Class groups of the quadratic fields found by F. Diaz y Diaz

Shanks, Daniel

doi:10.1090/S0025-5718-1976-0399039-9

Class groups of the quadratic fields found by F. Diaz y Diaz
HTML articles powered by AMS MathViewer

by Daniel Shanks PDF

Math. Comp. 30 (1976), 173-178 Request permission

Corrigendum: Math. Comp. 30 (1976), 900.
Corrigendum: Math. Comp. 30 (1976), 900.

Abstract:

F. Diaz y Diaz has discovered 99 discriminants d between $- 3321607$ and $- 60638515$ inclusive for which $Q(\sqrt d )$ have a 3-rank ${r_3} = 3$. These 99 imaginary quadratic fields are analyzed here and the class groups are given and discussed for all those of special interest. In 98 cases, the associated real quadratic fields have ${r_3} = 2$, but for $d = 44806173 = 3 \cdot 14935391,Q(\sqrt d )$ has a class group $C(3) \times C(3) \times C(3)$; and this is now the smallest known d for which a real quadratic field has ${r_3} = 3$.

References

Daniel Shanks and Peter Weinberger, A quadratic field of prime discriminant requiring three generators for its class group, and related theory, Acta Arith. 21 (1972), 71–87. MR 309899, DOI 10.4064/aa-21-1-71-87
Daniel Shanks, New types of quadratic fields having three invariants divisible by $3$, J. Number Theory 4 (1972), 537–556. MR 313220, DOI 10.1016/0022-314X(72)90027-3
Daniel Shanks and Richard Serafin, Quadratic fields with four invariants divisible by $3$, Math. Comp. 27 (1973), 183–187. MR 330097, DOI 10.1090/S0025-5718-1973-0330097-0
Carol Neild and Daniel Shanks, On the $3$-rank of quadratic fields and the Euler product, Math. Comp. 28 (1974), 279–291. MR 352042, DOI 10.1090/S0025-5718-1974-0352042-5

Séminaire Delange-Pisot-Poitou

R. J. Porter, On irregular negative determinants of exponent $9_n$, Math. Tables Aids Comput. 10 (1956), 22–25. MR 78057, DOI 10.1090/S0025-5718-1956-0078057-1

Tables in the UMT file, MTAC

T. Callahan, The $3$-class groups of non-Galois cubic fields. I, II, Mathematika 21 (1974), 72–89; ibid. 21 (1974), 168–188. MR 366876, DOI 10.1112/S0025579300005805
T. Callahan, The $3$-class groups of non-Galois cubic fields. I, II, Mathematika 21 (1974), 72–89; ibid. 21 (1974), 168–188. MR 366876, DOI 10.1112/S0025579300005805

Math. Comp.

Daniel Shanks, Calculation and applications of Epstein zeta functions, Math. Comp. 29 (1975), 271–287. MR 409357, DOI 10.1090/S0025-5718-1975-0409357-2
David W. Boyd and H. Kisilevsky, On the exponent of the ideal class groups of complex quadratic fields, Proc. Amer. Math. Soc. 31 (1972), 433–436. MR 289454, DOI 10.1090/S0002-9939-1972-0289454-4
P. J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117–124. MR 313221, DOI 10.4064/aa-22-2-117-124

Crelle’s J.