On pairs of 17-congruent elliptic curves

TL;DR

This paper explicitly computes equations for surfaces parametrizing pairs of 17-congruent elliptic curves, revealing new examples and structures, including the first non-trivial symplectically 17-congruent pair over rationals.

Contribution

It provides explicit equations for the surfaces Z(17,1) and Z(17,3), and constructs the first known non-trivial symplectically 17-congruent elliptic curves over rationals.

Findings

Explicit equations for Z(17,1) and Z(17,3) surfaces.

First non-trivial symplectically 17-congruent elliptic curves over rationals.

Associated genus 2 curve with (17,17)-splitting.

Abstract

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