Explicit construction of the hilbert class fields of imaginary quadratic fields with class numbers 7 and 11

Abstract

Motivated by a constructive realization of dihedral groups of prime degree as Galois group over the field of rational numbers, we give an explicit construction of the Hilbert class fields of some imaginary quadratic fields with class numbers 7 and 11. This was done by explicitly evaluating the elliptic modular j -invariant at each representative of the ideal class of an imaginary quadratic field, and then forming the class equation. In an appendix we determine the explicit form of the modular equation of order 5 and 7. All computations were carried out on the computer algebra system MACSYMA. We also point out what computational difficulties were encountered and how we resolved them. In this paper we summarize the progress made so far on using the Computer Algebra System MACSYMA [8] to explicitly calculate the defining equations of the Hilbert class fields of imaginary quadratic fields with prime class number. Our motivation for undertaking this investigation is to construct rational polynomials with a given finite Galois group. The groups we try to realize here are the dihedral groups Dp for primes p . These groups are non-abelian groups of order 2p and are generated by two elements σ = (1 2 3. . .p ) and τ = (1)(2 p )(3 p −1). . .( 2 p +1 _ ___ 2 p +3 _ ___) _ ______________ * This research was partially supported by the National Science and Engineering Research Council of Canada under grant 3-643-126-90 (the first author) and under grant 3-661-114-30 (the second author). AMS (MOS) Mathematics Subject Classifications (1980); Firstly: 10-04, 14-04; Secondly: 12A55, 12A65, 10D25.