Constructing all genus 2 curves with supersingular Jacobian.

L. Moret-Bailly constructed families C → 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. [ABSTRACT FROM AUTHOR]