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Trinomials ax7 + bx + c and ax8 + bx + c with Galois Groups of Order 168 and 8 · 168

  • Nils Bruinco
  • Noam D. Elkies
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2369)

Abstract

We obtain the curves of genus 2 parametrizing trinomials ax 7 + bx + c whose Galois group is contained in the simple group G 168 of order 168, and trinomials ax 8 + bx + c whose Galois group is contained in G 1344 = (Z/2)3G 168. In the degree-7 case, we find rational points of small height on this curve over Q and recover four inequivalent trinomials: the known x 7 − 7x + 3 (Trinks-Matzat) and x 7 − 154x + 99 (Erbach-Fischer-McKay), and two new examples, 372 x 7 − 28x + 9 and 4992 x 7 − 23956x + 34113. We prove that there are no further rational points, and thus that every trinomial ax 7 + bx + c with Galois group ⊆ G 168 over Q is equivalent to one of those four examples. In the degree-8 case, we again find some rational points of small height and compute the associated trinomials. This time all our examples are new: x 8 + 16x + 28, x 8 + 576x + 1008, and 19453x 8 + 19x + 2, each with Galois group G 1344; and x 8 + 324i + 567, with Galois group G 168 acting transitively on the eight roots. We conjecture, but do not prove, that there are no further rational points, and thus that every trinomial ax 8 + bx + c with Galois group ⊆ G 1344 over Q is equivalent to one of those four examples.

Keywords

Rational Point Elliptic Curf Galois Group Abelian Variety Hyperelliptic Curve 
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.

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References

  1. BCP.
    Wieb Bosma, John Cannon, and Catherine Playoust: The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3–4):235–265, 1997. Computational algebra and number theory (London, 1993).zbMATHCrossRefMathSciNetGoogle Scholar
  2. B1.
    Nils Bruin: Chabauty Methods and Covering Techniques applied to Generalised Fermat Equations. PhD thesis, Universiteit Leiden, 1999.Google Scholar
  3. B2.
    Nils Bruin: Chabauty methods using elliptic curves. Technical Report W99-14, Leiden, 1999.Google Scholar
  4. B3.
    Nils Bruin: Chabauty methods using covers on curves of genus 2. Technical Report W99-15, Leiden, 1999.Google Scholar
  5. B4.
    Nils Bruin: On powers as sums of two cubes in Wieb Bosma (ed), Algorithmic Number Theory 4th International Symposium ANTS-IV Leiden, The Netherlands, July 2–7, 2000 Proceedings. Springer LNCS 1838.Google Scholar
  6. B5.
    Nils Bruin: Algae, a program for 2-Selmer groups of elliptic curves over number fields. see http://www.cecm.sfu.ca/~bruin/ell.shar.
  7. BE.
    Nils Bruin and Noam Elkies: Transcript of computations. available from http://www.math.harvard.edu/~elkies/trinomials_bruin.g, 2002.
  8. BF.
    Nils Bruin and E. Victor Flynn: Towers of 2-covers of hyperelliptic curves. PIMS-01-12, http://www.pims.math.ca/publications/#preprints, 2001.
  9. Ca.
    J. W. S. Cassels: Lectures on Elliptic Curves. LMS-ST 24. University Press, Cambridge, 1991.Google Scholar
  10. Co.
    Henri Cohen: A Course in Computational Algebraic Number Theory, GTM 138 Springer, Berlin-Heidelberg-New York, 1993.Google Scholar
  11. EFM.
    Erbach, D.W., Fischer J., and McKay, J.: Polynomials with Galois group PSL(2,7), J. Number Theory 11 (1979), 69–75.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Fa1.
    Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.CrossRefMathSciNetzbMATHGoogle Scholar
  13. Fa2.
    Faltings, G.: Diophantine approximation on Abelian varieties, Annals of Math. (2) 133 (1991) #3, 549–576.CrossRefMathSciNetGoogle Scholar
  14. Fl.
    E.V. Flynn: A flexible method for applying Chabauty’s theorem. Compositio Mathematica, 105:79–94, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  15. H.
    Florian Heß: Zur Klassengruppenberechnung in algebraischen Zahlkörpern. Diplomarbeit, Technische Universität Berlin, 1996. http://www.math.tu-berlin.de/~kant/publications/diplom/hess.ps.gz.
  16. K.
    M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, K. Wildanger: KANT V4, J. of Symbolic Comput., 3–4:267–283, 1997.CrossRefGoogle Scholar
  17. M.
    Matzat, B.H.: Konstruktive Galoistheorie., Springer Lect. Notes Math. 1284, 1987.Google Scholar
  18. Si.
    Joseph H. Silverman: The Arithmetic of Elliptic Curves. GTM 106. Springer-Verlag, 1986.Google Scholar
  19. St.
    Michael Stoll: Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith., 98(3):245–277, 2001.zbMATHMathSciNetCrossRefGoogle Scholar
  20. T.
    Trinks, W.: Ein Beispiel eines Zahlkörpers mit der Galoisgruppe iPSL(3,2) über Q, manuscript, Univ. Karlsruhe, Karlsruhe, 1968.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Nils Bruinco
    • 1
  • Noam D. Elkies
    • 2
  1. 1.Pacific Institute for Mathematical Sciences (SFU, UBC). Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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