On the rank of certain parametrized elliptic curves

TL;DR

This paper investigates a family of elliptic curves parametrized by prime numbers, establishing an upper bound on their rank and identifying conditions for various rank values, including the existence of curves with ranks 0 through 3.

Contribution

It provides the first known upper bound of 3 for the rank of these specific elliptic curves and characterizes conditions for different rank levels.

Findings

Maximal rank of the family is at most 3.

Conditions for rank 0, 1, and at least 2 are identified.

Existence of curves with ranks 0, 1, 2, and 3 is demonstrated.

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.