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Rational points in the moduli space of genus two

Abstract

We build a database of genus 2 curves defined over Q\mathbb Q which contains all curves with minimal absolute height h5h \leq 5, all curves with moduli height h20\mathfrak h \leq 20, and all curves with extra automorphisms in standard form y2=f(x2)y^2=f(x^2) defined over Q\mathbb Q with height h101h \leq 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 space M2\mathcal M_2 for which the field of moduli is a field of definition is discussed and some open problems are presented

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