  In this paper, we ask: for which $(g, n)$ is the rational Chow or cohomology
ring of $\overline{\mathcal{M}}_{g,n}$ generated by tautological classes? This
question has been fully answered in genus $0$ by Keel (the Chow and cohomology
rings are tautological for all $n$) and genus $1$ by Belorousski (the rings are
tautological if and only if $n \leq 10$). For $g \geq 2$, work of van Zelm
shows the Chow and cohomology rings are not tautological once $2g + n \geq 24$,
leaving finitely many open cases. Here, we prove that the Chow and cohomology
rings of $\overline{\mathcal{M}}_{g,n}$ are isomorphic and generated by
tautological classes for $g = 2$ and $n \leq 9$ and for $3 \leq g \leq 7$ and
$2g + n \leq 14$. For such $(g, n)$, this implies that the tautological ring is
Gorenstein and $\overline{\mathcal{M}}_{g,n}$ has polynomial point count.
