Parametrization of Algebraic Points of Low Degree on the Hyperelliptic Curves \(y^{2}=x(x^2-n^2)(x^2-4n^2)\)

We give a parametrization of the set of algebraic points of degree at most 3 over \(\mathbb {Q}\) on hyperelliptic curves \( y^{2}=x(x^2-n^2)(x^2-4n^2)\) with \(n \in \{1, 2, 3, q \text{ a prime number and } q \equiv 7 \pmod {24} \}\) . The result obtained extends the work of Heiden, Evink, and Evink, et al., who determined the set of \(\mathbb {Q}-\) rational points on the same curves.