  Let $F$ be a non-Archimedean locally compact field of residual characteristic
$p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of
the general linear group $\text{\rm GL}_n(F)$. We give a transparent
parametrization of the irreducible, totally ramified, cuspidal representations
of $G$ of parametric degree $n$. We show that the parametrization is respected
by the Jacquet-Langlands correspondence, relative to any other inner form. This
expresses the Jacquet-Langlands correspondence for such representations within
a single, compact formula.
