Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 161–171 | Cite as

Torsion Groups of a Family of Elliptic Curves Over Number Fields

  • Pallab Kanti DeyEmail author


We compute the torsion group explicitly over quadratic fields and number fields of degree coprime to 6 for a family of elliptic curves of the form E: y2 = x3 + c, where c is an integer.


torsion group elliptic curve number field 

MSC 2010

14H52 11R04 


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  1. [1]
    R. Ayoub: An Introduction to the Analytic Theory of Numbers. Mathematical Surveys 10, American Mathematical Society, Providence, 1963.zbMATHGoogle Scholar
  2. [2]
    A. Bourdon, P. L. Clark, J. Stankewicz: Torsion points on CM elliptic curves over real number fields, Trans. Am. Math. Soc. 369 (1996), 8457–8496.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    P. K. Dey: Elliptic curves with rank 0 over number fields, Funct. Approximatio, Comment. Math. 56 (2017), 25–37.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    E. González-Jiménez: Complete classification of the torsion structures of rational elliptic curves over quintic number fields, J. Algebra. 478 (2017), 484–505.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Jeon, C. H. Kim, E. Park: On the torsion of elliptic curves over quartic number fields, J. Lond. Math. Soc., II. Ser. 74 (2006), 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Kamienny: Torsion points on elliptic curves and q-coefficients of modular forms, Invent. Math. 109 (1992), 221–229.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. A. Kenku, F. Momose: Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125–149.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. W. Knapp: Elliptic Curves. Mathematical Notes (Princeton) 40, Princeton University Press, Princeton, 1992.Google Scholar
  9. [9]
    B. Mazur: Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes ´Etud. Sci. 47 (1977), 33–186.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    F. Najman: Complete classification of torsion of elliptic curves over quadratic cyclotomic fields, J. Number Theory. 130 (2010), 1964–1968.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. Najman: Torsion of elliptic curves over quadratic cyclotomic fields, Math. J. Okayama Univ. 53 (2011), 75–82.MathSciNetzbMATHGoogle Scholar
  12. [12]
    F. Najman: Torsion of rational elliptic curves over cubic fields and sporadic points on X1(n), Math. Res. Lett. 23 (2016), 245–272.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. D. Olson: Points of finite order on elliptic curves with complex multiplication, Manuscr. Math. 14 (1974), 195–205.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    L. C. Washington: Elliptic Curves. Number Theory and Cryptography. Chapman and Hall/CRC, Boca Raton, 2008.CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.Harish-Chandra Research InstituteJhunsi, AllahabadIndia

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