Integrality identity for the modular discriminant
= Integrality identity for the modular discriminant
{title2=$\Delta=(5S_3+7S_5)/12+100S_3^2+8000S_3^3-147S_5^2$}
Put $S_j=\sum_{n\geq1}\sigma_j(n)q^n$. Expanding $E_4=1+240S_3$ and $E_6=1-504S_5$ gives $\Delta=(5S_3+7S_5)/12+100S_3^2+8000S_3^3-147S_5^2$. For every integer $d$, $5d^3+7d^5$ is divisible by twelve, by separate congruences modulo three and four. Consequently every coefficient of the <modular discriminant> is integral, without using a product expansion.