On the family of elliptic curves $Y^2=X^3-T^2X+1$

Abstract

Let E be the elliptic curve over Q(T) given by the equation E : Y^2 = X^3 - T^2X + 1: We prove that the torsion subgroup of the group E(C(T)) is trivial, rank Q(T)(E) = 3 and rank C(T)(E) = 4. We find a parametrization of E of rank at least four over the function field Q(a, i, s, n, k) where s^2 = i^3 - a^2i. From this we get a family of rank >= 5 over the field of rational functions in two variables and a family of rank >= 6 over an elliptic curve of positive rank. We also found particular elliptic curves with rank >= 11.