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

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