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

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