A note on 8-division fields of elliptic curves

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 $\mathrm{Gal}(K(E[8])/K)$.

9 pages, 1 section, 13 references Changes made so that this article is as it appears in European Journal of Mathematics

References in corpus (2)

• The abelianization of the level 2 mapping class group
• The divisibility by 2 of rational points on elliptic curves