Skip to main content

Elliptic Curves with Good Reduction Outside of the First Six Primes

  • Conference paper
  • First Online:
Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))


We present a database of rational elliptic curves, up to -isomorphism, with good reduction outside {2, 3, 5, 7, 11, 13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

    However, work in progress by the second author gives the same set of curves using a different method.

  2. 2.

    Allan MacLeod has communicated to us the Mordell–Weil bases of 10 more of these curves, using his own implementations of similar techniques to those outlined above. Nevertheless a fast general method to find bases for all of our curves remains elusive.


  1. M. K. Agrawal, J. H. Coates, D. C. Hunt, and A. J. van der Poorten. Elliptic curves of conductor 11. Math. Comp., 35(151):991–1002, 1980.

    Article  MathSciNet  Google Scholar 

  2. Alan Baker. Experiments on the abc-conjecture. Publ. Math. Debrecen, 65(3–4):253–260, 2004.

    MathSciNet  MATH  Google Scholar 

  3. Michael A. Bennett, Adela Gherga, and Andrew Rechnitzer. Computing elliptic curves over . Math. Comp., 88(317):1341–1390, 2019.

    Google Scholar 

  4. Michael A. Bennett and Andrew Rechnitzer. Computing elliptic curves over \(\mathbb Q\): bad reduction at one prime. In Recent progress and modern challenges in applied mathematics, modeling and computational science, volume 79 of Fields Inst. Commun., pages 387–415. Springer, New York, 2017.

    Google Scholar 

  5. Nicolas Billerey. On some remarkable congruences between two elliptic curves, 2016. arXiv:1605.09205.

  6. Bryan J. Birch and Willem Kuyk, editors. Modular functions of one variable. IV. Lecture Notes in Mathematics, Vol. 476. Springer-Verlag, Berlin-New York, 1975.

    Google Scholar 

  7. Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14(4):843–939, 2001.

    Article  MathSciNet  Google Scholar 

  8. Armand Brumer and Oisín McGuinness. The behavior of the Mordell-Weil group of elliptic curves. Bull. Am. Math. Soc., 23(2):375–382, 1990.

    Article  MathSciNet  Google Scholar 

  9. Francis Coghlan. Elliptic Curves with Conductor 2m3n. Ph.D. thesis, Manchester, England, 1967.

    Google Scholar 

  10. Henri Cohen. Number Theory: Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics, Number Theory. Springer-Verlag, New York, 2007.

    Google Scholar 

  11. John E. Cremona. Elliptic curve data.

  12. John E. Cremona. Algorithms for modular elliptic curves. Cambridge University Press, second edition, 1997.

    Google Scholar 

  13. John E. Cremona and Mark P. Lingham. Finding all elliptic curves with good reduction outside a given set of primes. Experiment. Math., 16(3):303–312, 2007.

    Article  MathSciNet  Google Scholar 

  14. Bart de Smit. ABC@Home. ∼desmit/abc/.

  15. Benjamin M. M. de Weger. Solving exponential Diophantine equations using lattice basis reduction algorithms. J. Number Theory, 26(3):325–367, 1987.

    Google Scholar 

  16. The Sage Developers. SageMath (Version 9.0), 2020.

  17. Tom Fisher. Finding rational points on elliptic curves using 6-descent and 12-descent. arXiv:0711.3774, 2007.

  18. Tom Fisher. On families of 13-congruent elliptic curves, 2019. arXiv:1912.10777.

    Google Scholar 

  19. Rudolf Fueter. Ueber kubische diophantische Gleichungen. Comment. Math. Helv., 2(1):69–89, 1930.

    Article  MathSciNet  Google Scholar 

  20. Benedict H. Gross and Don B. Zagier. Heegner points and derivatives of l-series. Invent. Math., 84(2):225–320, 1986.

    Article  MathSciNet  Google Scholar 

  21. Jamie Weigandt ( Rational points on y 2 = x 3 − 860695. MathOverflow. (version: 2012-09-25).

  22. Rafael von Känel and Benjamin Matschke. Solving S-unit, Mordell, Thue, Thue–Mahler and generalized Ramanujan–Nagell equations via Shimura–Taniyama conjecture. arXiv:1605.06079,, 2016.

  23. Kiran S. Kedlaya and Andrew V. Sutherland. Computing L-Series of Hyperelliptic Curves. In Alfred J. van der Poorten and Andreas Stein, editors, Algorithmic Number Theory, Lecture Notes in Computer Science, pages 312–326, Berlin, Heidelberg, 2008. Springer.

    Google Scholar 

  24. Victor A. Kolyvagin. Euler systems. In The Grothendieck Festschrift, pages 435–483. Springer, 2007.

    Google Scholar 

  25. Angelos Koutsianas. Computing all elliptic curves over an arbitrary number field with prescribed primes of bad reduction. Exp. Math., 28(1):1–15, 2019.

    Article  MathSciNet  Google Scholar 

  26. The LMFDB Collaboration. The L-functions and modular forms database., 2020.

  27. Kurt Mahler. Zur Approximation algebraischer Zahlen. I. Math. Ann., 107(1):691–730, 1933.

    Google Scholar 

  28. J. Steffen Müller and Michael Stoll. Computing Canonical Heights on Elliptic Curves in Quasi-Linear Time. LMS J. Comput. Math., 19(A):391–405, 2016. arXiv:1509.08748.

  29. Andrew P. Ogg. Abelian curves of 2-power conductor. Math. Proc. Camb. Philos. Soc., 62(2):143–148, 1966.

    Article  MathSciNet  Google Scholar 

  30. Wolfgang M. Schmidt. Diophantine approximations and Diophantine equations, volume 1467 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991.

    Google Scholar 

  31. Igor R. Shafarevich. Algebraic number fields. In Proceedings of an International Congress on Mathematics, Stockholm, pages 163–176, 1962.

    Google Scholar 

  32. Carl L. Siegel. Über einige Anwendungen diophantischer Approximationen. Abh. Preuß. Akad. Wiss., Phys.-Math. Kl., 1929(1):70, 1929.

    Google Scholar 

  33. Joseph H. Silverman. Arithmetic of elliptic curves., volume 106 of Graduate Texts in Mathematics. Springer, 1986.

    Google Scholar 

  34. Joseph H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics. Springer, 1994.

    Book  Google Scholar 

  35. Denis Simon. Computing the rank of elliptic curves over number fields. LMS J. Comput. Math., 5:7–17, 2002. Program ellQ:

  36. William A. Stein and Mark Watkins. A Database of Elliptic Curves — First Report. In Claus Fieker and David R. Kohel, editors, Algorithmic Number Theory, Lecture Notes in Computer Science, pages 267–275, Berlin, Heidelberg, 2002. Springer.

    Google Scholar 

  37. Nelson M. Stephens. The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank. Ph.D. Thesis, Manchester, 1965.

    Google Scholar 

  38. Michael Stoll. Documentation for the ratpoints program. arXiv:math/0803.3165, 2014.

  39. The PARI Group, Univ. Bordeaux. PARI/GP version 2.11.2, 2019.

  40. Dave J. Tingley. Elliptic curves uniformized by modular functions. Ph.D. thesis, University of Oxford, 1975.

    Google Scholar 

  41. Mark Watkins. Searching for points using the Elkies ANTS-IV algorithm. ∼watkins/papers/

  42. Mark Watkins. Some Remarks on Heegner Point Computations. arXiv:math/0506325, 2006.

  43. Thomas Womack. Explicit Descent on Elliptic Curves. PhD thesis, University of Nottingham, July 2003.

    Google Scholar 

Download references


It is our pleasure to thank Edgar Costa for various useful comments and for computing the analytic ranks of all curves in our database, as well as the leading coefficients and root numbers of the associated L-series. They are available from the same GitHub repository. We would also like to thank the anonymous referees for their helpful comments on earlier versions of this article.

The first author was supported by Simons Foundation grant #550023 and a Hariri Institute Graduate Student Fellowship.

The second author was partially supported by Excellence Initiative of Université de Bordeaux (IdEx project DiGeMANT), and by Simons Foundation grant #550023.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Benjamin Matschke .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Best, A.J., Matschke, B. (2021). Elliptic Curves with Good Reduction Outside of the First Six Primes. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham.

Download citation

Publish with us

Policies and ethics