In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that :j\left(e^{\frac{2}{3}\pi i}\right) = 0, \quad j(i) = 1728. Rational functions of are modular, and in fact give all modular functions. Classically, the -invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this c...