engleski

The rank of certain subfamilies of the elliptic curve Y^2 = X^3 - X + T^2

Let E be the elliptic curve over Q(T) given by the equation E : Y^2 = X^3 - X + T^2. It is known that the torsion subgroup is trivial, rankC(T)(E) = 2 and rankQ(T)(E) = 2. We find a parametrization of rank >= 3 over the function field Q(a, i, s, n, k, l) where s^2 = i^3 + a^2. From this we get families of rank >= 3 and >= 4 over fields of rational functions in four variables and a family of rank >= 5 parametrized by an elliptic curve of positive rank. We also found a particular elliptic curve with rank >= 11.

parametrization ; elliptic surface ; elliptic curve ; function field ; rank ; family of elliptic curves ; torsion

nije evidentirano