Modular degrees of Elliptic curves

Modular degree is an interesting invariant of elliptic curves. It is computed by variety of methods. After computer calculations, Watkins conjectured that given E over the rational numbers of rank R, 2^R divides (Phi), where (Phi) : X_0(N) to E is the optimal map (up to isomorphism of E) and degree of (Phi) is the modular degree of E. In fact he observed that 2^{R+K} divides the degree of the modular degree and 2^K depends on {W}, where {W}is the group of Atkin-Lehner involutions, the cardinality of {W}=2^{omega(N)}, N is the conductor of the elliptic curve and omega(N) counts the number of distinct prime factors of N. The goal of this thesis is to study this conjecture. We have proved that 2^{R+K} divides the degree of (Phi) would follow from an isomorphism of complete intersection of a universal deformation ring and a Hecke ring, where 2^K is the cardinality of W^{prime}, the cardinality of a certain subgroup of the group of Atkin-Lehner involutions. I attempt to verify 2^{R+K} divides the degree of ({Phi}) for certain Ellipitic Curves E by using a computer algebra package Magma. I have verified when N is squarefree. Computations are in chapter 5.