A generalised Weber function is given by $w_N(z) = eta(z/N)/eta(z)$, where $eta(z)$ 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.
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