  L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow \mathbb{P}^1$
of genus 2 curves with supersingular jacobian. In this paper we first classify
the reducible fibers of a Moret-Bailly family using linear algebra over a
quaternion algebra. The main result is an algorithm that exploits properties of
two reducible fibers to compute a hyperelliptic model for any irreducible fiber
of a Moret-Bailly family.
