On the degree of the $p$-torsion field of elliptic curves over $\mathbb{Q}_\ell$ for $\ell \neq p$

Abstract: Let $\ell$ and $p \geq 3$ be distinct prime numbers. Let $E/\mathbb{Q}{\ell}$ be an elliptic curve with $p$-torsion module $E_p$. Let $\mathbb{Q}{\ell}(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree of the extension $\mathbb{Q}{\ell}(E_p)/\mathbb{Q}{\ell}$. As a consequence, we obtain a recipe to determine the discriminant ideal of the extension $\mathbb{Q}{\ell}(E_p)/\mathbb{Q}\ell$ in terms of standard information on $E$.