×

Computing elliptic curves over \(\mathbb{Q}\): bad reduction at one prime. (English) Zbl 1410.11045

Melnik, Roderick (ed.) et al., Recent progress and modern challenges in applied mathematics, modeling and computational science. Toronto: The Fields Institute for Research in the Mathematical Sciences; New York, NY: Springer. Fields Inst. Commun. 79, 387-415 (2017).
Summary: We discuss a new algorithm for finding all elliptic curves over \(\mathbb{Q}\) with a given conductor. Though based on (very) classical ideas, this approach appears to be computationally quite efficient. We provide details of the output from the algorithm in case of conductor \(p\) or \(p^{2}\), for \(p\) prime, with comparisons to existing data.
For the entire collection see [Zbl 1381.00028].

MSC:

11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11D59 Thue-Mahler equations
11E76 Forms of degree higher than two
11Y50 Computer solution of Diophantine equations
11Y65 Continued fraction calculations (number-theoretic aspects)
14H52 Elliptic curves

Software:

ecdata; CLN; Magma; PARI/GP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. K. Agrawal, J. H. Coates, D. C. Hunt and A. J. van der Poorten, Elliptic curves of conductor 11, Math. Comp. 35 (1980), 991-1002. · Zbl 0475.14031
[2] K. Belabas. A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237. · Zbl 0882.11070
[3] K. Belabas and H. Cohen, Binary cubic forms and cubic number fields, Organic Mathematics (Burnaby, BC, 1995), 175-204. CMS Conf. Proc., 20 Amer. Math. Soc. 1997. · Zbl 0904.11044
[4] M. A Bennett and A. Ghadermarzi, Mordell’s equation: a classical approach, L.M.S. J. Comput. Math. 18 (2015), 633-646. · Zbl 1371.11077
[5] M. A. Bennett and A. Rechnitzer, Computing elliptic curves over \(\mathbb{Q} \), submitted for publication. · Zbl 1410.11045
[6] W. E. H. Berwick and G. B. Mathews, On the reduction of arithmetical binary cubic forms which have a negative determinant, Proc. London Math. Soc. (2) 10 (1911), 43-53. · JFM 42.0243.03
[7] B. J. Birch and W. Kuyk (Eds.), Modular Functions of One Variable IV, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin and New York, 1975.
[8] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. Computational algebra and number theory (London, 1993). · Zbl 0898.68039
[9] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the Modularity of Elliptic Curves over \(\mathbb{Q} \): Wild 3-adic Exercises, J. Amer. Math. Soc. 14 (2001), 843-939. · Zbl 0982.11033
[10] A. Brumer and O. McGuinness, The behaviour of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. 23 (1990), 375-382. · Zbl 0741.14010
[11] A. Brumer and J. H. Silverman, The number of elliptic curves over \(\mathbb{Q}\) with conductorN, Manuscripta Math. 91 (1996), 95-102. · Zbl 0868.11029
[12] J. Coates, An effectivep-adic analogue of a theorem of Thue. III. The diophantine equationy^2 = x^3 + k, Acta Arith. 16 (1969/1970), 425-435.
[13] F. Coghlan, Elliptic Curves with Conductor 2^m3^n, Ph.D. thesis, Manchester, England, 1967.
[14] J. Cremona, Elliptic curve tables, http://johncremona.github.io/ecdata/ · JFM 01.0253.01
[15] J. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997. Available online at http://homepages.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html · Zbl 0872.14041
[16] J. Cremona, Reduction of binary cubic and quartic forms, LMS J. Comput. Math. 4 (1999), 64-94. · Zbl 0927.11020
[17] J. Cremona and M. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), 303-312. · Zbl 1149.11028
[18] H. Davenport, The reduction of a binary cubic form. I., J. London Math. Soc. 20 (1945), 14-22. · Zbl 0060.12006
[19] H. Davenport, The reduction of a binary cubic form. II., J. London Math. Soc. 20 (1945), 139-147. · Zbl 0060.12006
[20] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II., Proc. Roy. Soc. London Ser. A. 322 (1971), 405-420. · Zbl 0212.08101
[21] B. Edixhoven, A. de Groot and J. Top, Elliptic curves over the rationals with bad reduction at only one prime, Math. Comp. 54 (1990), 413-419. · Zbl 0756.14030
[22] N. D. Elkies, How many elliptic curves can have the same prime conductor?, http://math.harvard.edu/ elkies/condp_banff.pdf
[23] N. D. Elkies, and M. Watkins, Elliptic curves of large rank and small conductor, Algorithmic number theory, 42-56, Lecture Notes in Comput. Sci., 3076, Springer, Berlin, 2004. · Zbl 1125.11327
[24] T. Hadano, On the conductor of an elliptic curve with a rational point of order 2, Nagoya Math. J. 53 (1974), 199-210. · Zbl 0268.14008
[25] B. Haible, CLN, a class library for numbers, available from http://www.ginac.de/CLN/
[26] H. Hasse, Arithmetische Theorie der kubischen Zahlköper auf klassenkörpertheoretischer Grundlage, Math. Z. 31 (1930), 565-582. · JFM 56.0167.02
[27] C. Hermite, Note sur la réduction des formes homogènes à coefficients entiers et à deux indétermineées, J. reine Angew. Math. 36 (1848), 357-364. · ERAM 036.1015cj
[28] C. Hermite, Sur la réduction des formes cubiques à deux indéxtermineées, C. R. Acad. Sci. Paris 48 (1859), 351-357.
[29] G. Julia, Étude sur les formes binaires non quadratiques à indéterminďes rélles ou complexes, Mem. Acad. Sci. l’Inst. France 55 (1917), 1-293.
[30] J.-F. Mestre and J. Oesterlé. Courbes de Weil semi-stables de discriminant une puissancem-ième, J. reine angew. Math 400 (1989), 173-184. · Zbl 0693.14004
[31] G. L. Miller, Riemann’s hypothesis and tests for primality in Proceedings of seventh annual ACM symposium on Theory of computing, 234-239 (1975).
[32] L. J. Mordell, The diophantine equationy^2 − k = x^3, Proc. London. Math. Soc. (2) 13 (1913), 60-80.
[33] L. J. Mordell, Diophantine Equations, Academic Press, London, 1969. · Zbl 0188.34503
[34] T. Nagell, Introduction to Number Theory, New York, 1951. · Zbl 0042.26702
[35] O. Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten II, Math. Nach. 56 (1973), 269-280. · Zbl 0277.14011
[36] I. Papadopolous, Sur la classification de Néron des courbes elliptiques en caractéristique résseulé 2 et 3, J. Number Th. 44 (1993), 119-152. · Zbl 0786.14020
[37] The PARI Group, Bordeaux. PARI/GP version2.7.1, 2014. available at http://pari.math.u-bordeaux.fr/.
[38] A. Pethő, On the resolution of Thue inequalities, J. Symbolic Computation 4 (1987), 103-109. · Zbl 0625.10011
[39] A. Pethő, On the representation of 1 by binary cubic forms of positive discriminant, Number Theory, Ulm 1987 (Springer LNM 1380), 185-196.
[40] M. O. Rabin, Probabilistic algorithm for testing primality, J. Number Th. 12 (1980) 128-138. · Zbl 0426.10006
[41] B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. 10 (1975), 367-378. · Zbl 0324.14005
[42] I. R. Shafarevich, Algebraic number theory, Proc. Internat. Congr. Mathematicians, Stockholm, Inst. Mittag-Leffler, Djursholm (1962), 163-176.
[43] J. P. Sorenson and J. Webster, Strong Pseudoprimes to Twelve Prime Bases, arXiv preprint arXiv:1509.00864. · Zbl 1370.11140
[44] V. G. Sprindzuk, Classical Diophantine Equations, Springer-Verlag, Berlin, 1993.
[45] W. Stein and M. Watkins, A database of elliptic curve – first report, Algorithmic Number Theory (Sydney, 2002), Lecture Notes in Compute. Sci., vol. 2369, Springer, Berlin, 2002, pp. 267-275. · Zbl 1058.11036
[46] N. Tzanakis and B. M. M. de Weger, On the practical solutions of the Thue equation, J. Number Theory 31 (1989), 99-132. · Zbl 0657.10014
[47] N. Tzanakis and B. M. M. de Weger, Solving a specific Thue-Mahler equation, Math. Comp. 57 (1991) 799-815. · Zbl 0738.11029
[48] N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math., 84 (1992), 223-288. · Zbl 0773.11023
[49] B. M. M. de Weger, Algorithms for diophantine equations, CWI-Tract No. 65, Centre for Mathematics and Computer Science, Amsterdam, 1989. · Zbl 0687.10013
[50] B. M. M. de Weger, The weighted sum of twoS-units being a square, Indag. Mathem. 1 (1990), 243-262. · Zbl 0714.11017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.