Elliptic curves with maximal Galois action on their torsion points

Understand

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.