  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.
