  We use an invariant-theoretic method to compute certain twists of the modular
curves X(n) for n=7,9,11. Searching for rational points on these twists enables
us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs
of non-isogenous elliptic curves over Q whose n-torsion subgroups are
isomorphic as Galois modules. We also show by giving explicit non-trivial
examples over Q(T) that there are infinitely many examples over Q in the cases
n=9 and n=11.
