Advertisement

Enumeration of Totally Real Number Fields of Bounded Root Discriminant

  • John Voight
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5011)

Abstract

We enumerate all totally real number fields F with root discriminant δ F  ≤ 14. There are 1229 such fields, each with degree Open image in new window .

Keywords

Basic Bound Proper Face Generalize Riemann Hypothesis Precision Loss Lattice Point Counting 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguirre, J., Bilbao, M., Peral, J.C.: The trace of totally positive algebraic integers. Math. Comp. 75(253), 385–393 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Belabas, K.: A fast algorithm to compute cubic fields. Math. Comp. 66(219), 1213–1237 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bhargava, M.: Gauss composition and generalizations. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 1–8. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Cohen, H.: Advanced Topics in Computational Number Theory. In: Graduate Texts in Mathematics, vol. 193, Springer, New York (2000)Google Scholar
  5. 5.
    Cohen, H., Diaz y Diaz, F.: A polynomial reduction algorithm. Sém. Théor. Nombres Bordeaux 3(2), 351–360 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cohen, H., Diaz y Diaz, F., Olivier, M.: A table of totally complex number fields of small discriminants. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 381–391. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Cohen, H., Diaz y Diaz, F., Olivier, M.: Constructing complete tables of quartic fields using Kummer theory. Math. Comp. 72(242), 941–951 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Cohn, H., Elkies, N.: New upper bounds on sphere packings I. Ann. Math. 157, 689–714 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Conway, J.H.,, Sloane, N.J.A.: Sphere packings, lattices and groups. In: Grund. der Math. Wissenschaften, 3rd edn., vol. 290, Springer, New York (1999)Google Scholar
  10. 10.
    De Loera, J., Hemmecke, R., Tauzer, J., Yoshia, R.: Effective lattice point counting in rational convex polytopes. J. Symbolic Comput. 38(4), 1273–1302 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    De Loera, J.: LattE: Lattice point Enumeration (2007), http://www.math.ucdavis.edu/~latte/
  12. 12.
    Ellenberg, J.S., Venkatesh, A.: The number of extensions of a number field with fixed degree and bounded discriminant. Ann. of Math. 163(2), 723–741 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fincke, U., Pohst, M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comp. 44, 170, 463–471 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Hajir, F., Maire, C.: Tamely ramified towers and discriminant bounds for number fields. Compositio Math. 128, 35–53 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Hajir, F., Maire, C.: Tamely ramified towers and discriminant bounds for number fields. II. J. Symbolic Comput. 33, 415–423 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
  17. 17.
    Klüners, J., Malle, G.: A database for number fields, http://www.math.uni-duesseldorf.de/~klueners/minimum/minimum.html
  18. 18.
    Klüners, J., Malle, G.: A database for field extensions of the rationals. LMS J. Comput. Math. 4, 82–196 (2001)Google Scholar
  19. 19.
    Kreuzer, M., Skarke, H.: PALP: A Package for Analyzing Lattice Polytopes (2006), http://hep.itp.tuwien.ac.at/~kreuzer/CY/CYpalp.html
  20. 20.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Long, D.D., Maclachlan, C., Reid, A.W.: Arithmetic Fuchsian groups of genus zero. Pure Appl. Math. Q. 2, 569–599 (2006)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Malle, G.: The totally real primitive number fields of discriminant at most 109. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 114–123. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  23. 23.
    Martin, J.: Improved bounds for discriminants of number fields (submitted)Google Scholar
  24. 24.
    Martinet, J.: Petits discriminants des corps de nombres. In: Journées Arithmétiques (Exeter, 1980). London Math. Soc. Lecture Note Ser., vol. 56, pp. 151–193. Cambridge Univ. Press, Cambridge (1982)Google Scholar
  25. 25.
    Martinet, J.: Tours de corps de classes et estimations de discriminants. Invent. Math. 44, 65–73 (1978)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Martinet, J.: Methodes geométriques dans la recherche des petitis discriminants. In: Sem. Théor. des Nombres (Paris 1983–84), pp. 147–179. Birkhäuser, Boston (1985)Google Scholar
  27. 27.
    Odlyzko, A.M.: Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results. Sém. Théor. Nombres Bordeaux 2(2), 119–141 (1990)MathSciNetzbMATHGoogle Scholar
  28. 28.
    The PARI Group: PARI/GP (version 2.3.2), Bordeaux (2006), http://pari.math.u-bordeaux.fr/.
  29. 29.
    Pohst, M.: On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields. J. Number Theory 14, 99–117 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Roblot, X.-F.: Totally real fields with small root discriminant, http://math.univ-lyon1.fr/~roblot/tables.html.
  31. 31.
    Stein, W.: SAGE Mathematics Software (version 2.8.12). The SAGE Group (2007), http://www.sagemath.org/
  32. 32.
    Smyth, C.J.: The mean values of totally real algebraic integers. Math. Comp. 42, 663–681 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Takeuchi, K.: Totally real algebraic number fields of degree 9 with small discriminant. Saitama Math. J. 17, 63–85 (1999)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Verschelde, J.: Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software 25, 251–276 (1999)CrossRefzbMATHGoogle Scholar
  35. 35.
    Voight, J.: Totally real number fields, http://www.cems.uvm.edu/~voight/nf-tables/

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • John Voight
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlington 

Personalised recommendations