Classical Modular Curve - Parametrization of The Modular Curve

Parametrization of The Modular Curve

When n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, X₀(n) has genus zero, and hence can be parametrized by rational functions. The simplest nontrivial example is X₀(2), where if

is (up to the constant term) the McKay–Thompson series for the class 2B of the Monster, and η is the Dedekind eta function, then

parametrizes X₀(2) in terms of rational functions of j₂. It is not necessary to actually compute j₂ to use this parametrization; it can be taken as an arbitrary parameter.