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If a prime p is decomposed as x2 + 4y2, the power 2m I lY 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. 1. THE POWER CONDUCTOR ALGORITHM The motivating problem for this work comes out of a type of algorithm [2] of ring class field theory. A prime p _ 1 mod 4 admits a (unique) representation in Z as a sum of squares. We shall consider only the additional power of 2 dividing the even square, namely the m for which (I.la) p = X2 + 4y2 2mIly. The (polynomial-time) algorithm for m generates a sequence {ao, a, am 1-} of length m as follows: (1. lb) aO = 9/8, rk ak, ak+I (rk +3)2/[8(rk + 1)] modp. The last k for which (akip) = 1 (or for which rk is definable) defines the length of the sequence as m = k + 1. (Incidentally, the method is independent of the choice of the sign of the square root in rk .) This is illustrated below for 89= 52+4.42 (where m=2): Received by the editor September 2, 1992. 1991 Mathematics Subject Classification. Primary 1 liF , 1 lY1 6.

How branching properties determine modular equations