Computing the Degree of the Modular Parametrization of a Modular Elliptic Curve

Abstract

The Weil-Taniyama conjecture states that every elliptic curve $E/\mathbb {Q}$ of conductor N can be parametrized by modular functions for the congruence subgroup ${\Gamma _0}(N)$ of the modular group $\Gamma = PSL(2,\mathbb {Z})$. Equivalently, there is a nonconstant map $\varphi$ from the modular curve ${X_0}(N)$ to E. We present here a method of computing the degree of such a map $\varphi$ for arbitrary N. Our method, which works for all subgroups of finite index in $\Gamma$ and not just ${\Gamma _0}(N)$, is derived from a method of Zagier published in 1985; by using those ideas, together with techniques which have recently been used by the author to compute large tables of modular elliptic curves, we are able to derive an explicit and general formula which is simpler to implement than Zagier’s. We discuss the results obtained, including a table of degrees for all the modular elliptic curves of conductors up to 200.