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Number Fields with Discriminant ±2a3b and Galois Group An or Sn

  • Gunter Malle (a1) and David P. Roberts (a2)
Abstract

The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, they obtain number fields ramified at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 11, 12, 17, 18, 25, 28, 30, and 33.

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References
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1.Birch, B., ‘Noncongruence subgroups, covers and drawings’, The Grothendieck theory of dessins d'enfants, London Math. Soc. Lecture Note Ser. 200 (ed. Schneps, L., Cambridge Univ. Press, 1994) 2546.
2.Jones, J. W., Ongoing computer searches for number fields; updated results periodically posted at http://math.la.asu.edu/~jj/numberfields/.
3.Jones, J. W. and Roberts, D. P., ‘Sextic number fields with discriminant —j2a3b, Number Theory (Ottawa, 1996), CRM Proc. Lecture Notes 19 (ed. Gupta, R. and Williams, K., Amer. Math. Soc, Providence, RI, 1999) 141172.
4.Jones, J. W. and Roberts, D. P., ‘Septic number fields with discriminant ±2a 3b, Math. Comp. 72 (2003) 244, 19751985.
5.Katz, N. M., Rigid local systems, Ann. of Math. Stud. 139 (Princeton Univ. Press, Princeton, NJ, 1996).
6.Kraft, H., ‘A result of Hermite and equations of degree 5 and 6’, preprint, http://xxx.lanl.gov/abs/math.AC/0403323+.
7.Malle, G., ‘Fields of definition of some three-point ramified field extensions’, The Grothendieck theory of dessins d'enfants, London Math Soc Lecture Note Series 200 (ed. Schneps, L., Cambridge Univ. Press, 1994) 147168.
8.Malle, G. and Matzat, B. H., Inverse Galois theory, Monogr. Math. (Springer, Berlin, 1999).
9. PARI/GP, Version 2.1.5, Bordeaux, 2004, http://pari.math.u-bordeaux.fr/.
10.Roberts, D. P., ‘An ABC construction of number fields’, Number Theory (Montréal, 2002), CRM Proc. Lecture Notes 36 (ed. Kisilevsky, H. and Goren, E. Z., Amer. Math. Soc, Providence, RI, 2004) 237267.
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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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