Computing the Modular Degree of an Elliptic Curve

We review previous methods of computing the modular degree of an elliptic curve, and present a new method (conditional in some cases), which is based upon the computation of a special value of the symmetric square L-function of the elliptic curve. Our method is sufficiently fast to allow large-scale experiments to be done. The data thus obtained on the arithmetic character of the modular degree show two interesting phenomena. First, in analogy with class number in the number field case, there seems to be a Cohen–Lenstra heuristic for the probability that an odd prime divides the modular degree. Secondly, the experiments indicate that 2 should always divide the modular degree, where r is the Mordell–Weil rank of the elliptic curve. We also discuss the size distribution of the modular degree, or more exactly of the special L-value which we compute, again relating it to the number field case.