On the rank of certain parametrized elliptic curves

In this paper the family of elliptic curves over Q given by the equation Ep : Y 2 = (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(Q)) = 0 or rank(Ep(Q)) = 1 or rank(Ep(Q)) ≥ 2 are given.