  We prove results on the Mordell--Weil rank of elliptic curves $y^2=x(x-\alpha
a^2)(x-\beta b^2)$ parametrized by binary quadratic forms $\alpha a^2+\beta
b^2=\gamma c^2$. We express our explicit lower bounds over number fields and
offer a detailed description of the corresponding Mordell-Weil group structure
in the function field case.
