J-invariant

Summary

In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) 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^{2\pi i/3}\right) = 0, \quad j(i) = 1728 = 12^3. Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). Definition The j-invariant can be defined as a function on the upper half-plane H = {τ ∈ C, Im(τ) > 0}, :j(\tau) = 1728 \frac{g_2(\tau)^3}{\Delta(\tau)} = 1728 \frac{g_2(\tau)^3}{g_2(\tau)^3 - 27g_3(\tau)^2} = 172

Official source

https://en.wikipedia.org/wiki/J-invariant

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