On the rank of elliptic curves over \(\mathbb Q(i)\) with torsion group \(\mathbb Z/4\mathbb Z\times \mathbb Z/4\mathbb Z\). (English) Zbl 1217.11058

Dujella, Andrej; Jukić Bokun, Mirela

Proc. Japan Acad., Ser. A 86, No. 6, 93-96 (2010).

Summary: We construct an elliptic curve over \(\mathbb Q(i)\) with torsion group \(\mathbb Z/4\mathbb Z \times\mathbb Z/4\mathbb Z\) and rank equal to 7 and a family of elliptic curves with the same torsion group and rank \(\geq 2\).

MSC:

11G05	Elliptic curves over global fields
11Y50	Computer solution of Diophantine equations

Keywords:

elliptic curve; torsion group; rank

Software:

APECS; PARI/GP; ecdata

References:

[1]	I. Connell, APECS. ftp://ftp.math.mcgill.ca/pub/apecs/
[2]	J. E. Cremona, Algorithms for modular elliptic curves , Second edition, Cambridge Univ. Press, Cambridge, 1997. · Zbl 0872.14041
[3]	A. Dujella, On Mordell-Weil groups of elliptic curves induced by Diophantine triples, Glas. Mat. Ser. III 42(62) (2007), no. 1, 3-18. · Zbl 1137.11038
[4]	S. Kamienny, Torsion points on elliptic curves and \(q\)-coefficients of modular forms, Invent. Math. 109 (1992), no. 2, 221-229. · Zbl 0773.14016
[5]	M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125-149. · Zbl 0647.14020
[6]	J.-F. Mestre, Construction de courbes elliptiques sur \(\mathbf{Q}\) de rang \(\geq 12\), C. R. Acad. Sci. Paris Ser. I 295 (1982), 643-644. · Zbl 0541.14027
[7]	J.-F. Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), no. 2, 209-232. · Zbl 0607.14012
[8]	B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129-162. · Zbl 0386.14009
[9]	K. Nagao, An example of elliptic curve over \(\mathbf{Q}\) with rank \(\geq 20\), Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 8, 291-293. · Zbl 0794.14014
[10]	F. Najman, Torsion of elliptic curves over quadratic cyclotomic fields, Math. J. Okayama Univ. (to appear). · Zbl 1222.11076
[11]	F. Najman, Complete classification of torsion of elliptic curves over quadratic cyclotomic fields, J. Number Theory. (to appear). · Zbl 1200.11039
[12]	PARI/GP, version 2.3.3, Bordeaux, 2008. http://pari.math.u-bordeaux.fr/.
[13]	F. P. Rabarison, Torsion et rang des courbes elliptiques definies sur les corps de nombres algébriques, Doctorat de Université de Caen, 2008.
[14]	U. Schneiders and H. G. Zimmer, The rank of elliptic curves upon quadratic extension, in Computational number theory (Debrecen, 1989) , 239-260, de Gruyter, Berlin. · Zbl 0743.14023