On the Family of Elliptic Curves $y^2=x^3-5pqx$
Arkabrata Ghosh
TL;DR
This work analyzes the two-parameter family $E_{pq}: y^2 = x^3 - 5 p q x$ of elliptic curves, focusing on the rank and torsion over $\mathbb{Q}$ and the Gaussian field $\mathbb{Q}(i)$. Using a $2$-descent framework with isogenous curves and descent maps $\alpha$ and $\overline{\alpha}$, the authors derive precise rank results under congruence conditions on $p$ and $q$: rank $0$ over $\mathbb{Q}$ and $\mathbb{Q}(i)$ when $p \equiv 33 \pmod{40}$ and $q \equiv 7 \pmod{40}$; and rank $1$ over $\mathbb{Q}$ and rank $2$ over $\mathbb{Q}(i)$ when $p = 40k + 33$, $q = 40l + 27$ with $(25k + 5l + 21)$ a perfect square. In all cases, the torsion over $\mathbb{Q}$ is $\mathbb{Z}/2\mathbb{Z}$. These results are supported by detailed descent computations and modular arguments, with implications for the distribution of ranks in families of quadratic twists. The paper also highlights open questions about achieving higher ranks in this family.
Abstract
This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the rank of $E_{pq}$ is zero over both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the primes $ p $ and $q$ are of the form $ 40k + 33 $ and $ 40l + 27$, where $k,l \in \mathbb{Z}$ such that $(25k+ 5l +21)$ is a perfect square, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Finally, we have shown that torsion of $E_{pq}$ over $\mathbb{Q}$ is isomorphic to $ \mathbb{Z}/ 2\mathbb{Z}$.