  We implement two-cover descent for plane quartics over Q with all 28
bitangents rational and show that on a significant collection of test cases, it
resolves the existence of rational points. We also review a classical
description of the relevant moduli space and use it to generate examples. We
observe that local obstructions are quite rare for such curves, and only seem
to occur in practice at primes of good reduction. In particular, having good
reduction at 11 implies having no rational points. We also gather numerical
data on two-Selmer ranks of Jacobians of these curves, which suggests that
these often have non-trivial Tate-Shafarevich groups.
  We implement two-cover descent for plane quartics over Q with all 28
bitangents rational and show that on a significant collection of test cases, it
resolves the existence of rational points. We also review a classical
description of the relevant moduli space and use it to generate examples. We
observe that local obstructions are quite rare for such curves and only seem to
occur in practice at primes of good reduction. In particular, having good
reduction at 11 implies having no rational points. We also gather numerical
data on two-Selmer ranks of Jacobians of these curves, providing evidence these
behave differently from those of general abelian varieties due to the frequent
presence of an everywhere locally trivial torsor.
