Non-surjective prime (reviewed)

Let $E$ be an elliptic curve defined over $\Q$. A prime $p$ is called a non-surjective prime for $E$ if the mod-$p$ Galois representation \[ \rho_{E,p}: \Gal(\overline{\Q}/\Q) \to \GL(2,\F_p) \] attached to $E$ is not surjective.

There are only finitely many non-surjective primes for any elliptic curve without CM by a result of Serre. Conjecturally, the largest possible non-surjective prime is $37$.