Elliptic curves with j = 0, 1728 and low embedding degree. Issue 12 (1st December 2016) - Miret, J.M.; Sadornil, D.; Tena, J.
Abstract : Elliptic curves over a finite field F q with j -invariant 0 or 1728, both supersingular and ordinary, whose embedding degree k is low are studied. In the ordinary case we give conditions characterizing such elliptic curves with fixed embedding degree with respect to a subgroup of prime order ℓ. For k = 1, 2, these conditions give parameterizations of q in terms of ℓ and two integers m, n . We show several examples of families with infinitely many curves. Similar parameterizations for k ≥ 3 need a fixed k th root of the unity in the underlying field. Moreover, when the elliptic curve admits distortion maps, an example is provided.