Rocky Mountain Journal of Mathematics

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

A.S. Janfada, S. Salami, A. Dujella, and J.C. Peral

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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.

Article information

Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1867-1880.

First available in Project Euclid: 2 February 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]

$\theta$-congruent number elliptic curve Mordell-Weil rank


Janfada, A.S.; Salami, S.; Dujella, A.; Peral, J.C. On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves. Rocky Mountain J. Math. 44 (2014), no. 6, 1867--1880. doi:10.1216/RMJ-2014-44-6-1867.

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