Abstract

In their paper, "Arithmetic of Weil Curves", Mazur and Swinnerton-Dyer prove that the number of fundamental critical points of the normalized weight two modular form associated with an elliptic curve is an upper bound on the analytic rank of the curve. Their calculation of this quantity for all elliptic curves of conductor less than 424 confirmed that, with only 16 exceptions, the bound is sharp. It is the intention of this thesis to compute the number of fundamental critical points for all curves of conductor less than 4000.