j-invariant

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

${\displaystyle j\left(e^{2\pi i/3}\right)=0,\quad j(i)=1728=12^{3}.}$

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 ${\displaystyle \mathbb {C} }$, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).