Mordell-Weil ranks of families of elliptic curves parametrized by binary quadratic forms

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.