A family of monogenic $S_4$ quartic fields arising from elliptic curves

We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic $S_4$ fields given by $T^4 - 6T^2 - \alpha T - 3$ for $\alpha \in \mathbb{Z}$ such that $\alpha \pm 8$ are squarefree, are monogenic.