Cubic symmetroids and vector bundles on a quadric surface

We investigate the jumping conics of stable vector bundles $\text{Ee}$ of rank 2 on a smooth quadric surface $Q$ with the Chern classes $c_1={\text{Oo}}_Q(-1,-1)$ and $c_2=4$ with respect to the ample line bundle ${\text{Oo}}_Q(1,1)$. We describe the set of jumping conics of $\text{Ee}$, a cubic symmetroid in ${\text{PP}}_3$, in terms of the cohomological properties of $\text{Ee}$. As a consequence, we prove that the set of jumping conics, $S(\text{Ee})$, uniquely determines $\text{Ee}$. Moreover we prove that the moduli space of such vector bundles is rational.