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

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 = x³ – nx 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 = x³ – nx, 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 = x³ – nx, 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