Local points on twists of $X(p)$ with applications

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to~$E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb{Q}_\ell)$ is non-empty, for all primes $\ell$. We