Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Elliptic curves over the rationals with bad reduction at only one prime

Authors: Bas Edixhoven, Arnold de Groot and Jaap Top
Journal: Math. Comp. 54 (1990), 413-419
MSC: Primary 11G05; Secondary 11D25
MathSciNet review: 995209
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime.

References [Enhancements On Off] (What's this?)

  • [1] B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Lecture Notes in Mathematics, Vol. 476, Springer-Verlag, Berlin-New York, 1975. MR 0376533
  • [2] Armand Brumer and Kenneth Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), no. 4, 715–743. MR 0457453
  • [3] S. J. Edixhoven, Stable models of modular curves and applications, Ph.D. Thesis, Math. Inst., Univ. of Utrecht, 1989.
  • [4] Benedict H. Gross, Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics, vol. 776, Springer, Berlin, 1980. With an appendix by B. Mazur. MR 563921
  • [5] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230,
  • [6] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986) Nagoya Univ., Nagoya, 1986, pp. 217–242 (French). MR 891898
  • [7] T. Nagel, Über die rationalen Punkte auf einigen kubischen Kurven, Tôhoku Math. J. 24 (1925), 48-53.
  • [8] M. T. Nagell, L'analyse indéterminée de degré supérieur, Mémorial des Sciences Mathématiques 39 (1929).
  • [9] Bennett Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2) 10 (1975), 367–378. MR 0371904,
  • [10] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11G05, 11D25

Retrieve articles in all journals with MSC: 11G05, 11D25

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society