Open Access
2014 On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves
A.S. Janfada, S. Salami, A. Dujella, J.C. Peral
Rocky Mountain J. Math. 44(6): 1867-1880 (2014). DOI: 10.1216/RMJ-2014-44-6-1867


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.


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A.S. Janfada. S. Salami. A. Dujella. J.C. Peral. "On high rank $\pi/3$ and $2\pi/3$-congruent number elliptic curves." Rocky Mountain J. Math. 44 (6) 1867 - 1880, 2014.


Published: 2014
First available in Project Euclid: 2 February 2015

zbMATH: 1321.11062
MathSciNet: MR3310952
Digital Object Identifier: 10.1216/RMJ-2014-44-6-1867

Primary: 11G05

Keywords: $\theta$-congruent number , Elliptic curve , Mordell-Weil rank

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.44 • No. 6 • 2014
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