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Computing Elliptic Curves over \(\mathbb{Q}\): Bad Reduction at One Prime

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Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Part of the book series: Fields Institute Communications ((FIC,volume 79))

Abstract

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.

The authors were supported in part by grants from NSERC.

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Notes

  1. 1.

    Using the standard unix sort command and taking advantage of multiple cores.

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.

    MathSciNet  MATH  Google Scholar 

  2. K. Belabas. A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213–1237.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Google Scholar 

  4. M. A Bennett and A. Ghadermarzi, Mordell’s equation: a classical approach, L.M.S. J. Comput. Math. 18 (2015), 633–646.

    Google Scholar 

  5. M. A. Bennett and A. Rechnitzer, Computing elliptic curves over \(\mathbb{Q}\), submitted for publication.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Brumer and O. McGuinness, The behaviour of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. 23 (1990), 375–382.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Brumer and J. H. Silverman, The number of elliptic curves over \(\mathbb{Q}\) with conductorN, Manuscripta Math. 91 (1996), 95–102.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Google Scholar 

  13. F. Coghlan, Elliptic Curves with Conductor 2m3n, Ph.D. thesis, Manchester, England, 1967.

    Google Scholar 

  14. J. Cremona, Elliptic curve tables, http://johncremona.github.io/ecdata/

  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

  16. J. Cremona, Reduction of binary cubic and quartic forms, LMS J. Comput. Math. 4 (1999), 64–94.

    MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Davenport, The reduction of a binary cubic form. I., J. London Math. Soc. 20 (1945), 14–22.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Davenport, The reduction of a binary cubic form. II., J. London Math. Soc. 20 (1945), 139–147.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Google Scholar 

  24. T. Hadano, On the conductor of an elliptic curve with a rational point of order 2, Nagoya Math. J. 53 (1974), 199–210.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  28. C. Hermite, Sur la réduction des formes cubiques à deux indéxtermineées, C. R. Acad. Sci. Paris 48 (1859), 351–357.

    Google Scholar 

  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.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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).

    Google Scholar 

  32. L. J. Mordell, The diophantine equationy 2 − k = x 3, Proc. London. Math. Soc. (2) 13 (1913), 60–80.

    Google Scholar 

  33. L. J. Mordell, Diophantine Equations, Academic Press, London, 1969.

    MATH  Google Scholar 

  34. T. Nagell, Introduction to Number Theory, New York, 1951.

    Google Scholar 

  35. O. Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten II, Math. Nach. 56 (1973), 269–280.

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Papadopolous, Sur la classification de Néron des courbes elliptiques en caractéristique résseulé 2 et 3, J. Number Th. 44 (1993), 119–152.

    Article  Google Scholar 

  37. The PARI Group, Bordeaux. PARI/GP version 2.7.1, 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.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Pethő, On the representation of 1 by binary cubic forms of positive discriminant, Number Theory, Ulm 1987 (Springer LNM 1380), 185–196.

    Google Scholar 

  40. M. O. Rabin, Probabilistic algorithm for testing primality, J. Number Th. 12 (1980) 128–138.

    Article  MathSciNet  MATH  Google Scholar 

  41. B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. 10 (1975), 367–378.

    Article  MathSciNet  MATH  Google Scholar 

  42. I. R. Shafarevich, Algebraic number theory, Proc. Internat. Congr. Mathematicians, Stockholm, Inst. Mittag-Leffler, Djursholm (1962), 163–176.

    Google Scholar 

  43. J. P. Sorenson and J. Webster, Strong Pseudoprimes to Twelve Prime Bases, arXiv preprint arXiv:1509.00864.

    Google Scholar 

  44. V. G. Sprindzuk, Classical Diophantine Equations, Springer-Verlag, Berlin, 1993.

    Book  MATH  Google Scholar 

  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.

    Google Scholar 

  46. N. Tzanakis and B. M. M. de Weger, On the practical solutions of the Thue equation, J. Number Theory 31 (1989), 99–132.

    Article  MathSciNet  MATH  Google Scholar 

  47. N. Tzanakis and B. M. M. de Weger, Solving a specific Thue-Mahler equation, Math. Comp. 57 (1991) 799–815.

    Article  MathSciNet  MATH  Google Scholar 

  48. N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Compositio Math., 84 (1992), 223–288.

    MathSciNet  MATH  Google Scholar 

  49. B. M. M. de Weger, Algorithms for diophantine equations, CWI-Tract No. 65, Centre for Mathematics and Computer Science, Amsterdam, 1989.

    Google Scholar 

  50. B. M. M. de Weger, The weighted sum of twoS-units being a square, Indag. Mathem. 1 (1990), 243–262.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

    Michael A. Bennett & Andrew Rechnitzer

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  1. Michael A. Bennett
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  2. Andrew Rechnitzer
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Correspondence to Michael A. Bennett .

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Editors and Affiliations

  1. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roderick Melnik

  2. Department of Mathematics and MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, Ontario, Canada

    Roman Makarov

  3. Departement de Mathematiques, Universite de Montreal, Montreal, Québec, Canada

    Jacques Belair

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Bennett, M.A., Rechnitzer, A. (2017). Computing Elliptic Curves over \(\mathbb{Q}\): Bad Reduction at One Prime. In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6969-2_13

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