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

TL;DR

This paper investigates specific quartic $S_4$ fields derived from elliptic curves, demonstrating their monogenicity by analyzing torsion fields, reduction properties, and applying the Montes Algorithm.

Contribution

It introduces a family of monogenic quartic $S_4$ fields arising from elliptic curves and provides explicit power bases for their partial torsion fields.

Findings

The quartic fields are monogenic under certain conditions.

Explicit power bases are constructed for the partial 3-torsion fields.

The approach combines reduction analysis and the Montes Algorithm.

Abstract

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.