  Given an elliptic curve E over a number field k, the Galois action on the
torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k)
\to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E
for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly
disjoint from the cyclotomic extension of Q, then \rho_E will be surjective for
"most" elliptic curves over k.
