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Mathematics of Computation

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Higher descents on an elliptic curve with a rational 2-torsion point

Author: Tom Fisher
Journal: Math. Comp. 86 (2017), 2493-2518
MSC (2010): Primary 11G05, 11Y50
Published electronically: December 21, 2016
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Abstract: Let $ E$ be an elliptic curve over a number field $ K$. Descent calculations on $ E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of $ 4$-descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when $ K=\mathbb{Q}$) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when $ E$ has a rational $ 2$-torsion point. In particular, when $ E$ has full rational $ 2$-torsion, we describe a method for $ 8$-descent that is practical for elliptic curves $ E/\mathbb{Q}$ with large discriminant.

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Tom Fisher
Affiliation: University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Received by editor(s): September 15, 2015
Received by editor(s) in revised form: March 16, 2016
Published electronically: December 21, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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