Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Explicit 5-descent on elliptic curves

Tom Fisher

Abstract: We compute equations for genus one curves representing non-trivial 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.

Files available: paper (PDF), Magma code