Let $E$ be an elliptic curve over a number field $F$. As usual, let
$\bar{\rho}:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}(E[p])$ be the mod $p$ Galois representation associated to $E$. It is known that $\bar{\rho}$ is unramified at primes of good reduction. Let $\ell$ be a prime which $\bar{\rho}$ ramifies. Can we compute the ramification index for $\ell$ explicitly?