Generalised Weber functions

A generalised Weber function is given by w_N(z) - η(z/N)/η(z), where η(z) is the Dedekind function and N is any integer; the original function corresponds to N=2. We classify the cases where some power w_N^e evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating w_N(z) and j(z). Our ultimate goal is the use of these invariants in constructing reductions of elliptic curves over finite fields suitable for cryptographic use.