The 12-th roots of the discriminant of an elliptic curve and the torsion points

TL;DR

This paper provides an explicit description of the 12-th roots of an elliptic curve's discriminant using torsion points and Weil pairing, extending previous results and deepening understanding of elliptic curve invariants.

Contribution

It introduces a new explicit description of the $$-torsor associated with the discriminant in terms of 3- and 4-torsion points and the Weil pairing.

Findings

Explicit description of the $$-torsor in terms of torsion points

Connection between the discriminant's roots and Weil pairing

Generalization of Coates' result on 12-th roots of the discriminant

Abstract

Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $\mu_{12}$-torsor over the field. In this paper, we give an explicit description of this $\mu_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant of an elliptic curve.