On the rank of certain parametrized elliptic curves
Number Theory
2012-01-30 v1
Abstract
In this paper the family of elliptic curves over \Q given by the equation E_{p}: Y^2=(X-p)^3+X^3+(X+p)^3 where p is a prime number, is studied. It is shown that the maximal rank of the elliptic curves is at most 3 and some conditions under which we have rank(E_p(\Q))=0 or rank(E_p(\Q))=1 or rank(E_p(\Q))>=2 are given. Moreover It is shown that in the family there are elliptic curves with rank 0,1,2 and 3.
Cite
@article{arxiv.1201.5731,
title = {On the rank of certain parametrized elliptic curves},
author = {A. Astaneh-Asl},
journal= {arXiv preprint arXiv:1201.5731},
year = {2012}
}