On the rank of certain parametrized elliptic curves

Subject Areas : Applied Mathematics

Ali Hadavand ^{1 *}

1 - aDepartment of mathematics, Arak Branch, Islamic Azad university, Arak, Iran.

Received: 2016-08-24 Accepted : 2015-08-14 Published : 2016-12-01

Keywords: Elliptic Curve, Selmer Group,

Abstract :

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.