Constructing all genus 2 curves with supersingular Jacobian

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