How branching properties determine modular equations

If a prime p is decomposed as ${x^2} + 4{y^2}$, the power ${2^m}\vert\vert y$ can be determined by an algorithm of polynomial efficiency based on use of singular moduli from the modular equation of order 2. The properties of the modular functions required in this algorithm are simple branching and parametrization properties, which in turn define the modular functions and equations (essentially uniquely). The well-known equations of "Klein's Icosahedron" and their Hecke analogues come into play here, and to some extent they can be uniquely characterized in this fashion. The extraneous cases which arise are in some sense interesting analogues of modular equations.