  The space of vacua of many four-dimensional, $\mathcal{N}=2$ supersymmetric
gauge theories can famously be identified with a family of complex curves. For
gauge group $SU(2)$, this gives a fully explicit description of the low-energy
effective theory in terms of an elliptic curve and associated modular
fundamental domain. The two-dimensional space of vacua for gauge group $SU(3)$
parametrizes an intricate family of genus two curves. We analyze this family
using the so-called Rosenhain form for these curves. We demonstrate that two
natural one-dimensional subloci of the space of $SU(3)$ vacua, $\mathcal{E}_u$
and $\mathcal{E}_v$, each parametrize a family of elliptic curves. For these
elliptic loci, we describe the order parameters and fundamental domains
explicitly. The locus $\mathcal{E}_u$ contains the points where mutually local
dyons become massless, and is a fundamental domain for a classical congruence
subgroup. Moreover, the locus $\mathcal{E}_v$ contains the superconformal
Argyres-Douglas points, and is a fundamental domain for a Fricke group.
