Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 10, 2020

Generators and integral points on elliptic curves associated with simplest quartic fields

Sylvain Duquesne, Tadahisa Nara and Arman Shamsi Zargar
From the journal Mathematica Slovaca

Abstract

We associate to some simplest quartic fields a family of elliptic curves that has rank at least three over ℚ(m). It is given by the equation

Em:y2=x33636m4+48m2+2536m448m2+25x.

Employing canonical heights we show the rank is in fact at least three for all m. Moreover, we get a parametrized infinite family of rank at least four. Further, the integral points on the curve Em are discussed and we determine all the integral points on the original quartic model when the rank is three. Previous work in this setting studied the elliptic curves associated with simplest quartic fields of ranks at most two along with their integral points (see [2, 3]).

  1. Communicated by Milan Paštéka

References

[1] Bosma, W.—Cannon, J.—Playoust, C.: The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265.10.1006/jsco.1996.0125Search in Google Scholar

[2] Duquesne, D.: Integral points on elliptic curves defined by simplest cubic fields, Exp. Math. 10(1) (2001), 91–102.10.1080/10586458.2001.10504431Search in Google Scholar

[3] Duquesne, D.: Elliptic curves associated with simplest quartic fields, J. Théor. Nombres Bordeaux 19(1) (2007), 81–100.10.5802/jtnb.575Search in Google Scholar

[4] Fujita, Y.: Generators for the elliptic curvey2 = x3nx of rank at least three, J. Number Theory 133(5) (2013), 1645–1662.10.1016/j.jnt.2012.10.011Search in Google Scholar

[5] Fujita, Y.: Generators for congruent number curves of ranks at least two and three, J. Ramanujan Math. Soc. 29(3) (2014), 307–319.Search in Google Scholar

[6] Fujita, Y.—Nara, T.: Generators and integral points on twists of the Fermat cubic, Acta Arith. 168(1) (2015), 1–6.10.4064/aa168-1-1Search in Google Scholar

[7] Fujita, Y, —Terai, N.: Generators for the elliptic curvey2 = x3nx, J. Théor. Nombres Bordeaux 23(2) (2011), 403–416.10.5802/jtnb.769Search in Google Scholar

[8] Fujita, Y, —Terai, N.: Generators and integral points on the elliptic curvey2 = x3nx, Acta Arith. 160(4) (2013), 333–348.10.4064/aa160-4-3Search in Google Scholar

[9] Gras, M. N.: Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de ℚ, Publ. Math. Besançon Algébre Théorie Nr. 2 (1977/1978).10.5802/pmb.a-17Search in Google Scholar

[10] Gusić, I.—Tadić, P.: Injectivity of specialization homomorphism of elliptic curves, J. Number Theory 148 (2015), 137–152.10.1016/j.jnt.2014.09.023Search in Google Scholar

[11] Kim, H. K.: Evaluation of zeta functions ats = –1 of the simplest quartic fields, Proceedings of the 2003 Nagoya Conference “Yokoi-Chowla Conjecture and Related Problems”, Saga Univ., Saga, 2004, pp. 63–73.Search in Google Scholar

[12] Lazarus, A. J.: Class Numbers of Simplest Quartic Fields, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 313–323.10.1515/9783110848632-027Search in Google Scholar

[13] Lazarus, A. J.: On the class number and unit index of simplest quartic fields, Nagoya Math. J. 121 (1991), 1–13.10.1017/S0027763000003378Search in Google Scholar

[14] Louboutin, S.: The simplest quartic fields with ideal class groups of exponents less than or equal to 2, J. Math. Soc. Japan 56(3) (2004), 717–727.10.2969/jmsj/1191334082Search in Google Scholar

[15] Olajos, P.: Power integral bases in the family of simplest quartic fields, Exp. Math. 14(2) (2005) 129–132.10.1080/10586458.2005.10128916Search in Google Scholar

[16] Saad, Y.: Numerial Methods for Large Eigenvalue Problems, 2nd ed., SIAM, Philadelphia, 2011.10.1137/1.9781611970739Search in Google Scholar

[17] Sage software, available at http://sagemath.orgSearch in Google Scholar

[18] Siksek, S.: Infinite descent on elliptic curves, Rocky Mountain J. Math. 25(4) (1995), 1501–1538.10.1216/rmjm/1181072159Search in Google Scholar

[19] Silverman, J.: Computing heights on elliptic curves, Math. Comp. 51(183) (1988), 339–358.10.1090/S0025-5718-1988-0942161-4Search in Google Scholar

[20] Silverman, J.: The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 2009.10.1007/978-0-387-09494-6Search in Google Scholar

[21] Washington, L. C.: Elliptic Curves: Number Theory and Cryptography, 2nd ed., CRC Press, Taylor & Francis Group, Boca Raton, FL, 2008.10.1201/9781420071474Search in Google Scholar

Received: 2019-03-04
Accepted: 2019-09-19
Published Online: 2020-03-10
Published in Print: 2020-04-28

© 2020 Mathematical Institute Slovak Academy of Sciences