= Prime-level cusp orders of discriminant degeneracy forms
{title2=$\operatorname{ord}_{\infty}(\Delta,\Delta(p\tau))=(1,p),\quad\operatorname{ord}_{0}=(p,1)$}
For prime $p$, the two cusps of $\Gamma_0(p)$ have widths one and $p$. At infinity the <modular discriminant> product gives the first pair of orders. At zero use $S\tau=-1/\tau$ and the parameter $q_0=e^{2\pi i\tau/p}$. The weight-twelve transforms are $\Delta|_{12}S=\Delta(\tau)$ and $\Delta(p\tau)|_{12}S=p^{-12}\Delta(\tau/p)$, giving orders $p$ and one in $q_0$.
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