On the existence of elliptic curves of Mordell-Weil rank 6 with four parameters

We construct an elliptic curve defined over Q with Mordell-Weil rank greater than or equal to 6 as a generic twist by a certain quadratic extension. Moreover, since they have four independent parameters, they give us rather a large supply of elliptic curves defined over Q with rank greater than or equal to 6. As an application, we find infinitely many pairs (a, b) of rational numbers such that the elliptic curve by(2) = x(3) + ax(2) + ax + 1 is of Mordell-Weil rank greater than or equal to 6. (C) 1996 Academic Press. Inc.