Hopf-Galois module structure of quartic Galois extensions of Q

Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness ring integers $\mathcal{O}_L$ as module over associated order $\mathfrak{A}_H$ in $H$. For classical $H_c$, know by Leopoldt's theorem that is $\mathfrak{A}_{H_c}$-free. If cyclic, it admits unique non-classical structure, whereas if biquadratic, three such structures. In both cases, obtain depends solvability $\mathbb{Z}$ certain generalized Pell equations. We shall translate some results equations into $\mathfrak{A}_H$-freeness $\mathcal{O}_L$.