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

Abstract

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