| Modular degree is an interesting invariant of elliptic curves. It is computed by variety of methods. After computer calculations, Watkins conjectured that given E/ Q of rank R, 2R | deg(phi), where phi : X0(N) → E is the optimal map (up to isomorphism of E) and deg(phi) is the modular degree of E. In fact he observed that 2R+K divides the degree of the modular degree and 2K depends on ;The goal of this thesis is to study this conjecture. We have proved that 2R+K | deg(phi) would follow from an isomorphism of complete intersection of a universal deformation ring and a Hecke ring, where 2K = ;I attempt to verify 2R+ K | deg(phi) for certain Ellipitic Curves E by using a computer algebra package Magma. I have verified when N is squarefree, and N ≤ 250. Tables are in Section 5.1. |
