  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.
