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.
Read more about this topic: Classical Modular Curve
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“In philosophical inquiry, the human spirit, imitating the movement of the stars, must follow a curve which brings it back to its point of departure. To conclude is to close a circle.”
—Charles Baudelaire (1821–1867)