On pairs of 17-congruent elliptic curves

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.

24 pages

References in corpus (1)

• On families of n-congruent elliptic curves

Cited by in corpus (1)

• Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations