Imaginary quadratic fields with class groups of 3-rank at least 2


In this short note, we construct a family of imaginary quadratic fields whose class group has 3-rank at least 2. We show that, for every large X, there are \(\gg X^{\frac{1}{2}-\epsilon }\) such fields with the discriminant \(-D\) satisfying \(D\le X\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Byeon, D.: Imaginary quadratic fields with non-cyclic ideal class group. Ramanujan J. 11, 159–164 (2006)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Byeon, D., Koh, E.: Real quadratic fields with class number divisible by \(3\). Manuscr. Math. 111(2), 261–263 (2003)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chakraborty, K., Murty, R.: On the number of real quadratic fields with class number divisible by \(3\). Proc. AMS 131(1), 41–44 (2001)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cohen, H., Lenstra, H.W.: Heuristics on Class Groups of Number Fields. Lecture Notes in Mathematics, vol. 1068, pp. 33–62. Springer, Berlin (1984)

  5. 5.

    Craig, A.: A type of class group for imaginary quadratic fields. Acta Arith. 22, 449–459 (1973)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Craig, A.: A construction for irregular discriminants. Osaka Math. J. 14(2), 365–402 (1977); Corrigendum. Osaka Math. J. 15(2), 461 (1978)

  7. 7.

    Heath-Brown, D.R.: Counting Rational Points on Algebraic Varieties, Analytic Number Theory. Lecture Notes in Math., vol. 1891, pp. 51–95. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. 8.

    Heath-Brown, D.R.: Quadratic class numbers divisible by 3. Funct. Approx. Comment. Math. 37, 203–211 (2007); Corrigendum. Funct. Approx. Comment. Math. 43, 227 (2010)

  9. 9.

    Hooley, C.: On the power-free values of polynomials in two variables. In: Analytic Number Theory, pp. 235–266, Cambridge Univ. Press, Cambridge (2009)

  10. 10.

    Kishi, Y., Miyake, K.: Parametrization of the quadratic fields whose class numbers are divisible by three. J. Number Theory 80, 209–217 (2000)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Levin, A., Yan, S., Wiljanen, L.: Quadratic fields with a class group of large \(3\)-rank (2019). arXiv preprint arXiv:1910.12276

  12. 12.

    Luca, F., Pacelli, A.: Class groups of quadratic fields of \(3\)-rank at least \(2\): effective bounds. J. Number Theory 128, 796–804 (2008)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Murty, R.: Exponents of class groups of quadratic fields. In: Topics in Number Theory. University Park, PA (1997), Math. Appl., vol. 467, pp. 229–239. Kluwer Academic, Dordrecht (1999)

  14. 14.

    Nagell, T.: Über die Klassenzahl imaginar quadratischer Zahlkörper. Abh. Math. Semin. Univ. Hambg. 1, 140–150 (1922)

    Article  Google Scholar 

  15. 15.

    Soundararajan, K.: Divisibility of class numbers of imaginary quadratic fields. J. Lond. Math. Soc. 61(2), 681–690 (2000)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Weinberger, P.J.: Real quadratic fields with class numbers divisible by \(n\). J. Number Theory 5, 237–241 (1973)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Yamamoto, Y.: On unramified Galois extensions of quadratic number fields. Osaka J. Math 7, 57–76 (1970)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Yu, G.: A note on the divisibility of class numbers of real quadratic fields. J. Number Theory 97, 35–44 (2002)

    MathSciNet  Article  Google Scholar 

Download references


We thank Professor Michael Filaseta for helpful discussions. We are grateful to the anonymous referee for carefully reviewing this paper, correcting several issues, and suggesting some changes that improved the presentation of this work. We also thank the referee for bringing the preprint [11] to our attention.

Author information



Corresponding author

Correspondence to Gang Yu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yu, G. Imaginary quadratic fields with class groups of 3-rank at least 2. manuscripta math. 163, 569–574 (2020).

Download citation

Mathematics Subject Classification

  • 11R11
  • 11R29