# A note on 8-division fields of elliptic curves

## TL;DR

This paper explicitly describes the generators and Galois action on the 8-division fields of elliptic curves over fields of characteristic not 2, using roots of defining polynomials.

## Contribution

It provides explicit formulas for generators of the 8-division field and describes the Galois automorphism action in terms of polynomial roots.

## Key findings

- Explicit generators for the 8-division field are given.
- The Galois automorphism action is explicitly described.
- Results apply to elliptic curves defined by degree 3 or 4 polynomials.

## Abstract

Let $K$ be a field of characteristic different from $2$ and let $E$ be an elliptic curve over $K$, defined either by an equation of the form $y^{2} = f(x)$ with degree $3$ or as the Jacobian of a curve defined by an equation of the form $y^{2} = f(x)$ with degree $4$. We obtain generators over $K$ of the $8$-division field $K(E[8])$ of $E$ given as formulas in terms of the roots of the polynomial $f$, and we explicitly describe the action of a particular automorphism in $\mathrm{Gal}(K(E[8]) / K)$.