On Ramanujans Modular Equations and Hecke Groups

We study the relation between Hecke groups and the modular equations in Ramanujans theories of signature 2, 3 and 4. The solution $(alpha,beta)$ to the generalized modular equation satisfies a polynomial equation $P(alpha,beta)=0$ and we determine the degree in each of $alpha$ and $beta$ of the polynomial $P(alpha,beta)$ explicitly. We establish some mutually equivalent statements related to Hecke subgroups and modular equations, and prove that $(1-beta, 1-alpha)$ is also a solution to the generalized modular equation and $P(1-beta, 1-alpha)=0$.