Received March 4, 2026. Published online October 26, 2026.
Abstract: We consider two infinite parametric families of elliptic curves defined over $\mathbb{Q}$ given by the equations $E_{a,b} \colon y^2 = x^3 - a^2x + b^2$ and $E^{\prime}_{a,b} \colon y^2 = x^3 - a^2x + b^6$, where $a,b \in\mathbb{N}$ satisfy certain mild conditions. We prove that the torsion group of $E_{a,b}(\mathbb{Q})$ is trivial and the Mordell-Weil ranks of both $E_{a,b}(\mathbb{Q})$ and $E^{\prime}_{a,b}(\mathbb{Q})$ are at least $3$ for infinitely many choices of $a$ and $b$ by using the Néron-Tate height of a rational point and by exploiting the unit group of the ring of integers of $\mathbb{Q}(\sqrt3)$. This is an extension of the results of Brown-Myres and Fujita-Nara, where lower bounds of the ranks were provided under the assumption that $a = 1$ or $b = 1$. Also, our families of elliptic curves vastly generalize the curves recently investigated by Hatley-Stack and Walsh.
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