  We build a database of genus 2 curves defined over $\mathbb Q$ which contains
all curves with minimal absolute height $h \leq 5$, all curves with moduli
height $\mathfrak h \leq 20$, and all curves with extra automorphisms in
standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$.
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
$\mathcal M_2$ for which the field of moduli is a field of definition is
discussed and some open problems are presented.
