in this paper the family of elliptic curves over q given by the equation ep :y2 = (x - p)3 + x3 + (x + p)3 where p is a prime number, is studied. itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(ep(q)) = 0 or rank(ep(q)) = 1 orrank(ep(q))≥2 are given.

We characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely xy +y +x = 0. We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of type y + y = x via Class Field Theory.