Growth of torsion of elliptic curves with full 2-torsion over quadratic cyclotomic fields

Let K = Q(root-3) or Q(root-1) and let C-n, denote the cyclic group of order n. We study how the torsion part of an elliptic curve over K grows in a quadratic extension of K. In the case E(K)[2] approximate to C-2 circle plus C-2 we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure. (C) 2016 Elsevier Inc. All rights reserved.