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

Let $E$ be an elliptic curve over the rationals that does not have complex multiplication. For each prime $\mathrm{\ell }$, the action of the absolute Galois group on the $\mathrm{\ell }$-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 ${\rho }_{E,\mathrm{\ell }}$ is surjective for all sufficiently large $\mathrm{\ell }$. In this paper, we describe a simple algorithm based on Serre’s proof that can quickly determine the finite set of primes $\mathrm{\ell }$ for which ${\rho }_{E,\mathrm{\ell }}$ is not surjective. We will also give some improved bounds for Serre’s theorem.