We study N-congruences between quadratic twists of elliptic curves. If N
has exactly two distinct prime factors we show that these are parametrised by
double covers of certain modular curves. In many, but not all, cases the
modular curves in question correspond to the normaliser of a Cartan subgroup of
GL2(ℤ/Nℤ). By computing explicit models for
these double covers we find all pairs, (N, r), such that there exist
infinitely many j-invariants of elliptic curves E/ℚ which are
N-congruent with power r to a quadratic twist of E. We also find an
example of a 48-congruence over ℚ. We make a conjecture
classifying nontrivial (N, r)-congruences between quadratic twists of elliptic
curves over ℚ.
Finally, we give a more detailed analysis of the level 15 case. We use
elliptic Chabauty to determine the rational points on a modular curve of genus
2 and Jacobian of rank 2 which arises as a double cover of the modular curve
X(ns 3+, ns 5+). As a consequence we obtain a
new proof of the class number 1 problem.