#### 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 $\mathfrak{5}$ 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 $\mathfrak{5}$-isogeny.

##### Keywords
elliptic curves, descent, Selmer groups, Pfaffians
##### Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 14H52, 14H25

Magma code

##### Milestones
Published: 14 November 2013
##### Authors
 Tom Fisher DPMMS, Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom