent to E by searching for ℚ-rational points on these curves. It helps with this search if we first make a change of co-ordinates over ℚ to simplify the equations, i.e. so that they have small integer coefficients. Following [CFS] this task naturally falls into two parts, called minimisation and reduction. In minimisation one seeks to remove primes from a suitably defined invariant. Then reduction, which may be thought of as the analogue of minimisation at the infinite place, makes a final GL_{m}(ℤ)-transformation.

5.1. The invariant.

Definition 5.1. We split into the cases n = 7, 9, 11. Case n = 7. The invariant of a twisted form of F = x^{3}y + y^{3}z + z^{3}x is

Ψ(μ(F ∘ M)) = μ^{3}(det M)^{4}.

Case n = 9. The invariant of a twisted form of (F_{1}, F_{2}) = (x^{2}y + y^{2}z + z^{2}x, xy^{2} + yz^{2} + zx^{2} − t^{3}) is

Ψ((αF_{1} + βF_{2}) ∘ M, (γF_{1} + δF_{2}) ∘ M) = (αδ − βγ)^{6}(det M)^{9}.

Case n = 11. The invariant of a twisted form of F = v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v is

Ψ(μ(F ∘ M)) = μ^{5}(det M)^{3}.

Lemma 5.2. Let F be one of the twisted forms in Definition 5.1. Then (i) Ψ(F) is well-defined, i.e. it is independent of the choice of M ∈ GL_{m}(\overline{K}). (ii) If F has coefficients in K then Ψ(F) ∈ K.

Proof: (i) This is easy to check for M a scalar matrix. In general we use that Aut(X(n)) ≅ PSL_{2}(ℤ/nℤ). This is proved in [AR, Lemma 20.4] for n ≥ 7 prime, and the same proof works for n = 9. We are reduced to considering M = ρ(γ) for some γ ∈ SL_{2}(ℤ/nℤ). It now suffices to recall (see the proof of Proposition 2.3) that the only 1-dimensional characters of SL_{2}(ℤ/nℤ) are those of order 3 in the case n = 9. This explains why in the case n = 9 we defined the invariant as (αδ − βγ)^{6}(det M)^{9} and not just (αδ − βγ)^{2}(det M)^{3}. (ii) This follows from (i) by Galois theory. □

Remark 5.3. (i) In the case n = 7 it is shown in [PSS, Section 7.1] that Ψ(F) is an integer coefficient polynomial in the coefficients of F. We expect that similar formulae exist in the cases n = 9 and n = 11. (ii) The twisted forms in Theorems 1.2, 1.3 and 1.4 have the following invariants. These were computed by following the proofs in Section 3.

X_{E}(n)  X_{E}^{−}(n)
n = 7 −4(4a^{3} + 27b^{2}) 16(4a^{3} + 27b^{2})^{2}
n = 9 −2^{10}(4a^{3} + 27b^{2})^{4} 2^{8}(4a^{3} + 27b^{2})^{5}
n = 11 −4(4a^{3} + 27b^{2})^{2} 8(4a^{3} + 27b^{2}).

(iii) The diagonal twists studied in Section 4 have invariants

n = 7 X[ξ_{1}, ξ_{2}, ξ_{3}]  Ψ = ξ_{1}ξ_{2}ξ_{3}
n = 9 X[ξ_{1}, ξ_{2}, ξ_{3}; η]  Ψ = (ξ_{1}ξ_{2}ξ_{3})^{4}η^{3}
X^{−}[ξ_{1}, ξ_{2}, ξ_{3}; η] Ψ = (ξ_{1}ξ_{2}ξ_{3})^{5}η^{3}
n = 11 X[ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}] Ψ = ξ_{1}ξ_{2}ξ_{3}ξ_{4}ξ_{5}.

5.2. Minimisation. The level of a model F at a prime p is the p-adic valuation of the invariant, i.e. v_{p}(Ψ(F)). We seek to make a change of co-ordinates that minimises the level. This is a local problem.

Theorem 5.4. Let E be an elliptic curve over ℚ_{p} with p ≠ 2, 3. If n = 7, 9, 11 then X_{E}(n) and X_{E}^{−}(n) admit models (with coefficients in ℤ_{p}) with the following levels.

Kodaira Symbol X_{E}(7) X_{E}^{−}(7) X_{E}(11) X_{E}^{−}(11)
I_{m}, I_{m}^{∗} m ≡ 0 (mod n) 0  0  0  0
I_{m}, I_{m}^{∗} (m/n) = +1  1  2  2  1
I_{m}, I_{m}^{∗} (m/n) = −1  2  1  1  2
II, II^{∗}, III, III^{∗}, IV, IV^{∗}  2  2  2  2

Kodaira Symbol X_{E}(9) X_{E}^{−}(9) Kodaira Symbol X_{E}(9) X_{E}^{−}(9)

I_{m}, I_{m}^{∗} m ≡ 0 (mod 9)  0  0  II, IV^{∗}  5  7
I_{m}, I_{m}^{∗} m ≡ 3, 6 (mod 9) 3  3  III, III^{∗}  6  6
I_{m}, I_{m}^{∗} m ≡ 1 (mod 3)  4  5  IV, II^{∗}  7  5
I_{m}, I_{m}^{∗} m ≡ 2 (mod 3)  5  4

Proof: Replacing E by a quadratic twist does not change the curves X_{E}(n) and X_{E}^{−}(n). So we may assume that either (i) E has good reduction, or (ii) E has split multiplicative reduction, or (iii) E has additive reduction of type II, III or IV. We split into these three cases.

(i) If E has good reduction then by Remark 5.3(ii) the formulae in Theorems 1.2, 1.3 and 1.4 give models for X_{E}(n) and X_{E}^{−}(n) of level 0. (ii) If E has split multiplicative reduction then by the Tate parametrisation E[n] contains a copy of the Galois module μ_{n}. So we may apply the results of Section 4. More precisely if E has Kodaira symbol I_{m} then E(\overline{ℚ}_{p}) ≅ \overline{ℚ}_{p}^{×}/q^{ℤ} for some q ∈ ℚ_{p} with v_{p}(q) = m ≥ 1. It may be verified by computing the Weil pairing that the conditions of Lemma 4.1 are satisfied. So by Theorem 4.3 the curves X_{E}(n) and X_{E}^{−}(n) are the diagonal twists with θ = q or q^{−1}. If n = 7 or 11 then by Lemma 4.2 they admit models of the form X[ξ_{1}, ξ_{2}, …] where at most two of the ξ_{i} have p-adic valuation 1 and the rest are units. We read off the level from Remark 5.3(iii). The case n = 9 is similar. (iii) Finally suppose E has Kodaira symbol II, III or IV. In terms of a minimal Weierstrass equation for E, say y^{2} = x^{3} + ax + b, these are the cases

\begin{matrix}II & v_{p}(a) ≥ 1 & v_{p}(b) = 1, \\ III & v_{p}(a) = 1 & v_{p}(b) ≥ 2, \\ IV & v_{p}(a) ≥ 2 & v_{p}(b) = 2. \end{matrix}

Integer-coefficient models of the required level are obtained by modifying the formulae in Theorems 1.2, 1.3 and 1.4 as follows.

X_{E}(7)  X_{E}^{−}(7)  X_{E}(11)  X_{E}^{−}(11)
II F(x, y, z)  \frac{1}{p^{2}} G(x, y, pz) \frac{1}{p}F(pv + w, w, x, y, z)  G(v, w, x, y, z)
III \frac{1}{p^{3}} F(px, py, z) G(x, \frac{1}{p}y, z)  \frac{1}{p^{2}} F(pv, pw, x, y, z)  \frac{1}{p^{2}} G(pv, pw, px, y, z)
IV \frac{1}{p^{2}} F(x, py, z) \frac{1}{p^{2}} G(x, \frac{1}{p}y, pz) \frac{1}{p^{3}} F(pv, pw, x, py − x, z) \frac{1}{p}G(pv, w, px, y, \frac{1}{p}z)
X_{E}(9)  X_{E}^{−}(9)
II \begin{matrix}\frac{1}{p^{2}} F_{1}(px, py, pz, t) \\ \frac{1}{p^{3}} F_{2}(px, py, pz, t) \end{matrix}  \begin{matrix}\frac{1}{p}G_{1}(px, y, z, t) \\ \frac{1}{p}G_{2}(px, y, z, t) \end{matrix}

III \begin{matrix}\frac{1}{p^{2}} F_{1}(px, py, z, t) \\ \frac{1}{p^{2}} F_{2}(px, py, z, t) \end{matrix}  \begin{matrix}\frac{1}{p}G_{1}(px, y, z, t) \\ \frac{1}{p^{2}} G_{2}(px, y, z, t) \end{matrix}

IV \begin{matrix}\frac{1}{p^{4}} F_{1}(p^{2}x, p^{2}y, pz, t) \\ \frac{1}{p^{5}} F_{2}(p^{2}x, p^{2}y, pz, t) \end{matrix} \begin{matrix}\frac{1}{p^{2}} G_{1}(px, y, z, t) \\ \frac{1}{p^{2}} G_{2}(px, y, z, t) \end{matrix}
□

To compute integer-coefficient models with level as specified in Theorem 5.4 we could in principle follow the proof of the theorem. In practice however it is simpler to use a range of ad hoc tricks. We have not proved that these tricks always get down to the level specified in the theorem, but this does at least happen in all numerical examples we have tried.

Again on the basis of some numerical experimentation, we conjecture that the levels in Theorem 5.4 are the minimal levels. At the primes 2 and 3 it seems the minimal levels can be larger. See Section 7 for some examples.

5.3. Reduction. In Section 2.2 we saw that the action of SL_{2}(ℤ/nℤ) on X(n) ⊂ ℙ^{m−1} = ℙ(V) lifts to an irreducible representation ρ : SL_{2}(ℤ/nℤ) → GL(V). So by the Weyl unitary trick there is an SL_{2}(ℤ/nℤ)-invariant inner product on V, and this is unique up to scalars. To reduce our equations for X_{E}(n) and X_{E}^{−}(n) we run the LLL algorithm on the Gram matrix for this inner product. In the untwisted case it is clear that the SL_{2}(ℤ/nℤ)-invariant inner product is the standard one on ℂ^{m}, i.e. ρ is a unitary representation. So to compute the inner product it suffices to have numerical approximations to the matrices h_{1} and h_{2} in Lemma 1.6. Since our method for finding equations for X_{E}(n) and X_{E}^{−}(n) involved explicitly computing h_{1} and h_{2} this is obviously something we can do. To use these formulae we need numerical approximations for a, b, c … in Lemmas 3.4, 3.11 and 3.16. These are computed by evaluating suitable q-expansions. See [E] or [HK2] for the relevant formulae in the case n = 7.

6. Modular interpretation

In Section 3 we gave explicit formulae for X_{E}(n) and X_{E}^{−}(n) for n = 7, 9, 11. In this section we give equations for the families of curves they parametrise.

6.1. Computing the j-invariant. We first give formulae for the j-map X_{E}(n) → ℙ^{1} and X_{E}^{−}(n) → ℙ^{1}. This is sufficient for some applications: see for example [PSS]. In each case we adapt the formulae in Lemmas 3.2, 3.10 and 3.15 by writing them in an way that behaves well under all changes of co-ordinates.

Case n = 7. Let X = {F = 0} ⊂ ℙ^{2} be a twist of X(7). Starting with F in place of the Klein quartic F the formulae in Section 3.1 define polynomials H(F), c_{4}(F) and c_{6}(F). If F = μ(F ∘ M) then Ψ(F) = μ^{3}(det M)^{4} and

(23)

H(F) = μ^{3}(det M)^{2}(H ∘ M)
c_{4}(F) = μ^{8}(det M)^{6}(c_{4} ∘ M)
c_{6}(F) = μ^{12}(det M)^{9}(c_{6} ∘ M)

As noted in [PSS] the syzygy c_{4}^{3} − c_{6}^{2} ≡ 1728H^{7} (mod F) becomes

c_{4}(F)^{3} − c_{6}(F)^{2} ≡ 1728 Ψ(F) H(F)^{7} (mod F).

In particular the j-map X → ℙ^{1} is given by

j = \frac{c_{4}(F)^{3}}{Ψ(F)H(F)^{7}} .

Case n = 11. Let X ⊂ ℙ^{4} be a twist of X(11) given as the singular locus of the Hessian of a cubic form F = F(v, w, x, y, z). Starting with F in place of the cubic form F = v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v the formulae in Section 3.3 define polynomials H(F) and c_{4}(F). If F = μ(F ∘ M) then Ψ(F) = μ^{5}(det M)^{3} and

(24)  \begin{aligned}H(F) & = μ^{5}(det M)^{2}(H ∘ M) \\ c_{4}(F) & = μ^{17}(det M)^{8}(c_{4} ∘ M) \end{aligned}

By Lemma 3.15 the j-map X → ℙ^{1} is given by

j = \frac{c_{4}(F)^{3}}{Ψ(F)^{8}F^{11}} .

Case n = 9. Our construction of c_{4}(a, b, c, d) and c_{6}(a, b, c, d) in Section 3.3 does not immediately generalise to twists of X(9). Instead we exploit the fact that the pencil of cubics defining X(9) ⊂ ℙ^{3} is naturally a copy of X(3) ≅ ℙ^{1}. The notation X_{M} for a twist of X(n) was introduced in Section 1.1.

Theorem 6.1. Let X_{M} = {F_{1} = F_{2} = 0} ⊂ ℙ^{3} be a twist of X(9) where M is a symplectic Galois module with M ≅ (ℤ/9ℤ)^{2} as an abelian group. Then

(i) Writing H for the determinant of the matrix of second partial derivatives

we have

H(rF_{1} + sF_{2}) = f(r, s)D(a, b, c, d)

where f and D are homogeneous polynomials of degree 4. (ii) We have X_{M[3]} ≅ ℙ^{1} with cusps at the roots of f(r, s) = 0. (iii) For P ∈ X_{M} with tangent line P + tQ write F_{i}(P + tQ) = γ_{i}t^{2} + δ_{i}t^{3} for

i = 1, 2. Then the forgetful map X_{M} → X_{M[3]} is P ↦ (−γ_{2} : γ_{1}).

Proof: We first prove the theorem in the case M ≅ μ_{9} × ℤ/9ℤ.

(i) Taking F_{1} = a^{2}b + b^{2}c + c^{2}a and F_{2} = ab^{2} + bc^{2} + ca^{2} − d^{3} we compute

H(rF_{1} + sF_{2}) = 48(r^{3} + s^{3})s(a^{3} + b^{3} + c^{3} − 3abc)d.

(ii) Since^{2} M[3] ≅ μ_{3} × ℤ/3ℤ we have X_{M[3]} = X(3). We recall from Section 1.4 that X(3) ≅ ℙ^{1} with cusps at the roots of A(27A^{3} + B^{3}) = 0. Our two choices of co-ordinates on X(3) are now related by (r : s) = (B : 3A). (iii) We temporarily write a_{1}, a_{2}, a_{3}, a_{4} for a, b, c, d and let Λ_{2} be the 4 × 4 alternating matrix with (i, j) entry

\frac{∂F_{1}}{∂a_{l}} \frac{∂F_{2}}{∂a_{k}} − \frac{∂F_{1}}{∂a_{k}} \frac{∂F_{2}}{∂a_{l}}

where (i, j, k, l) is an even permutation of (1, 2, 3, 4). This is the same as the matrix Λ_{2} in Section 3.2. By direct calculation we find

Λ_{2} (\begin{matrix}\frac{∂^{2}F_{i}}{∂a^{2}} & … & \frac{∂^{2}F_{i}}{∂a∂d} \\ ⋮ & & ⋮ \\ \frac{∂^{2}F_{i}}{∂a∂d} & … & \frac{∂^{2}F_{i}}{∂d^{2}} \end{matrix} ) Λ_{2} ≡ γ_{i} D (a ; ⋮ ; d) (a ⋯ d) mod (F_{1}, F_{2})

2The pairing e_{9} on M induces a pairing e_{3} on M[3] by the rule e_{3}(3S, T) = e_{9}(S, T) for all S ∈ M and T ∈ M[3].

for i = 1, 2, where γ_{1} = 18d^{3}, γ_{2} = −6(a^{3} + b^{3} + c^{3} + 6abc) and D = −(a^{3} + b^{3} + c^{3} − 3abc)d.

In Corollary 6.5 below we show that the forgetful map X(9) → X(3) is

(25)  (a : b : c : d) ↦ (A : B) = (d^{3} : a^{3} + b^{3} + c^{3} + 6abc).

Thus (B : 3A) = (−γ_{2} : γ_{1}) as required.

Relaxing our restriction on M, the general case of the theorem follows since the forgetful map X_{M} → X_{M[3]} may be characterised geometrically, i.e. it quotients out by the symplectic automorphisms of M that act trivially on M[3]. □

Remark 6.2. If we only use Theorem 6.1 to compute the j-invariant, then we can replace the forward reference (25) in the above proof by the observation that the polynomials c_{4}(A, B) and c_{6}(A, B) in Section 1.4 are related to the polynomials c_{4}(a, b, c, d) and c_{6}(a, b, c, d) in Section 3.2 by

c_{4}(d^{3}, a^{3} + b^{3} + c^{3} + 6abc) ≡ c_{4}(a, b, c, d) mod (F_{1}, F_{2}),
c_{6}(d^{3}, a^{3} + b^{3} + c^{3} + 6abc) ≡ c_{6}(a, b, c, d) mod (F_{1}, F_{2}).

Since we already gave formulae for the universal families above Y_{E}(3) and Y_{E}^{−}(3) in Theorem 1.1, the following corollary gives formulae for the universal families above Y_{E}(9) and Y_{E}^{−}(9).

Corollary 6.3. Let E/K be the elliptic curve y^{2} = x^{3} − 27c_{4}x − 54c_{6}. Let X_{E}(3) and X_{E}^{−}(3) be as given in Theorem 1.1, and let X_{E}(9) and X_{E}^{−}(9) be as given in Theorem 1.3 with a = −27c_{4} and b = −54c_{6}. Then the forgetful maps X_{E}(9) → X_{E}(3) and X_{E}^{−}(9) → X_{E}^{−}(3) are given by (λ : μ) = (γ_{2} : 3γ_{1}) where γ_{1}, γ_{2} are computed by the tangent line construction in Theorem 6.1(iii).

Proof: (i) We have X_{E}(9) = {F_{1} = F_{2} = 0} ⊂ ℙ^{3} where F_{1} and F_{2} are as given in Theorem 1.3. We compute H(rF_{1} + sF_{2}) = f(r, s)D(x, y, z, t) with

(26)  f(r, s) = r^{4} + 2ar^{2}s^{2} − 4brs^{3} − \frac{1}{3}a^{2}s^{4}.

On the other hand the family of curves parametrised by X_{E}(3) in Theorem 1.1 has cusps at the roots of

(27)  D(λ, μ) = λ^{4} − 6c_{4}λ^{2}μ^{2} − 8c_{6}λμ^{3} − 3c_{4}^{2}μ^{4}.

Comparing (26) and (27) we see that our two choices of co-ordinates on X_{E}(3) ≅ ℙ^{1} are related by (λ : μ) = (r : −3s). Taking (r : s) = (−γ_{2} : γ_{1}) gives the result.

(ii) We have X_{E}^{−}(9) = {G_{1} = G_{2} = 0} ⊂ ℙ^{3} where G_{1} and G_{2} are as given in Theorem 1.3. We compute H(rG_{1} + sG_{2}) = f(r, s)D(x, y, z, t) with

(28)  f(r, s) = ar^{4} − 6br^{3}s − 2a^{2}r^{2}s^{2} + 2abrs^{3} − (\frac{1}{3}a^{3} + 3b^{2})s^{4}.

On the other hand the family of curves parametrised by X_{E}^{−}(3) in Theorem 1.1 has cusps at the roots of

(29)  c_{4}(λ, μ) = c_{4}λ^{4} + 4c_{6}λ^{3}μ + 6c_{4}^{2}λ^{2}μ^{2} + 4c_{4}c_{6}λμ^{3} − (3c_{4}^{3} − 4c_{6}^{2})μ^{4}.

Comparing (28) and (29) we see that our choices of co-ordinates on X_{E}^{−}(3) ≅ ℙ^{1} are related by (λ : μ) = (r : −3s). Taking (r : s) = (−γ_{2} : γ_{1}) gives the result. □

6.2. Modular interpretation of X(n). In Section 2.1 we gave equations for X(n). We recall from [F0] that (analogous to Definition 2.2) the elliptic curve E ⊂ ℙ^{n−1} above (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) ∈ Y (n) has equations

rank(a_{i−j}x_{i+j})_{i,j=0}^{n−1} ≤ 2.

Taking n = 7, 9, 11 we now put this curve in Weierstrass form. Notice that the Weierstrass equations have coefficients that are homogeneous polynomials of degree 4m and 6m for some integer m.

Theorem 6.4. We split into the cases n = 7, 9, 11.

(i) The family of curves parametrised by X(7) = {a^{3}b + b^{3}c + c^{3}a = 0} ⊂ ℙ^{2} is

(30)  y^{2} = x^{3} − 27(abc)^{2}c_{4}(a, b, c)x − 54(abc)^{3}c_{6}(a, b, c)

where c_{4}, c_{6} ∈ K[a, b, c] are as defined in Section 3.1. (ii) The family of curves parametrised by X(9) = {F_{1} = F_{2} = 0} ⊂ ℙ^{3} is

y^{2} = x^{3} − 27c_{4}(a, b, c, d)x − 54c_{6}(a, b, c, d)

where c_{4}, c_{6} ∈ K[a, b, c, d] are as defined in Section 3.2. (iii) The family of curves parametrised by X(11) ⊂ ℙ^{4} is

(31)  y^{2} = x^{3} − 27(abcde)c_{4}(a, b, c, d, e)x − 54(abcde)c̃_{6}(a, b, c, d, e).

where c_{4}, c̃_{6} ∈ K[a, b, c, d, e] are as defined in Section 3.3.

Proof: In Section 4 we wrote λ for a co-ordinate (or pair of co-ordinates) on X_{1}(n) for n = 7, 9, 11. With q(λ) as defined in (22), we see by Lemma 4.1 that X(n) is birational to {q(λ) = τ^{n}} ⊂ X_{1}(n) × 𝔾_{m}. In the case n = 7 an explicit birational map is given in [F1, Section 2.2]. Applying the same method for n = 9, 11 we obtain

n = 7  (a : b : c) ↦ (λ, τ) = (−ac^{2}/b^{3}, ac/b^{2}),
n = 9  (a : b : c : d) ↦ (λ, τ) = (−ac/b^{2}, a^{2}d/b^{2}c),
n = 11  (a : b : c : d : e) ↦ (λ, ν, τ) = (−abd/c^{2}e, ab^{3}/c^{3}e, −ab/c^{2}).

We checked directly that these are birational maps, and that the cusps of X(n), i.e. (1 : 0 : …: 0) and its translates under the action of SL_{2}(ℤ/nℤ), map to the cusps of X_{1}(n), i.e. the roots of (21).

Let c_{4}(λ) and c_{6}(λ) be the invariants of the Weierstrass equation for C_{λ} in Section 4. Using Magma we compute

n = 7 \begin{aligned}c_{4}(−ac^{2}/b^{2}) & ≡ ξ_{7}^{4}(abc)^{2}c_{4}(a, b, c) mod (a^{3}b + b^{3}c + c^{3}a) \\ c_{6}(−ac^{2}/b^{2}) & ≡ ξ_{7}^{6}(abc)^{3}c_{6}(a, b, c) mod (a^{3}b + b^{3}c + c^{3}a) \end{aligned}

n = 9 \begin{aligned}c_{4}(−ac/b^{2}) & ≡ ξ_{9}^{4}c_{4}(a, b, c, d) mod (F_{1}, F_{2}) \\ c_{6}(−ac/b^{2}) & ≡ ξ_{9}^{6}c_{6}(a, b, c, d) mod (F_{1}, F_{2}) \end{aligned}

n = 11 \begin{aligned}c_{4}(−abd/c^{2}e, ab^{3}/c^{3}e) & ≡ ξ_{11}^{4}(abcde)c_{4}(a, b, c, d, e) mod I \\ c_{6}(−abd/c^{2}e, ab^{3}/c^{3}e) & ≡ ξ_{11}^{6}(abcde)c̃_{6}(a, b, c, d, e) mod I \end{aligned}

where ξ_{7} = a/b^{5}c, ξ_{9} = a^{2}/b^{4}c and ξ_{11} = a^{3}b/c^{6}e^{2}.

See [HK2, Section 3] for a sketch of an alternative proof in the case n = 7. □

Corollary 6.5. The forgetful map X(9) → X(3) is given by

(a : b : c : d) ↦ (d^{3} : a^{3} + b^{3} + c^{3} + 6abc).

Proof: This follows from Theorem 6.4(ii) and Remark 6.2. □

6.3. An alternative projective embedding. We take p ≥ 5 a prime and let G = PSL_{2}(ℤ/pℤ) act on X(p) in the usual way.

Theorem 6.6 (Adler, Ramanan). The group of G-invariant divisor classes on X(p) is free of rank 1 generated by a divisor class [Λ] of degree (p^{2} − 1)/24.

Proof: See [AR, Theorem 24.1]. □

Let m = (p − 1)/2. Klein showed there are embeddings X(p) ⊂ ℙ^{m−1} and X(p) ⊂ ℙ^{m} with linear G-action. The images are called the z-curve and the Acurve respectively. The corresponding hyperplane sections are (m − 1)Λ and mΛ, and indeed the divisor Λ in Theorem 6.6 is constructed by taking the difference of these. It is conjectured that each of these embeddings is via a complete linear system (the WYSIWYG Hypothesis in [AR]) and this is certainly known for p = 7 and p = 11. The equations for X(p) we have used so far (introduced in Section 2.1) are for the z-curve. However in Sections 6.4 and 6.5 below we also need the Acurve.

Remark 6.7. Let π : 𝔛 → X be the universal family over X = X(p). It can be shown that π_{∗}(Ω_{𝔛/X}) ≅ O(pΛ). Hence the family of curves 𝔛 → X has Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} where

c_{k} ∈ H^{0}(X(p), O(kpΛ))

for k = 4, 6. If we realise X(p) ⊂ ℙ^{m−1} as the z-curve then to write c_{k} as a polynomial (in m variables) we need that m − 1 = (p − 3)/2 divides kp. Thus in the case p = 7 we were able to write c_{4} and c_{6} as polynomials in K[a, b, c]. In the case p = 11 we likewise constructed c_{4} ∈ K[a, b, c, d, e], but there was no polynomial c_{6}.

Case p = 7. The z-curve is the Klein quartic

X(7) = {x^{3}y + y^{3}z + z^{3}x = 0} ⊂ ℙ^{2}.

The cusps of X(7) are the 24 points of inflection. We recall from [PSS] that the cusps are naturally partitioned into eight sets of three {P_{1}, P_{2}, P_{3}} with

P_{1} + 3P_{2} ∼ P_{2} + 3P_{3} ∼ P_{3} + 3P_{1} ∼ H

where H ∼ 2Λ is the hyperplane section. We write T_{0}, …, T_{7} for the effective divisors of degree 3 of the form P_{1} + P_{2} + P_{3} and note that one of these divisors, say T_{0}, satisfies X(7)∩{xyz = 0} = 4T_{0}. We also recall from [PSS] that 2T_{i} ∼ 2T_{j} for all 0 ≤ i, j ≤ 7. It follows by Theorem 6.6 that 2T_{0} ∼ 3Λ. Since 3Λ ∼ 3H −2T_{0} and L(3H − 2T_{0}) has basis x^{2}y, y^{2}x, z^{2}x, xyz the A-curve is the image of

X(7) → ℙ^{3} ; (x : y : z) ↦ (t_{1} : t_{2} : t_{3} : t_{4}) = (x^{2}y : y^{2}z : z^{2}x : xyz)

with equations

rank (\begin{matrix}t_{1} & 0 & t_{4} & −t_{2} \\ t_{2} & −t_{3} & 0 & t_{4} \\ t_{3} & t_{4} & −t_{1} & 0 \end{matrix}) ≤ 2.

Case p = 11. The z-curve is the singular locus of the Hessian of

{F = v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v = 0} ⊂ ℙ^{4}.

We write H ∼ 4Λ for the hyperplane section. The cusps are the 60 points of intersection of X(11) with {F = 0}. They are naturally partitioned into twelve sets of five {P_{1}, …, P_{5}} with

P_{1} + 6P_{3} + 3P_{4} + 10P_{5} ∼ H

and likewise under all cyclic permutations of the P_{i}. We write T_{0}, …, T_{11} for the effective divisors of degree 5 of the form P_{1} + … + P_{5} and note that one of these divisors, say T_{0}, satisfies X(11) ∩ {vwxyz = 0} = 20T_{0}. It may be shown that 5T_{i} ∼ 5T_{j} for all 0 ≤ i, j ≤ 11 and hence 5T_{0} ∼ 5Λ by Theorem 6.6. Since 5Λ ∼ 5H − 15T_{0} we find by computing a basis for L(5H − 15T_{0}) that the A-curve is the image of the morphism X(11) → ℙ^{5} given by

(t_{1} : t_{2} : t_{3} : t_{4} : t_{5} : t_{6}) = (v^{2}wxz : vw^{2}xy : wx^{2}yz : vxy^{2}z : vwyz^{2} : vwxyz).

It is shown in [AR, Theorem 51.1], and we checked using Magma, that this is the singular locus of the quartic hypersurface

t_{6}^{4} − (t_{1}^{2}t_{2} + t_{2}^{2}t_{3} + t_{3}^{2}t_{4} + t_{4}^{2}t_{5} + t_{5}^{2}t_{1})t_{6} + t_{1}^{2}t_{3}t_{5} + t_{2}^{2}t_{4}t_{1} + t_{3}^{2}t_{5}t_{2} + t_{4}^{2}t_{1}t_{3} + t_{5}^{2}t_{2}t_{4} = 0.

6.4. Formulae in the case n = 7.

Theorem 6.8. Let X = {F = 0} ⊂ ℙ^{2} be a twist of the Klein quartic, with hyperplane section H. Let T = P_{1}+P_{2}+P_{3} where P_{1}, P_{2}, P_{3} are points of inflection on X with

P_{1} + 3P_{2} ∼ P_{2} + 3P_{3} ∼ P_{3} + 3P_{1} ∼ H.

Let d ∈ K[x, y, z] be a cubic form with {d = 0} meeting X in a divisor 2D with D ∼ 2T. Then there is a Gal(\overline{K}/K)-module M such that for every field extension L/K and rational point P = (x : y : z) ∈ X (L)\{d = 0}, not a point of inflection, the elliptic curve

(32)  Y^{2} = X^{3} − 27\frac{c_{4}(F)(x, y, z)}{d(x, y, z)^{2}}X − 54\frac{c_{6}(F)(x, y, z)}{d(x, y, z)^{3}}

has 7-torsion isomorphic to M as a Gal(\overline{L}/L)-module.

Proof: We first note that if d_{1}, d_{2} ∈ K[x, y, z] are cubic forms meeting X in divisors 2D_{1} and 2D_{2} with D_{1} ∼ D_{2} then d_{1}/d_{2} is the square of a rational function and hence the elliptic surfaces (32) with d = d_{1} and d = d_{2} are isomorphic over \overline{K}. Since X is a twist of the Klein quartic it follows (by taking D = 2T_{0} as defined in the last section) that the elliptic surfaces (30) and (32) are isomorphic over \overline{K}. (Notice it does not matter whether we write the terms d(x, y, z) in the numerator or in the denominator.) We are done by [RS1, Proposition 2.1]. □

In Theorem 6.10 below we determine rational functions d satisfying the hypothesis of Theorem 6.8 in the cases X = X_{E}(7) and X = X_{E}^{−}(7). We also show how to scale these functions to give the quadratic twist with M ≅ E[7].

Remark 6.9. Recall that X_{E}(7) has a trivial K-rational point corresponding to E itself. Following [RS1] one method for finding the right quadratic twist would be to specialise at this point. However this approach fails when d vanishes at the trivial point. Neither does the method generalise to X_{E}^{−}(7).

Theorem 6.10. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. If j(E) ≠ 0, 1728 then the families of elliptic curves parametrised by Y_{E}(7) and Y_{E}^{−}(7) are given by (32) with (F, d) = (F, d_{1}) and (G, d_{2}) where F and G are the quartics in Theorem 3.6 and

d_{1}(x, y, z) = −6(3x^{2} + c_{4}xz − 3c_{4}y^{2} + c_{6}yz)z
d_{2}(x, y, z) = 2Δ(4x^{3} + c_{4}x^{2}z − 12c_{4}xy^{2} − 2c_{6}xyz + 8c_{6}y^{3} + c_{4}^{2}y^{2}z + 200Δz^{3}).

Proof: We fix a symplectic isomorphism ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ and let (a : b : c) be the \overline{K}-point on X(7) corresponding to (E, ϕ). As in the proof of Theorem 3.6 we scale (a, b, c) so that c_{4}(a, b, c) = c_{4} and c_{6}(a, b, c) = c_{6}.

Consideration of the action of SL_{2}(ℤ/7ℤ) on both the z-curve and the A-curve suggests we start with the forms

s_{1}(x, y, z) = (a^{2}c^{3} − 2ab^{3}c)x^{2}y + (a^{3}b^{2} − 2abc^{3})y^{2}z
+ (b^{3}c^{2} − 2a^{3}bc)xz^{2} + (a^{3}c^{2} + a^{2}b^{3} + b^{2}c^{3})xyz,
s_{2}(x, y, z) = a^{2}bx^{2}y + b^{2}cy^{2}z + c^{2}az^{2}x + 2abcxyz.

We then let r_{1} and r_{2} be the unique cubic forms satisfying

(33)  r_{i}(x, y, z)xyz ≡ s_{i}(x, y, z)^{2} mod (x^{3}y + y^{3}z + z^{3}x)

for i = 1, 2. The coefficients of r_{1} and r_{2} are homogeneous polynomials in a, b, c of degrees 10 and 6. Recall that in the proof of Theorem 3.6 we put

F(x, y, z) = \frac{1}{H^{3}} F (xx + y(∇F × ∇H) + zHe),
G(x, y, z) = \frac{1}{H^{2}} F (x∇F + y(x × e) + zH^{2}∇H).

The cubics d_{1} and d_{2} in the statement of the theorem are likewise found by putting

(34)  \begin{aligned}d_{1}(x, y, z) & = \frac{1}{2abcH^{4}} r_{1}(xx + y(∇F × ∇H) + zHe), \\ d_{2}(x, y, z) & = \frac{2H^{5}}{abc} r_{2}(x∇F + y(x × e) + zH^{2}∇H). \end{aligned}

It is clear from these constructions that {d_{1} = 0} and {d_{2} = 0} meet the corresponding twists of the Klein quartic in divisors of the form specified in Theorem 6.8. Hence our formulae for the families of elliptic curves parametrised by Y_{E}(7) and Y_{E}^{−}(7) are correct up to quadratic twist, say by δ ∈ K^{×}. It remains to show that δ is a square. As noted in [HK2, Section 7.1] it suffices to check this in the case ϕ : E[7] ≅ μ_{7} × ℤ/7ℤ is defined over K. Then (a : b : c) is a K-rational point on X(7). We write (a, b, c) = (λa_{0}, λb_{0}, λc_{0}) with a_{0}, b_{0}, c_{0} ∈ K. By our earlier choice of scaling for a, b, c we have λ^{7} ∈ K. Comparing the Weierstrass equation (30) for E with that in the statement of the theorem we deduce that λ^{7}a_{0}b_{0}c_{0} ∈ (K^{×})^{2}. Hence a^{7}, b^{7}, c^{7} ∈ K and (abc)^{7} ∈ (K^{×})^{2}. Using (14) and (23) we compute

c_{k}(F)(x, y, z) = (2^{9}3^{6})^{k/2}c_{k}(xx + y(∇F × ∇H) + zHe)
c_{k}(G)(x, y, z) = (2^{9}3^{6}H^{7})^{k/2}c_{k}(x∇F + y(x × e) + zH^{2}∇H).

for k = 4, 6. It follows by (34) that

\frac{c_{k}(F)(x, y, z)}{d_{1}(x, y, z)^{k/2}} = ξ^{k} \frac{c_{k}(xHx + yH(∇F × ∇H) + zH^{2}e)}{((abc)^{6}r_{1}(xHx + yH(∇F × ∇H) + zH^{2}e))^{k/2}}
\frac{c_{k}(G)(x, y, z)}{d_{2}(x, y, z)^{k/2}} = η^{k} \frac{c_{k}(xH^{3}∇F + yH^{3}(x × e) + zH^{5}∇H)}{((abc)^{6}H^{10}r_{2}(xH^{3}∇F + yH^{3}(x × e) + zH^{5}∇H))^{k/2}}

for some ξ, η ∈ K^{×}. The covariant columns Hx, H(∇F × ∇H), H^{2}e have degrees 7, 14, 21 and the contravariant columns H^{3}∇F, H^{3}(x × e), H^{5}∇H have degrees 21, 28, 35. Since each column has degree a multiple of 7, its evaluation at (a, b, c) is K-rational. Thus the families of curves in the statement of the theorem are K-isomorphic to

Y^{2} = X^{3} − 27\frac{c_{4}(x, y, z)}{((abc)^{6}r_{1}(x, y, z))^{2}} X − 54\frac{c_{6}(x, y, z)}{((abc)^{6}r_{1}(x, y, z))^{3}}

and

Y^{2} = X^{3} − 27\frac{c_{4}(x, y, z)}{((abc)^{6}H^{10}r_{2}(x, y, z))^{2}} X − 54\frac{c_{6}(x, y, z)}{((abc)^{6}H^{10}r_{2}(x, y, z))^{3}} .

To identify these with (30) we note that the cubic forms (abc)^{3}s_{1}(x, y, z) and (abc)^{3}H^{5}s_{2}(x, y, z) have coefficients in K (since the degree of each coefficient is a multiple of 7) and then use (33). □

Making the change of co-ordinates in Remark 3.7 we can replace d_{1} and d_{2} by cubic forms that satisfy the conditions of Theorem 6.8 for X_{E}(7) = {F = 0} ⊂ ℙ^{2} and X_{E}^{−}(7) = {G = 0} ⊂ ℙ^{2} where F and G are the quartics in Theorem 1.2. Moreover having found one such form we can use the Riemann-Roch machinery in Magma to find further such forms.

In the case of X_{E}(7) we obtain a cubic form d_{11} with X_{E}(7) ∩ {d_{11} = 0} = 2D_{1} for some divisor D_{1} ∼ 2T. Then L(3H − D_{1}) has basis

d_{11} = −2(ax^{2} + 3bxz + 3y^{2} + 2ayz)z,
d_{12} = 2(ax^{2} + 3bxz + 3y^{2} + 2ayz)x,
d_{13} = 4(3bx^{2} − 2axy − 2a^{2}xz − 3byz − 2abz^{2})z,
d_{14} = 4(a^{2}x^{2} + 3bxy + 4abxz + ay^{2} + 3b^{2}z^{2})z.

More generally we compute cubic forms d_{ij} for 1 ≤ i, j ≤ 4 such that the matrix (d_{ij}) is symmetric and each 2 × 2 minor vanishes mod F. The remaining d_{ij} are computed using d_{11}d_{ij} ≡ d_{1i}d_{1j} (mod F). Then X_{E}(7) ∩ {d_{ij} = 0} = D_{i} + D_{j} where D_{1}, …, D_{4} are divisors all linearly equivalent to 2T. The family of elliptic curves parametrised by Y_{E}(7) is now given by (32) with (F, d) = (F, d_{ii}) for any 1 ≤ i ≤ 4. The A-curve is the image of X_{E}(7) → ℙ^{3}; (x : y : z) ↦ (d_{11} : …: d_{14}) with equations

rank (\begin{matrix}0 & t_{3} & −t_{4} & 2at_{1} + t_{4} \\ t_{1} & 2at_{1} + t_{4} & 2bt_{1} + at_{2} + at_{3} & 2at_{2} + at_{3} \\ t_{2} & 2bt_{1} + at_{3} & −a^{2}t_{1} + bt_{3} − at_{4} & 2bt_{2} − bt_{3} − at_{4} \end{matrix}) ≤ 2.

Our formula for the elliptic curve corresponding to P ∈ Y_{E}(7) fails at points P with d_{ii}(P) = 0. These are the points whose image on the A-curve lies on the co-ordinate hyperplane {t_{i} = 0}. Therefore for any given point P we have d_{ii}(P) ≠ 0 for some i. So unlike the treatment in [HK2, Theorem 5.2], where only the cubic form d_{11} was given, we have found formulae that cover all cases.

In the case of X_{E}^{−}(7) we likewise find cubic forms d′_{ij} for 1 ≤ i, j ≤ 4 such that the matrix (d′_{ij}) is symmetric and each 2 × 2 minor vanishes mod G. Explicitly

d′_{11} = −7ax^{2}y + 6x^{2}z + 3a^{2}y^{3} − 8ay^{2}z + 3yz^{2},
d′_{22} = 2ax^{3} + 12bx^{2}y − 2axyz − 3aby^{3} + 6by^{2}z,
d′_{33} = 2a^{2}xy^{2} − 10axyz + 6xz^{2} + 5aby^{3} − 12by^{2}z,
d′_{44} = 2a^{2}x^{2}y − 3ax^{2}z + 5abxy^{2} − 12bxyz − 3a^{2}y^{2}z + 8ayz^{2} − 3z^{3}.

The remaining d′_{ij} are computed using d′_{11}d′_{ij} ≡ d′_{1i}d′_{1j} (mod G). The family of elliptic curves parametrised by Y_{E}^{−}(7) is now given by (32) with (F, d) = (G, Δd′_{ii}) for any 1 ≤ i ≤ 4. Exactly as before these formulae cover all cases.

6.5. Formulae in the case n = 11. Our approach is similar to that in the last section. As one would expect the formulae in the case n = 11 are more complicated than those in the case n = 7. There are however two further complications. One as noted in Remark 6.7 is the absence of a polynomial c_{6}. The other is that the form we are looking for is no longer uniquely determined by its image in the co-ordinate ring. Indeed in the case n = 7 we were looking for a cubic form, and in the case n = 11 we are looking for a quintic form. But in both cases the homogeneous ideal is generated by quartics.

Consideration of the action of SL_{2}(ℤ/11ℤ) on both the z-curve and the A-curve suggests we start with the forms

s_{1}(v, w, x, y, z) = (a^{3}bc^{3} + b^{4}cd^{2} − ab^{2}c^{2}de − 2bc^{2}de^{3})v^{2}wxz
+ (b^{3}cd^{3} + c^{4}de^{2} − abc^{2}d^{2}e − 2a^{3}cd^{2}e)vw^{2}xy
+ (c^{3}de^{3} + a^{2}d^{4}e − abcd^{2}e^{2} − 2ab^{3}de^{2})wx^{2}yz
+ (a^{3}d^{3}e + ab^{2}e^{4} − a^{2}bcde^{2} − 2a^{2}bc^{3}e)vxy^{2}z
+ (ab^{3}e^{3} + a^{4}bc^{2} − a^{2}b^{2}cde − 2ab^{2}cd^{3})vwyz^{2}
+ 2(a^{2}b^{2}c^{2}e + a^{2}b^{2}de^{2} + a^{2}cd^{2}e^{2} + ab^{2}c^{2}d^{2} + bc^{2}d^{2}e^{2})vwxyz,
s_{2}(v, w, x, y, z) = a^{2}bcev^{2}wxz + ab^{2}cdvw^{2}xy + bc^{2}dewx^{2}yz + acd^{2}evxy^{2}z
+ abde^{2}vwyz^{2} + 2abcdevwxyz.

We then solve for r_{1} and r_{2} satisfying

(35)  r_{i}(v, w, x, y, z)(vwxyz)^{3} ≡ s_{i}(v, w, x, y, z)^{4} (mod I, I′)

where I and I′ are the homogeneous ideals for X(11) ⊂ ℙ^{4} with respect to the two sets of variables a, b, c, d, e and v, w, x, y, z. The coefficients of r_{1} and r_{2} are homogeneous polynomials of degrees 28 and 20 in a, b, c, d, e. It is important to note that r_{1} and r_{2} are not uniquely determined by (35). However by averaging over the group we were able to choose r_{i} = (abcde)^{3}r̃_{i} in such a way that the coefficients of

r̃_{1}(vx_{1} + wx_{4} + xx_{5} + yx_{9} + zx_{14})

and

r̃_{2}(v∇F + w∇I_{7} + x∇I_{8} + y∇I_{9} + z∇c_{4})

are congruent mod I to certain polynomials in F and c_{4}. The result is a pair of quintic forms d̃_{1}(v, w, x, y, z) and d̃_{2}(v, w, x, y, z) with coefficients in ℚ[F, c_{4}]. We then put

d_{1}(v, w, x, y, z) = d̃_{1}(F v, w, F^{7}x, F^{2}y, F^{4}z)
d_{2}(v, w, x, y, z) = \frac{1}{F^{4}} d̃_{2}(F^{2}v, F^{8}w, F^{4}x, y, F^{3}z)

and replace F^{11} by Δ so that d_{1} and d_{2} have coefficients in ℚ[c_{4}, Δ].

Remark 6.11. Unfortunately the polynomials r̃_{i} and d_{i} would take several pages to print out, so we must refer the reader to the accompanying Magma file [F4] for further details. We should also remark that the computation of d_{1} and d_{2} took several hours of computer time (whereas no other calculation up to this point took more than a few seconds).

Theorem 6.12. Let E/K be an elliptic curve with Weierstrass equation y^{2} = x^{3} − 27c_{4}x − 54c_{6} and let Δ = (c_{4}^{3} − c_{6}^{2})/1728. Assume j(E) ≠ 0, 1728 and let X = X_{E}(11), respectively X_{E}^{−}(11), be as given in Theorem 3.17. If (v : w : x : y : z) ∈ X(K) \ {d_{i} = 0}, not a cusp, then the corresponding elliptic curve E′/K satisfies

c_{4}(E′) ≡ d_{1}(v, w, x, y, z) c_{4}(F)(v, w, x, y, z) mod (K^{×})^{4},

respectively

c_{4}(E′) ≡ d_{2}(v, w, x, y, z) c_{4}(G)(v, w, x, y, z) mod (K^{×})^{4}.

Proof: As noted in [HK2, Section 7.1] we are free to extend our field K so that ϕ : E[11] ≅ μ_{11} × ℤ/11ℤ is defined over K. Let (a : b : c : d : e) be the corresponding K-point on X(11). We scale a, b, c, d, e so that c_{4}(a, b, c, d, e) = c_{4}. Then a^{11}, …, e^{11} ∈ K and by comparing the Weierstrass equation for E in the statement of the theorem with (31) we deduce (abcde)^{11} ∈ (K^{×})^{4}. The polynomials F and G were computed in Section 3.3 as twists of F. Putting

(v′, w′, x′, y′, z′)^{T} = vF^{7}x_{1} + wF^{6}x_{4} + xF^{13}x_{5} + yF^{8}x_{9} + zF^{10}x_{14},
(v′′, w′′, x′′, y′′, z′′)^{T} = vF^{3}∇F + wF^{9}∇I_{7} + xF^{5}∇I_{8} + yF ∇I_{9} + zF^{4}∇c_{4},

it follows by (18), (19) and (24) that

c_{4}(F)(v, w, x, y, z) = \frac{(c_{4}^{3} − 1728F^{11})^{8}}{F^{22}} c_{4}(v′, w′, x′, y′, z′),
c_{4}(G)(v, w, x, y, z) = \frac{(55(c_{4}^{3} − 1728F^{11}))^{8}}{F^{11}} c_{4}(v′′, w′′, x′′, y′′, z′′).

By construction of d_{1} and d_{2} we have

d_{1}(v, w, x, y, z) = \frac{1}{(abcde)^{3}F^{30}} r_{1}(v′, w′, x′, y′, z′),
d_{2}(v, w, x, y, z) = \frac{1}{(abcde)^{3}F^{9}} r_{2}(v′′, w′′, x′′, y′′, z′′).

In view of Theorem 6.4 our aim is to show that equivalently

d_{1}(v, w, x, y, z) c_{4}(F)(v, w, x, y, z) ≡ v′w′x′y′z′c_{4}(v′, w′, x′, y′, z′) mod (K^{×})^{4},

d_{2}(v, w, x, y, z) c_{4}(G)(v, w, x, y, z) ≡ v′′w′′x′′y′′z′′c_{4}(v′′, w′′, x′′, y′′, z′′) mod (K^{×})^{4},

(abcde)^{8}F^{36}r_{1}(v′, w′, x′, y′, z′) ≡ v′w′x′y′z′ mod (K^{×})^{4},
(abcde)^{8}F^{24}r_{2}(v′′, w′′, x′′, y′′, z′′) ≡ v′′w′′x′′y′′z′′ mod (K^{×})^{4}.

To finish the proof we note that the quintic forms

(abcde)^{2}F^{9}s_{1}(v′, w′, x′, y′, z′) and (abcde)^{2}F^{6}s_{2}(v′′, w′′, x′′, y′′, z′′)

have coefficients in K (since the degree of each coefficient is a multiple of 11) and then use (35). □

We already gave a formula for the j-invariant in Section 6.1. So (assuming j(E′) ≠ 0) Theorem 6.12 determines E′ up to quadratic twist by −1. In the case K = ℚ it is easy to decide which of the remaining two possibilities is correct by looking at traces of Frobenius.

In principle it should be possible to find alternative quintic forms to be used at points where d_{1} or d_{2} vanishes. (The quintic forms in question are those meeting the z-curve in a divisor 4D where D is a hyperplane section for the A-curve.) In the case n = 7 we managed to find the alternative forms using the Riemann-Roch machinery in Magma. Unfortunately the analogue of this in the case n = 11 does not appear to be practical. In the case of X_{E}^{−}(11) this is not a problem, since the 25 points with d_{2} = 0 correspond to the elliptic curves ℓ-isogenous to E for ℓ = 2, 7, 13. We can also account for 7 of the points on X_{E}(11) with d_{1} = 0 as corresponding to the elliptic curve E itself and the elliptic curves 5-isogenous to E. We are yet to encounter an example (over K = ℚ) where one of the remaining points with d_{1} = 0 is rational.

7. Examples

We use the formulae in Theorems 1.2, 1.3 and 1.4 to give examples of non-trivial n-congruences for n = 7, 9, 11 over ℚ and ℚ(T). By “non-trivial” we mean that the elliptic curves are not isogenous. The examples over ℚ illustrate the value of minimising and reducing as described in Section 5. The examples over ℚ(T) were found by setting a = b = −27j/(4(j − 1728)) to obtain a surface fibred over the j-line and then intersecting with one of the co-ordinate hyperplanes in the hope of finding a rational curve. We refer to elliptic curves over ℚ by their labels in Cremona’s tables [C]. For elliptic curves beyond the range of Cremona’s tables we simply write the conductor followed by a ∗.

7.1. Examples in the case n = 7.

Example 7.1. Let E be the elliptic curve 162c1. Let F and G be the equations for X_{E}(7) and X_{E}^{−}(7) as given in Theorem 1.2 with a = 3645 and b = −13122. These have invariants Ψ(F) = −2^{11} · 3^{18} and Ψ(G) = 2^{22} · 3^{36}. Minimising and reducing suggests that we substitute

F (x, y, z) = \frac{1}{2^{10}3^{14}} F(36y − 9z, 1944x − 972y − 1215z, z)
G(x, y, z) = \frac{1}{2^{12}3^{20}} G(18x + 18y + 9z, z, −486x + 1458y + 1944z)

to give quartics

F (x, y, z) = 3x^{3}z + 3x^{2}y^{2} − 6x^{2}yz + 3x^{2}z^{2} − 3xy^{3}
+ 3xz^{3} + 2y^{4} − y^{3}z − 9y^{2}z^{2} + 4yz^{3} − 5z^{4}
G(x, y, z) = −x^{3}y − x^{3}z − 6x^{2}z^{2} + 6xy^{2}z − 6xyz^{2}
+ 6xz^{3} + 2y^{4} + 2y^{3}z − 6y^{2}z^{2} − 38yz^{3} − 8z^{4}

with invariants Ψ(F) = −2 · 3^{4} and Ψ(G) = 2^{2} · 3^{4}. We find rational points P_{1} = (1 : 0 : 0), P_{2} = (3 : −2 : −1) on {F = 0} ⊂ ℙ^{2}, and rational points P_{3} = (1 : 0 : 0), P_{4} = (1 : 1 : −1), P_{5} = (4 : −1 : 1) on {G = 0} ⊂ ℙ^{2}. The corresponding elliptic curves 7-congruent to E are

P_{1} 162c1  y^{2} + xy = x^{3} − x^{2} + 3x − 1
P_{2} 293706∗ y^{2} + xy = x^{3} − x^{2} − 62930562x − 192134303740
P_{3} 162c2  y^{2} + xy = x^{3} − x^{2} − 42x − 100
P_{4} 17334f1 y^{2} + xy = x^{3} − x^{2} − 5473977x − 4956193171
P_{5} 624186∗ y^{2} + xy = x^{3} − x^{2} − 11751402282x + 360746315347508.

The fact 162c1 and 162c2 are reverse 7-congruent is already clear since they are 3-isogenous and (3/7) = −1.

It is shown in [HK2] that there are infinitely many 6-tuples of directly 7congruent non-isogenous elliptic curves over ℚ. The following example shows that there are infinitely many pairs of reverse 7-congruent non-isogenous elliptic curves over ℚ.

Example 7.2. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + ax + b where a = b = −27j/(4(j − 1728)) and j = 27T^{3}(5T − 56)/(T − 1). Then on X_{E}^{−}(7), with equation as given in Theorem 1.2, we find the rational point

(x : y : z) = (0 : −4(T^{2} − 12T + 8)(5T^{2} + 4T + 8) : 9T^{2}(T + 4)(5T − 56)).

Specialising T (and taking quadratic twists by d as indicated) we obtain the following pairs of reverse 7-congruent elliptic curves E_{1} and E_{2}.

\begin{matrix}T & d & E_{1} & E_{2} \\ −16 & −38 & 361a1 & 361a2 \\ 8 & −10 & 700g1 & 2100q1 \\ 2 & −2 & 2116b1 & 10580h1 \\ 16/5 & −42 & 24255r1 & 24255m2 \end{matrix}

The existence of specialisations E_{1} and E_{2} that are not isogenous is enough to show that there are infinitely many such specialisations.

7.2. Examples in the case n = 9.

Example 7.3. Let E be the elliptic curve 47775z1. Let F_{1} = F_{2} = 0 be the equations for X_{E}(9) ⊂ ℙ^{3} as given in Theorem 1.3 with a = −41489280 and b = 102867483600. The invariant is Ψ(F_{1}, F_{2}) = −2^{42}·3^{60}·5^{12}·7^{32}·13^{4}. Minimising and reducing suggests that we substitute

(\begin{matrix}x \\ y \\ z \\ t \end{matrix}) ← (\begin{matrix}2520473760 & 937149484320 & −1998984627360 & −152410870080 \\ 0 & 79644600 & −185343480 & −3827880 \\ 0 & −22932 & 47040 & 6468 \\ 0 & −6 & 13 & 1 \end{matrix}) (\begin{matrix}x \\ y \\ z \\ t \end{matrix})

so that X_{E}(9) is defined by

−x^{2}z + x^{2}t + 4xyz + 2xyt − 3xz^{2} + 2xzt − 3xt^{2} + 6y^{3} + 14y^{2}z

+ y^{2}t + 6yz^{2} − 4yzt + 9yt^{2} − 6z^{3} + 27z^{2}t − 13zt^{2} − t^{3} = 0

−3x^{2}y + 4x^{2}z + 3x^{2}t + 3xy^{2} + 20xyz − 12xyt − 3xz^{2} − 32xzt + 25xt^{2} + 21y^{3}

+16y^{2}z − 24y^{2}t − 12yz^{2} + 100yzt + 34yt^{2} + 39z^{3} − 21z^{2}t − 56zt^{2} − 11t^{3} = 0

with invariant −3^{3} · 5^{6} · 7^{5} · 13^{4}. We find rational points P_{1} = (1 : 0 : 0 : 0), P_{2} = (4 : −1 : −1 : 0) and P_{3} = (1 : 2 : −1 : 0). The corresponding elliptic curves directly 9-congruent to E are

P_{1} 47775z1  y^{2} + y = x^{3} − x^{2} − 32013x + 2215478
P_{2} 429975∗  y^{2} + y = x^{3} − 314688780x − 2148671872069
P_{3} 494901225∗ y^{2} + y = x^{3} − 23634650164230x − 21037908383222056594

Since X_{E}^{−}(9) is not locally soluble at p = 7 there are no elliptic curves reverse 9-congruent to E.

In addition to Example 7.3 we have found two further triples of directly 9congruent non-isogenous elliptic curves over ℚ. These are

\begin{matrix}4650j1 & y^{2} + xy = x^{3} + x^{2} − 2700x + 54000 \\ 553350∗ & y^{2} + xy = x^{3} + x^{2} − 10472207700x − 455228489646000 \\ 1966950∗ & y^{2} + xy = x^{3} − x^{2} − 20654522386242x − 36130051534030639084 \\ 27606c1 & y^{2} + xy = x^{3} − 10289707x + 12703497719 \\ 358878∗ & y^{2} + xy = x^{3} + 2940333x − 1416695391 \\ 1242270∗ & y^{2} + xy + y = x^{3} − x^{2} − 359912x − 322105301 \end{matrix}

The elliptic curves 1701a1, 1701g1 and 22113c1 are also 9-congruent but only the last two of these are directly 9-congruent.

Example 7.4. Let E be the elliptic curve 201c1. Let G_{1} = G_{2} = 0 be the equations for X_{E}^{−}(9) ⊂ ℙ^{3} as given in Theorem 1.3 with a = −1029699 and b = 402173694. The invariant is Ψ(G_{1}, G_{2}) = 2^{48} · 3^{85} · 67^{5}. Minimising and reducing suggests that we substitute

(\begin{matrix}x \\ y \\ z \\ t \end{matrix}) ← (\begin{matrix}−26471709 & −23136696 & 20106774 & −20376135 \\ −45147 & −39828 & 33990 & −34509 \\ 90294 & 79332 & −68304 & 69342 \\ 77 & 68 & −58 & 59 \end{matrix}) (\begin{matrix}x \\ y \\ z \\ t \end{matrix})

so that X_{E}^{−}(9) is defined by

−x^{3} + 4x^{2}y + 3x^{2}z − x^{2}t + 6xy^{2} + 2xyz − 2xyt − 6xz^{2} + 4xzt
−11xt^{2} + y^{3} + 7y^{2}t − 2yz^{2} + 4yzt − 4yt^{2} + 6z^{3} − 7z^{2}t + 4zt^{2} + t^{3} = 0
2x^{3} − x^{2}y + 5x^{2}t − 10xy^{2} − 2xyz + 16xyt − 3xz^{2} + 4xzt + 8xt^{2}
−5y^{3} − y^{2}z − 3y^{2}t − yz^{2} − 2yzt + 12yt^{2} + 3z^{3} − 4z^{2}t + 2zt^{2} − 3t^{3} = 0

with invariant −3^{4} · 67^{5}. On this curve we find the rational point (1 : −2 : −1 : 0). The corresponding elliptic curve reverse 9-congruent to E is

374865∗  y^{2} + xy = x^{3} + x^{2} − 60068738107x + 4858035498982726

The following example shows that there are infinitely many pairs of directly 9-congruent non-isogenous elliptic curves over ℚ.

Example 7.5. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + a(T)x + b(T) where

a(T) = \frac{1}{2}(39T^{4} − 60T^{3} − 162T^{2} + 60T + 39),
b(T) = 47T^{6} + 120T^{5} + 21T^{4} + 21T^{2} − 120T + 47.

Then on X_{E}(9), with equations as given in Theorem 1.3, we find the rational point

(x : y : z : t) = (\frac{15}{2}(3T^{4} + 8T^{3} − 2T^{2} − 8T + 3) : T^{2} + 1 : 1 : 0).

The corresponding curve directly 9-congruent to E is the curve directly 3-congruent to E constructed in Theorem 1.1 with c_{4} = −a(T)/27, c_{6} = −b(T)/54 and

(λ : μ) = (47T^{6} − 78T^{5} − 153T^{4} + 244T^{3} + 153T^{2} − 78T − 47
: 18(T^{2} + 1)(T^{2} + 6T − 1)).

Specialising to T = 0 gives a pair of curves with conductors 80640 and 5886720. In particular these curves are not isogenous.

Next we give an example to show there are infinitely many non-trivial pairs of reverse 9-congruent elliptic curves over ℚ.

Example 7.6. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + a(T)x + b(T) where

a(T) = 3(3T + 1)(6T^{3} − 3T − 1)(9T^{3} − 9T − 4)^{2},
b(T) = 2(3T^{3} + 27T^{2} + 21T + 4)(6T^{3} − 3T − 1)^{2}(9T^{3} − 9T − 4)^{2}.

Then X_{E}^{−}(9), with equations as given in Theorem 1.3, has rational point

(x : y : z : t) = (−(6T^{3} − 3T − 1)(9T^{3} − 9T − 4) : T : 1 : 0).

The corresponding curve reverse 9-congruent to E is the curve reverse 3-congruent to E constructed in Theorem 1.1 with c_{4} = −a(T)/27, c_{6} = −b(T)/54 and

(λ : μ) = ((3T + 1)(9T^{3} − 9T − 4)(6T^{3} − 3T − 1)(180T^{4} + 321T^{3} + 216T^{2} + 66T + 8)
: 3(369T^{6} + 1107T^{5} + 1431T^{4} + 1017T^{3} + 414T^{2} + 90T + 8)).

Specialising T (and taking quadratic twists by d as indicated) we obtain the following pairs of reverse 9-congruent elliptic curves E_{1} and E_{2}.

\begin{matrix}T & d & E_{1} & E_{2} \\ −1 & 6 & 24a4 & 24a5 \\ −1/3 & 6 & 243a1 & 243b2 \\ −1/4 & 3 & 768d1 & 114432o1 \\ −1/2 & −6 & 6400u1 & 6400u2 \\ −2/3 & 6 & 23814v1 & 23814i1 \end{matrix}

7.3. Examples in the case n = 11.

Example 7.7. Let E be the elliptic curve 1782b1. Let F be the cubic form describing X_{E}(11) ⊂ ℙ^{4} as given in Theorem 1.4 with a = 765 and b = 15102. The invariant is Ψ(F) = −2^{28} · 3^{12} · 11^{6}. Minimising and reducing suggests that we substitute

(\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix}) ← (\begin{matrix}984 & 12900 & −9093 & −34056 & 13689 \\ −2040 & −24252 & −3315 & 0 & −16857 \\ 328 & 164 & −435 & 0 & −57 \\ −352 & 88 & −264 & 264 & −1056 \\ −8 & −4 & −13 & 0 & 25 \end{matrix}) (\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix})

so that X_{E}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

−v^{2}w + v^{2}x − v^{2}y + 2v^{2}z − vw^{2} + 4vwz − 4vx^{2} − 8vxy + 2vxz + 6vyz

+3vz^{2} + 2w^{3} − 3w^{2}x − 2w^{2}y + 8w^{2}z + 6wx^{2} + 2wxy + 2wxz + 6wy^{2} − 6wyz

+9wz^{2} − x^{3} − x^{2}z − 3xy^{2} − 6xyz − 9xz^{2} − 6y^{3} + 9y^{2}z + 3yz^{2} − 7z^{3} = 0

with invariant 2^{2} · 3^{4} · 11^{2}. We find rational points P_{1} = (−1 : 5 : 1 : 2 : 1), P_{2} = (0 : 0 : 0 : 1 : 0) and P_{3} = (1 : 1 : −1 : 0 : −4). The corresponding elliptic curves directly 11-congruent to E are

P_{1} 1782b1  y^{2} + xy = x^{3} − x^{2} + 48x + 224
P_{2} 1782b2  y^{2} + xy = x^{3} − x^{2} − 447x − 7795
P_{3} 447282∗ y^{2} + xy = x^{3} − x^{2} − 17552171922x − 227953575178678

The fact 1782b1 and 1782b2 are directly 11-congruent is already clear since they are 3-isogenous and (3/11) = 1.

Example 7.8. Let E be the elliptic curve 4466c1. Let G be the cubic form describing X_{E}^{−}(11) ⊂ ℙ^{4} as given in Theorem 1.4 with a = 85 and b = −83162. The invariant is Ψ(F) = 2^{21} · 7 · 11^{2} · 29^{2}. Minimising and reducing suggests that we substitute

(\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix}) ← (\begin{matrix}4096 & −1408 & 128 & −1312 & 45088 \\ 0 & 128 & 128 & 32 & 110 \\ 0 & 0 & −256 & −96 & −103 \\ 0 & 0 & 0 & −32 & −11 \\ 0 & 0 & 0 & 0 & −1 \end{matrix}) (\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix})

so that X_{E}^{−}(11) ⊂ ℙ^{4} is the singular locus of the Hessian of

−2v^{2}z − 4vwy + 12vxy + 4vxz + 5vy^{2} + 6vyz − 43vz^{2} − w^{2}x + w^{2}y −4wxy − 2wxz − 3wy^{2} + 196wyz + 83wz^{2} − 11x^{3} − 12x^{2}y − 9x^{2}z

−11xy^{2} + 366xyz + 125xz^{2} + 322y^{3} + 447y^{2}z + 275yz^{2} + 632z^{3} = 0

with invariant −2^{2} · 7 · 11^{2} · 29^{2}. We find rational points P_{1} = (−7 : 11 : 3 : 1 : 1) and P_{2} = (7830 : −3553 : 510 : −281 : 71). The corresponding elliptic curves reverse 11-congruent to E are

P_{1} 4466c2 y^{2} + xy + y = x^{3} − x^{2} − 1755x − 27349
P_{2} 1174558∗ y^{2} + xy + y = x^{3} − x^{2} + 117885809240x + 16240157710556505

The fact 4466c1 and 4466c2 are reverse 11-congruent is already clear since they are 2-isogenous and (2/11) = −1.

We did not find any triples of 11-congruent non-isogenous elliptic curves over ℚ. The following example shows that there are infinitely many pairs of directly 11congruent non-isogenous elliptic curves over ℚ.

Example 7.9. Let E/ℚ(T) be the elliptic curve y^{2} = x^{3} + a(T)x + b(T) where

a(T) = −3(T − 3)(T^{4} − 5T^{2} − 24T − 92)/(T^{3} − T^{2} + 4T + 24)
b(T) = −2(T − 3)(T^{5} − T^{4} − 11T^{3} − 43T^{2} − 62T − 316)/(T^{3} − T^{2} + 4T + 24).

Then X_{E}(11), with equations as given in Theorem 1.4, has rational point

(\begin{matrix}v \\ w \\ x \\ y \\ z \end{matrix}) = (\begin{matrix}T^{6} + T^{5} + 31T^{4} + 259T^{3} + 520T^{2} + 676T + 1248 \\ −(T − 3)(T^{5} + 4T^{4} + 43T^{3} + 100T^{2} − 44T − 320) \\ −(T^{2} + 3T + 14)(T^{3} − T^{2} + 4T + 24) \\ 0 \\ (T + 4)(T^{3} − T^{2} + 4T + 24) \end{matrix})

Specialising T (and taking quadratic twists by d as indicated) we obtain the following pairs of directly 11-congruent elliptic curves E_{1} and E_{2}.

\begin{matrix}T & d & E_{1} & E_{2} \\ 2 & −6 & 11a3 & 11a2 \\ 1 & 42 & 49a1 & 49a4 \\ −3 & −2 & 216b1 & 1512c1 \\ 11 & −426 & 10082c1 & 70574h1 \end{matrix}

The elliptic curve 11-congruent to E is y^{2} = x^{3} + A(T)x + B(T) where

A(T) = −3(T − 3)(T^{2} − 8T − 17)(T^{3} − T^{2} + 4T + 24)(T^{12} − 250T^{11} + 3473T^{10}
− 23824T^{9} + 106654T^{8} − 354556T^{7} + 890186T^{6} − 1710568T^{5}
+ 2386357T^{4} − 2054170T^{3} + 1799781T^{2} + 956680T + 3570796),
B(T) = −2(T − 3)(T^{3} − T^{2} + 4T + 24)^{2}(T^{20} + 476T^{19} − 27815T^{18} + 556718T^{17}
− 6046664T^{16} + 42450848T^{15} − 213832636T^{14} + 823702888T^{13}
− 2497998850T^{12} + 5954643736T^{11} − 10798748818T^{10} + 13644339892T^{9}
− 7927895108T^{8} − 10398245632T^{7} + 25581636532T^{6} − 10366268760T^{5}

− 60876061719T^{4} + 164062110060T^{3} − 98120800447T^{2} + 262948421518T

+ 141270230564).

These elliptic curves have discriminants

2^{12}3^{6}(T − 5)(T − 3)^{2}(T + 1)^{5}(T^{2} + 7)/(T^{3} − T^{2} + 4T + 24)^{3},

−2^{12}3^{6}(T − 5)^{4}(T − 3)^{2}(T + 1)^{3}(T^{2} + 7)(T^{3} − T^{2} + 4T + 24)^{3}(T^{3} − T^{2} + 15T − 31)^{11}.

We did not find any pairs of reverse 11-congruent non-isogenous elliptic curves over ℚ(T). We note that according to [KS, Theorem 4] the modular diagonal surface in this case is of general type.

7.4. Tables. We have written a program in Magma that given an elliptic curve E/ℚ and n ∈ {7, 9, 11} searches for elliptic curves n-congruent to E. For n ∈ {9, 11} we have run these programs for all elliptic curves in the Cremona database (up to conductor 130000 at the time of our calculation). The resulting list of pairs of n-congruent elliptic curves is available from the author’s website [F4]. We have been careful to remove all pairs that could be deduced from earlier entries by any combination of the following observations.

• n-congruence of elliptic curves is an equivalence relation.

• If ϕ : E → E′ is a rational isogeny of degree coprime to n then E and E′ are n-congruent.

• If E and E′ are n-congruent then so are their twists by the same quadratic

character. There is no guarantee that our tables are complete, since we have only searched

for points of small height on the corresponding curves of genus 10 and 26. However the points we found (on minimised and reduced models) were generally much smaller than the search bound used. So in any given case it seems likely that we have found all the rational points, but there is little prospect of proving this for curves of such large genus.

54 T.A. FISHER

Table 1 : Pairs of 9-congruent elliptic curves

17a^{2}  + 493a  430b  + 20210b  968a  + 132616∗  1950h  − 122850l
33a^{2}  − 297c  434c^{2}  + 62062h  1050j2 − 11550bg2  1950m2 + 17550ba2
35a2  − 77b2  446b  − 150302∗  1066a − 7462e  1952c  + 44896e
35a3  − 1015b1  456c  − 20520e  1106c + 2281678∗  1963a  + 374933∗
66a1^{2} + 3102c1  459c1 + 2295a1  1110n2 + 13098q2  1998d  − 3774c
84a3^{2} − 1932b2  506b  + 2530b  1111a + 121215655∗ 2040h  + 4073880∗
91a  + 5005c  525c^{2}  + 4725r  1134d1 + 5670h1  2072c^{2} − 184408∗
106d  + 2438a  537b  + 148749∗  1155j1 + 47355s1  2093c  + 2576483∗
110c2 + 60610m2 570k3^{2} − 1254j2  1176a + 10584n  2118b1 − 7660806∗
115a  + 366505∗ 573c  − 21491511∗ 1176b + 34104b  2135d  + 207095∗
118c  + 27494c  600c^{2}  + 5400r  1190c2 − 393890∗  2190b  + 116070i
123b  + 3813a  606f^{5} + 155742∗  1209a + 1357707∗  2190e  + 15330k
131a  − 136633∗ 627b2 + 5643d2  1215b − 1215g  2190l  + 24090k
140a2 − 3220c2  648b  + 26568a  1215c + 23085b  2198b1 + 964922∗
142c^{2} + 232454∗ 696a  + 355656∗  1218h^{2} + 1218i  2209a  − 11045b
153a  − 117351a 702f  + 30186h  1275d + 263925∗  2211c  + 134871∗
162a1 + 810c1  710a  + 137030∗  1281d + 4209c  2314b  − 363298∗
166a  + 848426∗ 714c  + 6426n  1288e + 240856∗  2373f  − 21357c
174a1 + 17574f1 715b  + 1080365∗  1330d1 + 860510∗  2418a  + 2884674∗
174c  + 1914g  741b  + 3335241∗  1470e2 + 4998j2  2443c  + 925897∗
195c  + 1755a  741d  + 65949e  1470n^{2} + 13230dv  2450l^{13} + 129850r
200b^{2} − 20600v  768b^{2,5} + 114432j  1482e1 + 16302l1  2451b^{2} + 2451c
201a  + 82209a  781a  − 1349b  1482e2 + 117078p2  2530e  + 908270∗
201c  − 374865∗ 805b^{2}  + 10465a  1482f2 + 8128770∗  2537b  + 210571∗
229a  + 2357555∗ 805c^{2}  + 24955b  1518r + 13662m  2541a  − 22869o
235b  + 329a  808a  + 891224∗  1617b + 289443∗  2568e  + 7408680∗
238d^{2} − 91154b  810a2 + 208170∗  1653a − 520695∗  2571a  + 12749589∗
243a1 − 243b2  810d2 + 23490n2  1701a − 1701g  2616c  + 6574008∗
246c^{2} + 2214a  843a  + 1600857∗  1701a − 22113c  2646a1 + 60858h1
249a  + 42579e  850c  + 16150q  1725i^{2} − 343275∗  2646e1 + 13230e1
258c  + 1290h  858d2 + 4290l2  1757a + 335587∗  2674e  + 259378∗
270a1 + 2970i1  861b  + 192003∗  1771c1 + 2807035∗  2678a  − 20806a
302b^{2} + 226802∗ 861d  + 469245∗  1782c + 290466∗  2699b  + 5249555∗
338a^{7} + 22646d  867b^{2}  + 141321∗  1806a + 2775822∗  2706i^{2} + 65855922∗
350b2 − 6650bh2 897d^{2} + 8073a  1830h + 16470w  2728d  + 832040∗
354a^{2} + 55578e  906b  + 610222710∗ 1848g + 9097704∗  2758b  + 1069527578∗
364a  + 26572d  906e  + 28086d  1862a2 + 102410n2  2787d  + 393187173∗
390d3^{2} + 3510f2  930d^{2} + 564510∗  1870e + 136510∗  2793b  + 790419∗
406b2 − 62930k2 930i2  + 160890∗  1887b − 1054833∗  2835a1 + 65205g1
430a  + 7310e  930l  + 8370n  1911a − 17199o  2835d  + 53865d

ON FAMILIES OF n-CONGRUENT ELLIPTIC CURVES 55

Table 2 : Pairs of 11-congruent elliptic curves

138b^{2,3} + 17526b  3910l  + 43010o  16354h + 703222∗  55470m^{2} + 1275810∗
190b  + 2470a  3990z  + 3990ba  16390m + 23585210∗ 55594a  − 10062514∗
216b  + 1512c  4046n  + 416738∗  17550a + 298350∗  56100j^{2} − 15539700∗
238b^{2}  + 4522b  4158f  + 54054m  17550z + 122850r  56154t  − 27346998∗
258b^{2}  + 461562∗ 4200g^{2} + 79800h  17556d + 9778692∗  57354j  + 3498594∗
294a^{7} + 4998ba  4275a^{2} − 175275∗  17710h + 726110∗  58968k  + 58968l
325a^{3} + 23075e  4466c^{2} − 1174558∗ 18850i  + 131950∗  60088b  + 53658584∗
329a  + 59549a  4598b  + 32186g  19950bl + 618450∗  64596b  + 53162508∗
426b^{2}  + 77106c  4704a  − 192864∗  20838f − 10273134∗ 65220c  + 534086580∗
497a  + 148603∗ 4760b  + 775880∗  21175b + 21175g  66300q  + 1259700∗
513a  + 77463a  5070d  − 55770j  21760a + 805120∗  66930e  + 101265090∗
600b  + 4200c  5265b  + 194805∗  22950bi + 849150∗  67158c  + 3559374∗
645f  − 70305g  5454b  + 59994e  23595g − 165165∗  67650v  + 16168350∗
648b  − 4536c  5577b  − 39039b  24906a + 70558698∗ 70890w + 2197590∗
700e^{2}  − 11900d  5880b  − 417480∗  25110j − 426870∗  71610bj + 109634910∗
1080d + 39960h  6710h  + 4381630∗ 26026d − 2836834∗  73255b  − 89151335∗
1115a + 125995a 7315e  + 8214745∗ 27610f^{2} + 5273510∗  73326t  + 3886278∗
1134b − 12474f  7350n  + 95550ch  27650j + 1023050∗  73370h  + 130790∗
1155b − 42735a  7830f^{3} + 477630∗  27885p − 2704845∗  78650w^{2} − 78650bc
1176a − 15288a  8096a  − 13984a  28420d + 5087180∗  79800s  + 16837800∗
1210d + 1210f  8120d  + 332920∗  28830a − 2219910∗  80325u  + 80325v
1254b + 264594∗ 8410c  − 58870e  28840c + 30253160∗ 81675q  + 3348675∗
1470j  − 16170bl  8670o  − 8670p  30774h + 11417154∗ 83148g  − 582036∗
1666a + 18326b  9438d  + 1009866∗ 30855a + 215985∗  84270j  + 589890∗
1782b^{3} + 447282∗ 9450l  + 9450m  30888g + 216216∗  84930m + 59196210∗
1848f + 97944h  9450o  + 292950∗  31746q + 222222∗  84930p  + 31509030∗
1870c + 24310c  9450bo + 160650∗  32802d^{2} − 117660774∗ 85050bf + 2466450∗
1925g + 140525∗ 9555a  − 181545∗  32856f + 229992∗  87210l  + 3226770∗
2093b + 26611d  9966h  + 827178∗  34385a + 378235∗  87990d  + 5895330∗
2184c + 2184d  10010t + 670670∗  35574b − 35574e  88842l  + 120198k
2755b + 476615∗ 10082b + 70574c  36498s + 56406i  90986e  + 423175886∗
3234e + 947562∗ 11774a + 24172022∗ 37446i  − 4456074∗  92950s  + 9016150∗
3322c + 235862∗ 12376j + 12376k  37830o + 31285410∗ 93795r  + 2157285∗
3325a + 1173725∗ 12696b + 12696c  38070k − 5291730∗  95370ch + 667590∗^{2}
3610d + 25270j  13650ba + 2716350∗ 39732a + 24514644∗ 95550cb + 95550cc
3718d + 107822e 14196i − 269724∗ 45738b + 869022∗ 95700b + 2201100∗
3822u − 42042ce 14844a + 40182708∗ 47974d^{2} − 48885506∗ 98070z  + 270771270∗
3850a + 65450b 14910ba + 40450830∗ 53290a^{2} + 53290b 98982c − 19697418∗
3850i^{2} + 65450i  14950l + 1599650∗ 53865e − 10395945∗ 104910d + 3252210∗
3900e − 66300t  16150p + 4731950∗ 54813a − 1589577∗  105522e + 139183518∗

The entries are given as E_{1}±E_{2} where the ± indicates whether the n-congruence respects the Weil pairing. In all cases except when n = 9 and the curves admit a rational 3-isogeny we have omitted the final number from the Cremona reference (which may be taken to be 1). A superscript ℓ indicates an elliptic curve that admits a rational ℓ-isogeny where ℓ is a prime not dividing n. If ℓ is not a square mod n then we may change the sign of the congruence by passing to an isogenous curve. For elliptic curves beyond the range of Cremona’s tables we have again written the conductor followed by a ∗. The extended version of our tables [F4] also gives the Weierstrass equations.

References

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[F0] T.A. Fisher, On 5 and 7 descents for elliptic curves, PhD thesis, University of Cambridge, 2000.

[F1] T. A. Fisher, Some examples of 5 and 7 descent for elliptic curves over Q, J. Eur. Math. Soc. 3 (2001), no. 2, 169–201.

[F2] T. A. Fisher, The Hessian of a genus one curve, preprint, arXiv:math/0610403v2

[F3] T.A. Fisher, Invariant theory for the elliptic normal quintic, I. Twists of X(5), in preparation.

[F4] T.A. Fisher, On families of n-congruent elliptic curves, electronic data accompanying this

article, http://www.dpmms.cam.ac.uk/^{∼}taf1000/papers/highercongr.html

[Fr] G. Frey, On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2, in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), J. Coates and S.-T. Yau (eds.), Ser. Number Theory I, Int. Press, Cambridge, MA, (1995) 79–98.

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[HK2] E. Halberstadt and A. Kraus, Sur la courbe modulaire X_{E}(7), Experiment. Math. 12 (2003), no. 1, 27–40.

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preprint, http://www.mast.queensu.ca/^{∼}kani/mdqs.htm

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[K2] F. Klein, Über die Transformationen elfter Ordnung der elliptischen Funktionen, Math. Ann. 15 (1879), Reprinted in Gesammelte Mathematische Abhandlungen III, R. Fricke et al (eds.), Springer (1923) 140–168.

[K3] F. Klein, Über die elliptischen Normalkurven der n-ten Ordnung (1885). Reprinted in Gesammelte Mathematische Abhandlungen III, R. Fricke et al (eds.), Springer (1923) 198–254.

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[RS1] K. Rubin and A. Silverberg, Families of elliptic curves with constant mod p representations, in Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), J. Coates and S.-T. Yau (eds.), Ser. Number Theory I, Int. Press, Cambridge, MA, (1995) 148–161.

[RS2] K. Rubin and A. Silverberg, Mod 2 representations of elliptic curves, Proc. Amer. Math. Soc. 129 (2001), no. 1, 53–57.

[Sa] G. Salmon, A treatise on the higher plane curves, Third edition, Hodges, Foster and Figgis, Dublin, 1879.

[S] A. Silverberg, Explicit families of elliptic curves with prescribed mod N representations, in Modular forms and Fermat’s last theorem (Boston, MA, 1995), G. Cornell, J.H. Silverman and G. Stevens (eds.), Springer-Verlag, New York, (1997) 447–461.

[Si] J.H. Silverman, The Arithmetic of Elliptic Curves, Springer GTM 106 (1986).

[V1] J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sc. Paris, 273 (1971) 238-241.

[V2] J. Vélu, Courbes elliptique munies d’un sous-group ℤ/nℤ × μ_{n}, Bull. Math. Soc. math. France, Mémoire 57 (1978).

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

E-mail address: T.A.Fisher@dpmms.cam.ac.uk