Hopf-Galois module structure of monogenic orders in cubic number fields

For a cubic number field $L$, we consider the $\mathbb{Z}$-order in $L$ of the form $\mathbb{Z}[\alpha ]$, where $\alpha$ is a root of a polynomial of the form $x^3-ax+b$ and $a,b\in \mathbb{Z}$ are integers such that $v_p(a)\le 2$ or $v_p(b)\le 3$ for all prime numbers $p$. We characterize the freeness of $\mathbb{Z}[\alpha ]$ as a module over its associated order in the unique Hopf-Galois structure $H$ on $L$ in terms of the solvability of at least one between two generalized Pell equations in terms of $a$ and $b$. We determine when the equality ${\mathcal{O}}_L=\mathbb{Z}[\alpha ]$ is satisfied in terms of congruence conditions for $a$ and $b$. For such cases, we specialize our result so as to obtain criteria for the freeness of ${\mathcal{O}}_L$ as a module over its associated order in $H$.