  Consider the Drinfeld modular curve $X_0(\mathfrak{p})$ for $\mathfrak{p}$ a
prime ideal of $\mathbb{F}_q[T]$. It was previously known that if $j$ is the
$j$-invariant of a Weierstrass point of $X_0(\mathfrak{p})$, then the reduction
of $j$ modulo $\mathfrak{p}$ is a supersingular $j$-invariant. In this paper we
show the converse: Every supersingular $j$-invariant is the reduction modulo
$\mathfrak{p}$ of the $j$-invariant of a Weierstrass point of
$X_0(\mathfrak{p})$.
