Elliptic curves with 2-torsion contained in the 3-torsion field

TL;DR

This paper characterizes elliptic curves over Q whose 2-torsion field is contained in their 3-torsion field by providing an explicit model of the modular curve X'(6), contributing to the understanding of non-Serre curves and entanglement fields.

Contribution

The paper explicitly models the modular curve X'(6), linking elliptic curves with specific torsion field containment, and explores its implications for non-Serre curves and entanglement fields.

Findings

Explicit model of X'(6) provided

Infinite family of elliptic curves with non-abelian entanglement fields identified

Characterization of j-invariants with 2-torsion in 3-torsion field

Abstract

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). Our motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, X'(6)(Q) gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q.