  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 GL_2(Z/ell 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, quasi-linear 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/Q without CM. Under this conjecture we determine a
complete list of 160 exceptional Galois images G_E(ell) the 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.
