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A generalised Weber function is given by , 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 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 and j(z). Our ultimate goal is the use of these invariants in constructing...
We examine a class of modular functions for whose values generate ring class fields of imaginary quadratic orders. This fact leads to a new algorithm for constructing elliptic curves with complex multiplication. The difficulties arising when the genus of is not zero are overcome by computing certain modular polynomials.
Being a product of four -functions, the proposed modular functions can be viewed as a natural generalisation of the functions examined by Weber and usually employed...
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