Table of Contents
Journal of Numbers
Volume 2015, Article ID 501629, 4 pages
http://dx.doi.org/10.1155/2015/501629
Research Article

On the Rank of Elliptic Curves in Elementary Cubic Extensions

College of International Management, Ritsumeikan Asia Pacific University, Oita 874-8577, Japan

Received 21 July 2015; Accepted 6 September 2015

Academic Editor: Andrej Dujella

Copyright © 2015 Rintaro Kozuma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a method for explicitly constructing an elementary cubic extension over which an elliptic curve has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of and a generic polynomial for cyclic cubic extensions. In our method, the extension degree often becomes small.