Elliptic mod \ell Galois representations which are not minimally elliptic

Abstract

In a recent preprint, F. Calegari has shown that for $\ell = 2, 3, 5$ and 7 there exist 2-dimensional surjective 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 $\ell >7$, concretely, we will show that for any such $\ell$ the elliptic curve $$E^\ell: \qquad Y^2 = X (X- 3^\ell ) (X - 3^\ell - 1) $$ is semistable, has bad reduction at 3, the associated $\mod \ell$ Galois representation $\rho$ is surjective, unramified at 3, and there is no elliptic curve with good reduction at 3 whose associated $\mod \ell$ representation is isomorphic to $\rho$.