Skip to main content
SpringerLink
Log in

Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

In this paper, we present several methods for the construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp., 60 (1993), 399–405.

    Article  MathSciNet  MATH  Google Scholar 

  2. B. J. Birch, Elliptic curves and modular functions, Symposia Mathematica, Vol. IV, Academic Press, London, 1970, 27–32.

    Google Scholar 

  3. J. Bosman, P. Bruin, A. Dujella and F. Najman, Ranks of elliptic curves with prescribed torsion over number fields, preprint.

  4. É. Brier and C. Clavier, New families of ECM curves for Cunningham Numbers, Proceedings of ANTS IX., LNCS 6197, Springer, Heidelberg, 2010, 96–109.

    Google Scholar 

  5. A. Dujella, High rank elliptic curves with prescribed torsion, http://web.math.hr/_duje/tors/tors.html.

  6. A. Dujella, Infinite families of elliptic curves with high rank and prescribed torsion, http://web.math.hr/_duje/tors/generic.html.

  7. A. Dujella and M. Jukić Bokun, On the rank of elliptic curves over Q(i) with torsion group Z/4Z × Z/4Z, Proc. Japan Acad. Ser. A Math. Sci., 86 (2010), 93–96.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Jeon, C. H. Kim and Y. Lee, Families of elliptic curves over quartic number fields with prescribed torsion subgroups, Math. Comp., 80 (2011), 2395–2410.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Jukić Bokun, On the rank of elliptic curves over \(Q\left( {\sqrt { - 3} } \right)\) with torsion groups Z/3Z×Z/3Z and Z/3Z×Z/6Z, Proc. Japan Acad. Ser. A Math. Sci., 87 (2011), 61–64.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. W. Lenstra Jr., Factoring integers with elliptic curves, Ann. of Math., 126 (1987), 649–673.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp., 48 (1987), 243–264.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. P. Rabarison, Torsion et rang des courbes elliptiques definies sur les corps de nombres algébriques, Doctorat de Université de Caen, 2008.

  13. J. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 2009.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Andrej Dujella & Filip Najman

Authors
  1. Andrej Dujella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filip Najman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrej Dujella.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dujella, A., Najman, F. Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization. Period Math Hung 65, 193–203 (2012). https://doi.org/10.1007/s10998-012-8257-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-012-8257-7

Mathematics subject classification numbers

Key words and phrases

Search

Navigation