|On these pages, we present most recent results of our joint research, directly from the lab.|
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237
A. Derhem, Capitulation dans les extensions quadratiques de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988
D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831-847 and S55-S58
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I would like to know if there exist cyclic cubic fields of
conductor divisible by two primes
and whose 3-class number (the highest power of 3 dividing the class number of the field) is 81 (respectively 243).
I was told years ago that Professor ENNOLA had tables on these fields with such information
and I hope you may access that tables.
Dan (2001/12/24): One of the first authors who dealt with cyclic cubic
 Helmut HASSE, Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern,
Abh. Deutsch. Akad. Wiss. Berlin, math.-naturw. Kl. 1948, No. 2 (1950).
Dan: However, the first tables of cyclic cubics seem to
appeared as a numerical supplement to the paper
 M.-N. GRAS, Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q,
J. reine angew. Math. 277 (1975), 89-116.
I have got a copy of these tables in private communication.
The main table contains the first 630 cyclic cubic fields
ordered by increasing conductor m below the upper bound 4000, i. e., 7 <= m <= 3997.
The table lists m, the prime factors of m, two parameters a and b,
the coefficients tr(e) and tr(1/e) in the minimal polynomial P(X) = X^3 - tr(e)X^2 + tr(1/e)X - 1 of a norm-positive fundamental unit e,
and the class number h.
Three additional tables list fields with
4000 < m < 10000, 9 | h
4000 < m < 10000, 4 | h
4000 < m < 20000, m = (a^2+27)/4, (1+27b^2)/4, (9+27b^2)/4
In the last of these additional tables I found:
1 field with m = 17563 = 7*13*193 and with h = 81.
But, by the multiplicity-theory, there must exist three further fields with m = 17563
(probably their class number is only 9, I suppose)
Address of the author:
Marie-Nicole GRAS, 11 Rue de la fontaine, Pelousey, 25170 Recologne, France
Dan: An extension of these tables is due to
 V. ENNOLA and R. TURUNEN, On totally real cubic fields,
Math. Comp. 44 (1985), 495-518.
As before, I got a copy of these tables in private communication.
The tables continue the main table of M.-N. Gras listing analogous information
but with parameters -a and 3b in the normalization introduced by .
The first table (for03.dat.88 from August 11, 1982)
contains the cyclic cubic fields with ordinal numbers 631 <= n <= 1268 and conductors 4003 <= m <= 7999
In this table I found:
2 fields with m = 4711 = 7*673 and both with h = 27
2 fields with m = 5383 = 7*769 and both with h = 27
4 fields with m = 7657 = 13*19*31, one field with h = 81 and three fields with h = 9
(These fields occur also in the first additional table of M.-N. Gras.)
The second table (for03.dat.228 from September 10, 1982)
contains the cyclic cubic fields with ordinal numbers 1269 <= n <= 1904 and conductors 8001 <= m <= 11997
In this table I found:
2 fields with m = 11167 = 13*859 and both with h = 27
Address of the authors:
Veikko ENNOLA and Reino TURUNEN, Department of Mathematics, University of Turku,
SF-20500 Turku 50, Finland
I also observed several cases of families with 4 fields (quadrupletts) and thus with 3 prime factors of m
where two fields have h = 27 but the others have only 9 | h.
However, unfortunately I did not find any example of fields
whose conductor m has exactly 2 prime factors (whence these fields must necessarily occur as doubletts)
and where 81 | h, let alone 243 | h.
I am quite sure that such cases will appear for bigger values of m.