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

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.