Klein j-invariant ($j(\tau )$)

The Klein j-invariant is the weight-zero level-one modular functionIt is holomorphic on the upper half-plane and has a simple pole at infinity.

The valence formula for the modular group shows that $\Delta =(E_4^3-E_6^2)/1728$ has no zeros in the half-plane. For every $a\in \mathbb{C}$, the weight-twelve form $E_4^3-a\Delta$ has total weighted zero order one. At $a=0$ its sole zero is the order-three zero at the cubic elliptic orbit; at $a=1728$ its sole zero is the order-two zero at the quadratic elliptic orbit. Otherwise it has one simple ordinary zero orbit. Thus each value of the Klein j-invariant occurs on exactly one modular orbit and classifies one homothety of complex lattices.

The function $j(\tau )/j(2\tau )$ has weight zero and level ${\Gamma }_1(2)={\Gamma }_0(2)$. It has a simple zero at the cusp at infinity and a simple pole at the cusp zero.