Joint research 2002

of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer


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On these pages, we present most recent results of our joint research, directly from the lab.
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Basic bibliography:
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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Web master's e-mail address:
contact@algebra.at
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Class Numbers of Cyclic Cubic Fields (2002/04/01)
Dan (02/03/30): Again, I am starting a series of reports.
This time the topic is: the cyclic cubic fields K
and the distribution of their class numbers h.
For the computation, I perform the following three steps:

1. First, I construct a generating polynomial
P(X) = X^3 - CX - D with rational integer coefficients C > 0, D > 0
for K by a method due to M.-N. GRAS,
taking into account the multiplicity m(f) = 2^{n-1} of the conductor f = q_1*...*q_n,
followed by a TSCHIRNHAUSEN reduction, if 9 does not divide f.

2. Next, different from GRAS' techniques, I fire cannon balls on sparrows,
using the algorithm of VORONOI to determine the regulator R of K.
The reason is, that I also want to gain insight into the
geometry of lattices associated with cyclic cubics and their minima.

3. Finally, I approximate the temperament T of K,
T = lim_{s-->1}(zeta_K(s) / zeta(s)),
by means of an EULER product.
Then the class number h of K can be calculated
by means of the analytic class number formula,

h = [T * w * sqrt(|disc|)] / [2^s * (2*pi)^t *R],

where in our case
w = number of roots of unity in K = 2 (only +1 and -1),
disc = f^2 (square of the conductor),
(s,t) = signature = (3,0) (totally real cubic K),
and thus

h = [T * 2 * f] / [2^3 * R] = T * (f / 4R).
Dan (02/03/31):
Yesterday I (resp. my computers) have finished
two tables of cyclic cubic fields:

(*) with all prime conductors f = q < 20000,
(**) with all 2-prime-conductors f = q1*q2 < 10000.

1. The first table (*) even extends those of
ENNOLA and TURUNEN, which were bounded by f < 16000.

I found the following distribution of class numbers
among these 1125 fields with 3-class rank rho = 0:

class number | 1 | 2^2 | 2^4 | 2^8 | 7 | 7^2 | 13 | 19
----------------------------------------------------------
frequency | 926 | 118 | 7 | 1 | 28 | 4 | 7 | 8

class number | 5^2 | 31 | 37 | 43 | 61 | 73 | 109 | 127
--------------------------------------------------------
frequency | 3 | 4 | 2 | 2 | 2 | 1 | 2 | 1

class number | 2^2*7 | 2^2*13 | 2^2*5^2
----------------------------------------
frequency | 5 | 3 | 1


In the most cases (h = 1,4), the EULER product can be terminated
after 8 iterations with 50 rational primes each (last prime: 2741).
However, for bigger class numbers we need more primes.
The HiChamp in this respect was h = 127 for f = 15013,
which needed 50 iterations and primes up to 48611.

2. The second table (**) extends the main table of GRAS (f < 4000)
and the first sub-table of ENNOLA / TURUNEN (f < 8000).

Here we must distinguish two cases among the 788 fields,
according to Marie-Nicole's husband George GRAS:

a) the 714 fields with 3-class rank rho = 1
for which q1 and q2 are not cubic residues with respect to each other:

class number | 3 | 3*2^2 | 3*2^4 | 3*7 | 3*13 | 3*19 | 3*5^2 | 3*2^2*7 | 3*2^2*19
------------------------------------------------------------------------------------
frequency | 590 | 79 | 4 | 27 | 5 | 3 | 2 | 2 | 2


None of the h > 3 occurs for both fields with conductor f simultaneously.

b) the 74 fields with 3-class rank rho = 2,
for which q1 and q2 are mutual cubic residues:

class number | 3^2 | 3^3 | 3^2*2^2 | 3^2*7 | 3^2*13
----------------------------------------------------
frequency | 60 | 4 | 5 | 4 | 1


Again, none of the h > 9 occurs for both fields with conductor f simultaneously,
with the single exception of h = 27,
which occurs for two pairs of fields sharing the same conductor (f = 4711,5383).

However, the 2-prime-conductors are very time consuming,
since the splitting-identification is much slower than finding cubic residuacity.

The ranges 10000 < f < 20000 and 20000 < f < 30000 are running already
on different machines. But I cannot predict when they will finish.

I wonder if h = 3^4 or 3^5 will appear,
but it doesn't make sense to be impatient.
Dan (02/04/01): Historical Remark:
The above mentioned tables extend the results of

[1] 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).

[2] 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.

[3] V. ENNOLA and R. TURUNEN, On totally real cubic fields,
Math. Comp. 44 (1985), 495-518.

[4] V. ENNOLA and R. TURUNEN, On cyclic cubic fields,
Math. Comp. 45 (1985), 585-589.

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