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

Author:
Daniel Shanks

Journal:
Math. Comp. **30** (1976), 173-178

MSC:
Primary 12A25; Secondary 12A50

DOI:
https://doi.org/10.1090/S0025-5718-1976-0399039-9

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

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

MathSciNet review:
0399039

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Abstract | References | Similar Articles | Additional Information

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 ${r_3} = 3$.

- 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), 71â€“87. MR**309899**, DOI https://doi.org/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), 537â€“556. MR**313220**, DOI https://doi.org/10.1016/0022-314X%2872%2990027-3 - Daniel Shanks and Richard Serafin,
*Quadratic fields with four invariants divisible by $3$*, Math. Comp.**27**(1973), 183â€“187. MR**330097**, DOI https://doi.org/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), 279â€“291. MR**352042**, DOI https://doi.org/10.1090/S0025-5718-1974-0352042-5
F. DIAZ Y DIAZ, "Sur les corps quadratiques imaginaires dont le 3-rang du groupe des classes est supĂ©rieur Ă 1", - R. J. Porter,
*On irregular negative determinants of exponent $9_n$*, Math. Tables Aids Comput.**10**(1956), 22â€“25. MR**78057**, DOI https://doi.org/10.1090/S0025-5718-1956-0078057-1
R. J. PORTER, - T. Callahan,
*The $3$-class groups of non-Galois cubic fields. I, II*, Mathematika**21**(1974), 72â€“89; ibid. 21 (1974), 168â€“188. MR**366876**, DOI https://doi.org/10.1112/S0025579300005805 - T. Callahan,
*The $3$-class groups of non-Galois cubic fields. I, II*, Mathematika**21**(1974), 72â€“89; ibid. 21 (1974), 168â€“188. MR**366876**, DOI https://doi.org/10.1112/S0025579300005805
DANIEL SHANKS, "Review of Angellâ€™s table," - Daniel Shanks,
*Calculation and applications of Epstein zeta functions*, Math. Comp.**29**(1975), 271â€“287. MR**409357**, DOI https://doi.org/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), 433â€“436. MR**289454**, DOI https://doi.org/10.1090/S0002-9939-1972-0289454-4 - P. J. Weinberger,
*Exponents of the class groups of complex quadratic fields*, Acta Arith.**22**(1973), 117â€“124. MR**313221**, DOI https://doi.org/10.4064/aa-22-2-117-124
A. SCHOLZ, "Ăśber die Beziehung der Klassenzahlen quadratischer KĂ¶rper zueinander,"

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Article copyright:
© Copyright 1976
American Mathematical Society