Let$E$be an elliptic curve without complex multiplication (CM) over a number field$K$, and let$G_{E}(\ell )$be the image of the Galois representation induced by the action of the absolute Galois group of $K$on the$\ell$-torsion subgroup of $E$. We present two probabilistic algorithms to simultaneously determine$G_{E}(\ell )$up to local conjugacy for all primes$\ell$by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine$G_{E}(\ell )$up to one of at most two isomorphic conjugacy classes of subgroups of$\mathbf{GL}_{2}(\mathbf{Z}/\ell \mathbf{Z})$that have the same semisimplification, each of which occurs for an elliptic curve isogenous to$E$. Under the GRH, their running times are polynomial in the bit-size$n$of an integral Weierstrass equation for $E$, and for our Monte Carlo algorithm, quasilinear in$n$. We have applied our algorithms to the non-CM elliptic curves in Cremona’s tables and the Stein–Watkins database, some 140 million curves of conductor up to $10^{10}$, thereby obtaining a conjecturally complete list of 63 exceptional Galois images$G_{E}(\ell )$that arise for$E/\mathbf{Q}$without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images$G_{E}(\ell )$that arise for non-CM elliptic curves over quadratic fields with rational$j$-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the$j$-invariant is irrational.