On the rank of certain parametrized elliptic curves
Abstract
In this paper the family of elliptic curves over given by the equation Ep: 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(Ep())=0 or rank(Ep())=1 or rank(Ep())>=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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