## Functiones et Approximatio Commentarii Mathematici

### Points of order $13$ on elliptic curves

#### Abstract

We study elliptically parametrized families of elliptic curves with a point of order $13$ that do not arise from rational parametrizations. We also show that no elliptic curve over $\mathbb{Q}(\zeta_{13})^+$ can possess a rational point of order $13$.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 58, Number 1 (2018), 121-129.

Dates
First available in Project Euclid: 2 December 2017

https://projecteuclid.org/euclid.facm/1512183758

Digital Object Identifier
doi:10.7169/facm/1666

Mathematical Reviews number (MathSciNet)
MR3780039

Zentralblatt MATH identifier
06924921

#### Citation

Kamienny, Sheldon; Newman, Burton. Points of order $13$ on elliptic curves. Funct. Approx. Comment. Math. 58 (2018), no. 1, 121--129. doi:10.7169/facm/1666. https://projecteuclid.org/euclid.facm/1512183758

#### References

• S. Anni, S. Siksek, Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$, Algebra Number Theory 10 (2016), no. 6, 1147–1172.
• J. Bosman, P. Bruin, A. Dujella, F. Najman, Ranks of elliptic curves with prescribed torsion over number fields, Int. Math. Res. Not. IMRN 11 (2014), 2885–2923.
• J. Cremona, Tables of elliptic curves over number fields, University of Warwick, March 2014, available online at http://hobbes.la.asu.edu/lmfdb-14/cremona.pdf.
• M. Derickx, S. Kamienny, B. Mazur, Rational families of 17-torsion points of elliptic curves over number fields, available online at http://www.math.harvard.edu/~mazur/papers/For.Momose20.pdf.
• B. Gross, Lectures on the Conjecture of Birch and Swinnerton-Dyer, available online at http://www.math.harvard.edu/~gross/preprints/lectures-pcmi.pdf.
• M. Hoeij, Low Degree Places on the Modular Curve $X_1(N)$, arXiv:1202.4355 (Version 5), (2014).
• The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org, 2013, [Online; accessed 15 September 2016].
• B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.
• B. Mazur, J. Tate, Points of order $13$ on elliptic curves, Invent. Math. 22 (1973/74), 41–49.
• L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–44.
• F. Najman, Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$, arXiv:1211.2188 (Version 3), (2013).
• A.P. Ogg, Rational points on certain elliptic modular curves, Analytic number theory (Proc. Sympos. Pure Math., Vol XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, R.I., 1973, 221–231.
• T. Shaska, H. V ölklein, Elliptic subfields and automorphisms of genus $2$ function fields, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, 703–723.
• M. Yasuda, M. Torsion points of elliptic curves with good reduction, Kodai Math. J. 31 (2008), no. 3, 385–403.
• X. Yuan, S. Zhang, W. Zhang, The Gross-Zagier formula on Shimura curves, Annals of Mathematics Studies 184, Princeton University Press, Princeton, NJ, 2013.
• S. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), no. 1, 27–147.