Klein j-invariant classifies complex lattice homothety ($S{L}_2(\mathbb{Z})\backslash \mathbb{H}\stackrel{j}{\to }\mathbb{C}$)

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.