On the rank of certain parametrized elliptic curves

2012/01/27 by Astaneh-Asl, A.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1201.5731

Abstract

In this paper the family of elliptic curves over \Q given by the equation E_p: Y²=(X-p)³+X³+(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.

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