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N-congruences of elliptic curves and equations for Z(N,r)

A pair of elliptic curves E/ℚ and F/ℚ are said to be (N,r)-congruent if there exists an isomorphism E[N] ≅ F[N] such that the Weil pairing is raised to the power of r. A congruence is said to be nontrivial if E and F are not geometrically isogenous. The moduli space parametrising pairs of (N,r)-congruent elliptic curves is the Hilbert modular surface Z(N,r) of discriminant N² (these are also called modular diagonal quotient surfaces by Kani--Schanz). If E/ℚ and F/ℚ are nontrivially (N,-1)-congruent then there exists a genus 2 curve C/ℚ with optimal maps C → E and C → F of degree N (i.e., maps which do not factor through a subcover).

At this point we have examples of (N,r)-congruent elliptic curves for:

all N ≤ 15 and N = 17 for all r, and
(N,r) = (16,1), (16,3), (16,7).

From these we have genus 2 curves C/ℚ with optimal maps of degree N to an elliptic curve E/ℚ for all N ≤ 17.

We have computed equations (these results should not be treated as theorems except for the published cases) for Z(N,r) for:

All N ≤ 15 and N = 17, 18, 19 for all r.
(N,r) = (16,1), (16,3), (16,7), (20,1), (20,11), (22,1), (24,11), (24,23).

This is not work due only to me. Attributions are given in the README.md files. I will consider this project complete if/when I compute models for Z(N, r) with (N,r) = (16,5), (20,3), (21,2), (21,5).

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