On the high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves

TL;DR

This paper investigates elliptic curves related to the $ heta$-congruent number problem, specifically for $ heta=rac{ ext{ extpi}}{3}$ and $rac{2 ext{ extpi}}{3}$, discovering subfamilies with high rank and examples with rank up to 7.

Contribution

It identifies new subfamilies of these elliptic curves with high rank and provides explicit examples, advancing understanding of their rank distribution.

Findings

Found subfamilies with rank at least 3 over $\

Discovered subfamilies with rank 4 parametrized by elliptic curve points

Examples of curves with rank up to 7 over $\

Abstract

Consider the elliptic curves given by $ E_{n,\theta}:\quad y^2=x^3+2s n x^2-(r^2-s^2) n^2 x $ where $0 < \theta< \pi$, $\cos(\theta)=s/r$ is rational with $0\leq |s| <r$ and $\gcd (r,s)=1$. These elliptic curves are related to the $\theta$-congruent number problem as a generalization of the congruent number problem. For fixed $\theta$ this family corresponds to the quadratic twist by $n$ of the curve $E_{\theta}: \,\, y^2=x^3+2s x^2-(r^2-s^2) x.$ We study two special cases $\theta=\pi/3$ and $\theta=2\pi/3$. We have found a subfamily of $n=n(w)$ having rank at least $3$ over ${\mathbb Q}(w)$ and a subfamily with rank $4$ parametrized by points of an elliptic curve with positive rank. We also found examples of $n$ such that $E_{n, \theta}$ has rank up to $7$ over $\mathbb Q$ in both cases.