The Galois Group of 8-Torsion Points on Elliptic Curves

Abstract

In this paper, we explore the Galois group of 8-torsion points on elliptic curves over Q. In particular, we present algorithms that allow us to find the subgroup of GL2(Z/8Z) that best represents the 8-torsion group of a fixed elliptic curve E. We first give a general introduction to elliptic curves, algebraic number theory, and representation theory, which together provide a theoretical background for computations. We present the algorithms at the end of the paper, along with some computational results of its application.