We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with 12-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of 12-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves.
A key ingredient in the proof is to construct simple (algebraic) conditions for the 2, 3, or 4-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j-invariants of the pair of elliptic curves.