The Elliptic Curve Database for Conductors to 130000

  • John Cremona
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)


Tabulating elliptic curves has been carried out since the earliest days of machine computation in number theory. After some historical remarks, we report on significant recent progress in enlarging the database of elliptic curves defined over ℚ to include all those of conductor N≤130000. We also give various statistics, summarize the data, describe how it may be obtained and used, and mention some recent work regarding the verification of Manin’s “c=1” conjecture.


Elliptic Curve Elliptic Curf Modular Parametrization Positive Rank Heegner Point 
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  1. 1.
    Agashe, A., Ribet, K., Stein, W.A. (with an appendix by J. E. Cremona): The Manin Constant, Congruence Primes and the Modular Degree (preprint, 2006)Google Scholar
  2. 2.
    Birch, B.J., Kuyk, W. (eds.): Modular Functions of One Variable IV. Lecture Notes in Mathematics, vol. 476. Springer, Heidelberg (1975), zbMATHGoogle Scholar
  3. 3.
    Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on Elliptic Curves I. J. Reine Angew. Math. 212, 7–25 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Brumer, A., McGuinness, O.: The behaviour of the Mordell-Weil group of elliptic curves. Bull. Amer. Math. Soc (N.S.) 23(2), 375–382 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cremona, J.E.: Modular symbols for Γ1(N) and elliptic curves with everywhere good reduction. Math. Proc. Cambridge Philos. Soc. 111(2), 199–218 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cremona, J.E.: Algorithms for modular elliptic curves. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  7. 7.
    Cremona, J.E.: Algorithms for modular elliptic curves, 2nd edn. Cambridge University Press, Cambridge (1997), zbMATHGoogle Scholar
  8. 8.
    Cremona, J.E.: Computing the degree of the modular parametrization of a modular elliptic curve. Math. Comp. 64(211), 1235–1250 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cremona, J.E.: Tables of Elliptic Curves,
  10. 10.
    Cremona, J.E.: mwrank, a program for 2-descent on elliptic curves over ℚ,
  11. 11.
    Cremona, J.E., Siksek, S.: Computing a Lower Bound for the Canonical Height on Elliptic Curves over ℚ. In: ANTS VII Proceedings 2006. Springer, Heidelberg (2006)Google Scholar
  12. 12.
    Duke, W.: Elliptic curves with no exceptional primes. C. R. Acad. Sci. Paris Sér. I Math. 325(8), 813–818 (1997)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Edixhoven, B.: On the Manin constants of modular elliptic curves. In: Arithmetic algebraic geometry (Texel, 1989), Progr. Math., vol. 89, pp. 25–39. Birkhäuser, Boston (1991)Google Scholar
  14. 14.
    LiDIA: A C++ Library For Computational Number Theory,
  15. 15.
    NTL: A Library for doing Number Theory,
  16. 16.
    The MAGMA Computational Algebra System, version 2.12-16,
  17. 17.
    pari/gp, version 2.2.13, Bordeaux (2006),
  18. 18.
    Stein, W.A., Joyner, D.: SAGE: System for Algebra and Geometry Experimentation. Comm. Computer Algebra 39, 61–64 (2005)Google Scholar
  19. 19.
    Stein, W.A.: SAGE, Software for Algebra and Geometry Experimentation,
  20. 20.
    Stein, W.A., Grigorov, G., Jorza, A., Patrikis, S., Patrascu, C.: Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves (preprint, 2006)Google Scholar
  21. 21.
    Stein, W.A., Watkins, M.: A database of elliptic curves—first report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Stevens, G.: Stickelberger elements and modular parametrizations of elliptic curves. Invent. Math. 98(1), 75–106 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Tingley, D.J.: Elliptic curves uniformized by modular functions, University of Oxford D. Phil. thesis (1975)Google Scholar
  24. 24.
    Vélu, J.: Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris Sér. A-B 273, A238–A241 (1971)zbMATHGoogle Scholar
  25. 25.
    Watkins, M.: Computing the modular degree of an elliptic curve. Experiment. Math. 11(4), 487–502 (2002)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • John Cremona
    • 1
  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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