Bielliptic curves of genus 3 in the hyperelliptic moduli

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space ${\text{M}}_3^b$ of such curves with the hyperelliptic moduli \H_3. Such intersection $\mathrm{\S }$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2,k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $\mathrm{\S }$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\text{p}\in \mathrm{\S }$, such that $\mathrm{\mid }\text{Aut}(\text{p})\mathrm{\mid }>4$, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the $Gl(2, k)$-invariants.