joint with John Cremona, Catherine O'Neil, Denis Simon and Michael Stoll
This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n = 3 for elliptic curves over the rationals, and have been implemented in Magma.
One important ingredient of our work is an algorithm for trivialising central simple algebras. This is of independent interest: for example, it could be used for parametrising Brauer-Severi surfaces.
A Magma file accompanying the examples in Section 8 is available here.
The other papers in this series are Paper I. Algebra and Paper II. Geometry.