  Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. By
Serre's open image theorem, the mod $\ell$ Galois representation
$\overline{\rho}_{E, \ell}$ of $E$ is surjective for each prime number $\ell$
that is sufficiently large. Under the generalized Riemann hypothesis, we give
an explicit upper bound on the largest prime $\ell$, linear in the logarithm of
the conductor of $E$, such that $\overline{\rho}_{E, \ell}$ is nonsurjective.
