Rational points in the moduli space of genus two

Abstract

We build a database of genus 2 curves defined over Q which contains all curves with minimal absolute height h ≤ 5, all curves with moduli height h ≤ 20, and all curves with extra automorphisms in standard form y2 = f(x2) defined over Q with height h ≤ 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli spaceM2 for which the field of moduli is a field of definition is discussed and some open problems are presented.