## Rocky Mountain Journal of Mathematics

### 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.

#### Article information

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

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1422885098

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-1867

Mathematical Reviews number (MathSciNet)
MR3310952

Zentralblatt MATH identifier
1321.11062

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

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1422885098

#### References

• B.J. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves, II, J. reine angew. Math. 218 (1965), 79–108.
• W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997), 235–265.
• G. Campbell, Finding elliptic curves and families of elliptic curves over $\Q$ of large rank, Ph.D. thesis, Rutgers University, 1999.
• J. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997.
• A. Dujella, High rank elliptic curves with prescribed torsion, http:// www.maths.hr/~duje/tors.html.
• A. Dujella, A.S. Janfada and S. Salami, A search for high rank congruent number elliptic curves, J. Int. Seq. 12 (2009), 09.5.8.
• A. Dujella and M. Jukić Bokun, On the rank of elliptic curves over $\Q(i)$ with torsion group $\Z/4\Z \times \Z/4\Z$, Proc. Japan Acad. Math. Sci. 86 (2010), 93–96.
• N.D. Elkies, Three lectures on elliptic surfaces and curves of high rank, Lecture notes, Oberwolfach, 2007, arXiv:0709.2908.
• M. Fujiwara, $\ta$-congruent numbers, in Number theory, K. Győry, A. Pethő and V. Sós, eds., de Gruyter, Berlin, 1997.
• M. Fujiwara, Some properties of $\ta$-congruent numbers, Natural Science Report, Ochanomizu University, 118 (2001), 1–8.
• F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), 1–23.
• M. Kan, $\ta$-congruent numbers and elliptic curves, Acta Arith. 94 (2000), 153–160.
• N. Koblitz, Introduction to elliptic curves and modular forms, Grad. Texts Math. 97, 2nd edition, Springer-Verlag, Berlin, 1993.
• J.-F. Mestre, Construction de courbes elliptiques sur $\Q$ de rang $\geq 12$, C.R. Acad. Sci. Paris 295 (1982), 643–644.
• ––––, Formules explicites et minorations de conducteurs de variétés algébriques, Comp. Math. 58 (1986), 209–232.
• ––––, Rang de certaines familles de courbes elliptiques d' invariant donné, C.R. Acad. Sci. Paris 327 (1998), 763-764.
• K. Nagao, An example of elliptic curve over $\Q$ with rank $\geq 21$, Proc. Japan Acad. Math. Sci. 70 (1994), 104–105.
• PARI/GP, version 2.3.3, Bordeaux, 2008, http://pari.math.u-bordeaux.fr
• N. Rogers, Rank computations for the congruent number elliptic curves, Exp. Math. 9 (2000), 591–594.
• ––––, Elliptic curves $x^3+y^3=k$ with high rank, Ph.D. thesis, Harvard University, Cambridge, 2004.
• K. Rubin and A. Silverberg, Twists of elliptic curves of rank at least four, in Ranks of elliptic curves and random matrix theory, Cambridge University Press, 2007.
• ––––, Rank frequencies for quadratic twists of elliptic curves, Exp. Math. 10 (2001), 559–569.