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arXiv · 1704.06190

A note on 8-division fields of elliptic curves

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

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Jeffrey Yelton. 2017-04-20. A note on 8-division fields of elliptic curves. https://arxiv.org/abs/1704.06190

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