  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
$E$.
