  We show that the Burkhardt quartic threefold is rational over any field of
characteristic distinct from 3. We compute its zeta function over finite
fields. We realize one of its moduli interpretations explicitly by determining
a model for the universal genus 2 curve over it, as a double cover of the
projective line. We show that the j-planes in the Burkhardt quartic mark the
order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse
pencil on a j-plane gives rise to the universal curve as a discriminant of a
cubic genus one cover.
