  For $\ell = 3$ and 5 it is known that every odd, irreducible, 2-dimensional
representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant
equal to the cyclotomic character must "come from" the $\ell$-torsion points of
an elliptic curve defined over $\Q$. We prove, by giving concrete
counter-examples, that this result is false for every prime $\ell >5$.
