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

Abstract

Consider the elliptic curves given by $E_{n,\theta }:y^2=x^3+2sn{x}^2-(r^2-s^2)n^{2}x$ where $0<\theta <\pi$, $\cos (\theta )=s/r$ is rational with $0\le |s|<r$ 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.

11 pages, to appear in the Rocky mountain journal of mathematics