Elliptic $\mod \ell$ Galois representations which are not minimally elliptic

Abstract

In a recent preprint (see [C]), F. Calegari has shown that for $\ell = 2, 3, 5$ and $7$ there exist $2$-dimensional irreducible representations $\rho$ of Gal$(\bar{\Q}/\Q)$ with values in $\F_\ell$ coming from the $\ell$-torsion points of an elliptic curve defined over $\Q$, but not minimally, i.e., so that any elliptic curve giving rise to $\rho$ has prime-to-$\ell$ conductor greater than the (prime-to-$\ell$) conductor of $\rho$. In this brief note, we will show that the same is true for any prime $\ell >7$