  Given a Diophantine triple $\{c_1(t),c_2(t),c_3(t)\}$, the elliptic curve
over Q(t) induced by this triple, i.e. $y^2=(c_1(t) x+1) (c_2(t) x+1) (c_3(t)
x+1)$, can have as torsion group one of the non-cyclic groups in Mazur's
theorem, i.e. Z/2Z x Z/2Z, Z/2Z x Z/4Z, Z/2Z x Z/6Z or Z/2Z x Z/8Z.
  In this paper we present results concerning the rank over Q(t) of these
curves improving in some of the cases the previously known results.
