Rational configuration problems and a family of curves

Jonathan R. Love

Given $\eta =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\text{GL}}_2(\mathbb{Q})$, we consider the number of rational points on the genus one curve \[H_η:y^2=(a(1-x^2)+b(2x))^2+(c(1-x^2)…