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Elliptic curves with a point of order 13 defined over cyclic cubic fields

Peter Bruin, Maarten Derickx, Michael Stoll

Abstract

We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $\mathbb Q$ with a $K$-rational point of order 13 and such that $E$ is not defined over $\mathbb Q$.

Elliptic curves with a point of order 13 defined over cyclic cubic fields

Abstract

We show that there is essentially a unique elliptic curve defined over a cubic Galois extension of with a -rational point of order 13 and such that is not defined over .

Paper Structure

This paper contains 3 sections, 1 theorem, 11 equations.

Table of Contents

  1. Introduction
  2. The result
  3. Proof of the theorem

Key Result

Theorem 1

Let $K$ be a cubic Galois extension of $\mathbb{Q}$ and let $E$ be an elliptic curve defined over $K$ with $E(K)[13] \neq 0$. Then either $E$ is defined over $\mathbb{Q}$, or else $K = \mathbb{Q}(\alpha)$ with and $E$ is isomorphic to a Galois conjugate of the curve where

Theorems & Definitions (1)

  • Theorem 1