Elliptic curves play important roles in mathematics as well as other areas of science. In cryptography, they are used for enhanced security and shorter key length. In cryptography, we look for curves having points of large prime order, so that the corresponding cryptosystem resists against Pohlig-Hellman attack. In the second chapter of this thesis we first investigate this problem based on a paper of Lenstra. Then we will study more general problems using Howe's paper and Castryck and Hubrechts's paper. In the last chapter, we study a family of elliptic curves of cryptographic interest and for the first time we present formulae for isomorphism ltr"
