Computing the degree of a modular parametrization

Abstract

The Weil-Taniyama conjecture states that every elliptic curve E/ℚ of conductor N can be parametrized by modular functions for the congruence subgroup Γ₀(N) of the modular group Γ = PSL(2, ℤ). Equivalently, there is a non-constant map ϕ from the modular curve X ₀ (N) to E. We present here a method of computing the degree of such a map ϕ for arbitrary N. Our method, which works for all subgroups of finite index in Γ and not just Γ₀(N), is derived from a method of Zagier in [2]; by using those ideas, together with techniques which have been used by the author to compute large tables of modular elliptic curves (see [1]), we are able to derive an explicit and general formula which is simpler to implement than Zagier's. We discuss the results obtained, including several examples.