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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 ( k̄/k ) → GL 2 (ℤ̂) 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  ≠ ℚ is linearly disjoint from the cyclotomic extension of ℚ, then  ρ   E   will be surjective for ‘most’ elliptic curves over  k .

Elliptic curves with maximal Galois action on their torsion points