Elliptic curves with maximal Galois action on their torsion points

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρ_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of ρ_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 ρ_E will be surjective for "most" elliptic curves over k.

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