The Dedekind eta-function, a Hauptmodul for Gamma(0)(13), and invariant theory

We give an explicit formula for the Hauptmodul (eta(tau)/eta(13 tau))(2) of the level-13 Hecke modular group Gamma(0)(13) as a quotient of theta constants, together with some related explicit formulas. Similar results for primes p = 2, 3, 5, 7 (the other p for which Gamma(0)(p) has genus zero) are well known, and date back to Klein and Ramanujan. Moreover, we find an exotic modular equation, i.e., it has the same form as Ramanujan's modular equation of degree 13, but with different kinds of modular parameterizations.SCI(E)ARTICLE3689-7124