Ali Hadavand - aDepartment of mathematics, Arak Branch, Islamic Azad university, Arak, Iran.

In this paper the family of elliptic curves over Q given by the equation Ep :Y۲ = (X - p)۳ + X۳ + (X + p)۳ where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most ۳ and someconditions under which we have rank(Ep(Q)) = ۰ or rank(Ep(Q)) = ۱ orrank(Ep(Q))≥۲ are given.