On the discriminant of elliptic curves with non-trivial torsion

TL;DR

This paper explores the relationship between the torsion subgroup of elliptic curves over rationals and an associated Galois group, revealing a tight connection that enhances understanding of their algebraic structure.

Contribution

It establishes a precise link between torsion subgroups and Galois groups for elliptic curves over rationals, providing new insights into their algebraic properties.

Findings

Identifies a tight relationship between torsion subgroup and Galois group

Provides a characterization of Galois groups from torsion data

Enhances understanding of elliptic curve symmetries

Abstract

For those elliptic curves defined over the rational with non--trivial torsion subgroup, we find a tight relationship between the torsion subgroup itself and a Galois group naturally arising from the curve.