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 $\ell >7$