of totally real cubic fields
Introduction
The unconventional form of my
table
[1]
of all
10015 totally real cubic fields L
with
discriminant d(L) < 200000
(including 70 cyclic fields)
results from the joint theoretical experience of my papers
"Multiplicities of dihedral discriminants"
[2]
and
"Classification of dihedral fields"
[3].
The discriminant d(L) of a noncyclic cubic field L
can be written in the shape
d(L) = f^{2}d(k),
where f denotes the
conductor
of the normal closure N of L
over its quadratic subfield k with discriminant d(k) and 3class rank r.
The abovementioned papers consider
the rational prime factors of the conductor f
from two different viewpoints.

The first of them derives
information about the number m of nonconjugate fields L
which share a common discriminant d(L)
from a partition of the prime factors
p_{1},...,p_{t} (t >= 0)
of the conductor f into
u = # { 1 <= i <= t  delta(d(k),p_{i}) = 0 }
free primes
and
v = # { 1 <= i <= t  delta(d(k),p_{i}) = 1 } = tu
restrictive primes,
where the
deltainvariants
describe the local behaviour of the
principal ideal cubes
(including the fundamental unit) of k
modulo the primes pi.
(For the exact definition of these invariants, see our first paper.)
For f = 1 we have the wellknown simple formula
m = (3^{r}1) / 2,
but in the case f > 1
the formula for the
multiplicity m
of a discriminant
is more complicated
m = 3^{r+x}2^{u}(2^{v1}(1)^{v1}) / 3,
where x = 1 if
9  f, d(k) congruent 3 (modulo 9), delta(d(k),3) = 0
and x = 0 otherwise,
and u = t, v = 0 if
9  f, d(k) congruent 3 (modulo 9), delta(d(k),3) = 1,
but otherwise u,v have the meaning explained above.
In the technical treatise
"Dihedral fields from quadratic infrastructure"
we have proved the equivalence
delta(d(k),p) = 0 if, and only if, v_{3}(R(p) / R(k)) < v(p),
which is crucial for the
actual computability
of the deltainvariants
by means of rapid regulator calculations in orders of real quadratic fields.

In the second paper we classify the totally real cubic fields into
10 principal factor types
alpha1, alpha2, alpha3, beta1, beta2,
gamma, delta1, delta2, epsilon, zeta.
This
classification
contains complete information
on the principalization of ideal classes of k in N
and on the ambiguous principal ideals (principal factors) in L and N.
In many cases the type of a given field L can be determined readily
from the partition of the t prime factors of the conductor f into
those which do not split
in k, q_{1},...,q_{n} (n >= 0),
and
the others which split
in k, l_{1},...,l_{s} (s >= 0).
Computational techniques
The table was constructed twice by two independent methods.

First, by the time consuming classical procedure of
determining the indices i(P) of lots of cubic polynomials
in a sufficiently large set.
Given an upper bound B,
we can find a generating polynomial
P(X) = X^{3}CXD
of discriminant
d(P) = 4C^{3}27D^{2} > 0
for each field with discriminant
d(L) <= i(P)^{2}d(L) = d(P) < B
within the range
0 < C <= B^{1/2}, 0 < D < 2(C^{3} / 27)^{1/2}.
We have chosen
B = 200000
and thus had to investigate all the polynomials
with coefficients
0 < C <= 447, 0 < D <= 3637.
These generating polynomials were used in the algorithm of G. F. Voronoi
to compute the fundamental units
and regulators of those fields which could not be classified simply by
factoring the conductor f.

Second, by the fast new method of computing the deltainvariants
of real quadratic fields for rational prime conductors.
For f = 1, we have
d(k) = d(L) < B,
whence we had to determine all real quadratic fields k
with discriminant
d(k) < 200000
and positive 3class rank
by
computing cycles of reduced indefinite binary quadratic forms.
For prime conductors f >= 2,
we have
d(k) = d(L) / f^{2} <= d(L)/4 < B/4.
Therefore we had to determine the deltainvariants delta(d(k),f)
of all real quadratic fields k with discriminant
d(k) < 50000
and with arbitrary 3class rank
by
computing quotients of regulators of real quadratic orders,
and additionally investigating the local behaviour of principal ideal cubes
in the case of positive 3class rank.
Since d(k) >= 5 for real quadratic fields k,
we have
f = (d(L)/d(k))^{1/2} <= (d(L)/5)^{1/2} < (B/5)^{1/2}
and conductors are expected in the range
2 <= f < (40000)^{1/2} = 200.
For cyclic cubic fields L, however, the discriminant has the shape
d(L) = f^{2}. Therefore,
f = d(L)^{1/2} < B^{1/2}
and
conductors lie between the bounds
7 <= f < 200000^{1/2} (approx. 447).
The results of both methods were in perfect accordance with the table of
Ennola and Turunen
[4].
Acknowledgements
I am indebted to the staff of the
Department of Computer Science,
in particular to Professor H. C. Williams,
for making available the facilities of the
Computer Centre at the University of Manitoba.
I gratefully acknowledge that this research project was supported
by the Austrian Science Foundation, grant Nr.J0497PHY.
Bibliography

[1]
D. C. Mayer,
List of discriminants d_{L}<200000 of
totally real cubic fields L,
arranged according to their multiplicities m
and conductors f,
1991, Dept. of Comp. Sci., Univ. of Manitoba

[2]
D. C. Mayer,
Multiplicities of dihedral discriminants,
Math. Comp. 58 (1992), no. 198, 831847
and Supplements section S55S58

[3]
D. C. Mayer,
Classification of dihedral fields,
1991, Dept. of Comp. Sci., Univ. of Manitoba

[4]
V. Ennola and R.Turunen,
On totally real cubic fields,
Math. Comp. 44 (1985), no. 170, 495518
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