The group structure of frey elliptic curves over finite fields Fp

Abstract

Frey elliptic curves are the curves y(2) = x(3) - n(2)x and in this work the group structure E(F-p) of these curves over finite fields F-p is considered. This group structure and the number of points on these elliptic curves depend on the existence of elements of order 4. Therefore the cases where the group of the curves has such elements are determined. It is also shown that the number of such elements, if any, is either 4 or 12. Classification is made according to n is a quadratic residue or not.