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→Parametrization of the modular curve: eta function |
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When n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, or 25, X<sub>0</sub>(n) has [[geometric genus|genus]] zero, and hence can be parametrized by rational functions. The simplest nontrivial example is X<sub>0</sub>(2), where if
:<math>j_2(q) = q^{-1} - 24 + 276q -2048q^2 + 11202q^3 + \cdots = ((\eta(q)/\eta(q^2))^{24}</math>
is (up to the constant term) the [[Monstrous moonshine|McKay-Thompson series]] for the class 2B of the [[Monster group|Monster]], and η is the [[Dedekind eta function]], then
:<math>x = \frac{(j_2+256)^3}{j_2^2}, y = \frac{(j_2+16)^3}{j_2}</math>
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