Computing the degree of a modular parametrization
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 X0(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 ; by using those ideas, together with techniques which have been used by the author to compute large tables of modular elliptic curves (see ), 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.
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