Local points on twists of $X(p)$

Let $E$ be a rational elliptic curve and $p$ an odd prime. The modular curve $X_E^{-}(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. In this talk, I present my recent work with Nuno Freitas on a complete classification for when these curves admit points everywhere locally.
In the second half of the talk, we will see a Diophantine application of this result. More precisely, we use the modular method, together with our classification theorem to prove that certain generalized Fermat equations have no non-trivial, coprime solutions.