  We construct infinite families of pairs of (geometrically non-isogenous)
elliptic curves defined over $\mathbb{Q}$ 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.
