The parameter appearing in the definition of the incomplete elliptic integral is called the modulus. For fixed, can be regarded as a map from (most of) the complex plane into the elliptic curve with half-periods and . Thus from the modulus we obtain the invariant of the curve where the corresponding elliptic functions live. The modular function reverses this correspondence, that is, .
The function is invariant under the group of transformations of the complex upper half-plane generated by and .
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