  In this paper we are interested in the stratum H^{hyp}(4) of translation
surfaces, which consists of pairs (M,\omega), where M is a hyper-elliptic
Riemann surface of genus 3, and \omega is a holopmorphic 1-form on M having
only one zero. We first show that every surface in this stratum can be
decomposed into parallelograms following a unique model. We then single out a
condition on this decomposition, and show that if this condition is satisfied
then the SL(2,R) orbit of the surface is dense in the stratum. Using this
criterion, we show that there are generic surfaces in this stratum with
coordinates in any quadratic field, and that surfaces arising from the
Thurston-Veech construction with cubic trace field can be generic.
