If a prime p is decomposed as x2 + 4y2 , the power 2m\\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

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... nt they can be uniquely characterized in this fashion. The extraneous cases which arise are in some sense interesting analogues of modular equations.