Class groups of quadratic fields. II

Author:
Duncan A. Buell

Journal:
Math. Comp. **48** (1987), 85-93

MSC:
Primary 11R29; Secondary 11R11, 11Y40

MathSciNet review:
866100

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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**0480327****[2]**Duncan A. Buell,*Class groups of quadratic fields*, Math. Comp.**30**(1976), no. 135, 610–623. MR**0404205**, 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**0439802**, 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**0441914**, 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**, 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**, 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****[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**, 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**, 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**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1987-0866100-9

Article copyright:
© Copyright 1987
American Mathematical Society