Vol. 1, No. 1, 2013

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Explicit $5$-descent on elliptic curves

Tom Fisher

Vol. 1 (2013), No. 1, 395–411
Abstract

We compute equations for genus-one curves representing nontrivial elements of order 5 in the Tate-Shafarevich group of an elliptic curve. We explain how to write the equations in terms of Pfaffians and give examples for elliptic curves over the rationals both with and without a rational 5-isogeny.

Keywords
elliptic curves, descent, Selmer groups, Pfaffians
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 14H52, 14H25
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Published: 14 November 2013
Authors
Tom Fisher
DPMMS, Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
United Kingdom