  We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2
+ b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem
in this context and give a geometric proof of the fact that $P$ has
quaternionic multiplication (QM) by the quaternion order of discriminant $6$.
This allows us to describe the Galois action on the geometric endomorphism
algebra of $P$. As an application, we classify the torsion subgroups of the
Mordell-Weil groups $P(\mathbb{Q})$, as both abelian groups and
$\text{End}(P)$-modules.
