On the surjectivity of mod $\ell$ representations associated to elliptic curves

Let $E$ be an elliptic curve over the rationals that does not have complex multiplication. For each prime $\ell$, the action of the absolute Galois group on the $\ell$-torsion points of $E$ can be given in terms of a Galois representation $\rho_{E,\ell}\colon \operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to GL_2(\mathbb{F}_\ell)$. An important theorem of Serre says that $\rho_{E,\ell}$ is surjective for all sufficiently large $\ell$. In this paper, we describe a simple algorithm based on Serre's proof that can quickly determine the finite set of primes $\ell$ for which $\rho_{E,\ell}$ is not surjective. We will also give some improved bounds for Serre's theorem.