= Klein j-invariant classifies complex lattice homothety
{c}
{title2=$SL_2(\mathbb Z)\backslash\mathbb H\xrightarrow{\;j\;}\mathbb C$}
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>.
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