Class groups of quadratic fields. II
Author:
Duncan A. Buell
Journal:
Math. Comp. 48 (1987), 85-93
MSC:
Primary 11R29; Secondary 11R11, 11Y40
DOI:
https://doi.org/10.1090/S0025-5718-1987-0866100-9
MathSciNet review:
866100
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A computation has been made of the noncyclic class groups of imaginary quadratic fields for even and odd discriminants
from 0 to
. Among the results are that 95% of the class groups are cyclic, and that
and
are the first discriminants of imaginary quadratic fields for which the class group has rank three in the 5-Sylow subgroup. The latter was known to be of rank three; this computation demonstrates that it is the first odd discriminant of 5-rank three or more.
- [1] Josef Blass and Ray Steiner, On the equation 𝑦²+𝑘=𝑥⁷, Utilitas Math. 13 (1978), 293–297. MR 480327
- [2] Duncan A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), no. 135, 610–623. MR 404205, https://doi.org/10.1090/S0025-5718-1976-0404205-X
- [3] Duncan A. Buell, Small class numbers and extreme values of 𝐿-functions of quadratic fields, Math. Comp. 31 (1977), no. 139, 786–796. MR 439802, https://doi.org/10.1090/S0025-5718-1977-0439802-X
- [4] D. A. Buell, H. C. Williams, and K. S. Williams, On the imaginary bicyclic biquadratic fields with class-number 2, Math. Comp. 31 (1977), no. 140, 1034–1042. MR 441914, https://doi.org/10.1090/S0025-5718-1977-0441914-1
- [5] Duncan A. Buell, The expectation of success using a Monte Carlo factoring method—some statistics on quadratic class numbers, Math. Comp. 43 (1984), no. 167, 313–327. MR 744940, https://doi.org/10.1090/S0025-5718-1984-0744940-1
- [6] H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, https://doi.org/10.1007/BFb0099440
- [7] Franz-Peter Heider and Bodo Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 (1982), 1–25 (German). MR 671319, https://doi.org/10.1515/crll.1982.336.1
- [8] C.-P. Schnorr and H. W. Lenstra Jr., A Monte Carlo factoring algorithm with linear storage, Math. Comp. 43 (1984), no. 167, 289–311. MR 744939, https://doi.org/10.1090/S0025-5718-1984-0744939-5
- [9] R. J. Schoof, Class groups of complex quadratic fields, Math. Comp. 41 (1983), no. 163, 295–302. MR 701640, https://doi.org/10.1090/S0025-5718-1983-0701640-0
- [10] Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385
Retrieve articles in Mathematics of Computation with MSC: 11R29, 11R11, 11Y40
Retrieve articles in all journals with MSC: 11R29, 11R11, 11Y40
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1987-0866100-9
Article copyright:
© Copyright 1987
American Mathematical Society