My Extension of Angell's Table

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 non-cyclic cubic field L can be written in the shape
d(L) = f2d(k),
where f denotes the conductor of the normal closure N of L over its quadratic subfield k with discriminant d(k) and 3-class rank r.

The abovementioned papers consider the rational prime factors of the conductor f from two different viewpoints.

  1. The first of them derives information about the number m of non-conjugate fields L which share a common discriminant d(L) from a partition of the prime factors p1,...,pt (t >= 0) of the conductor f into
    u = # { 1 <= i <= t | delta(d(k),pi) = 0 }
    free primes and
    v = # { 1 <= i <= t | delta(d(k),pi) = 1 } = t-u
    restrictive primes, where the delta-invariants 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 well-known simple formula
    m = (3r-1) / 2,
    but in the case f > 1 the formula for the multiplicity m of a discriminant is more complicated
    m = 3r+x2u(2v-1-(-1)v-1) / 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, v3(R(p) / R(k)) < v(p),
    which is crucial for the actual computability of the delta-invariants by means of rapid regulator calculations in orders of real quadratic fields.
  2. 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, q1,...,qn (n >= 0), and the others which split in k, l1,...,ls (s >= 0).

Computational techniques

The table was constructed twice by two independent methods.
  1. 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) = X3-CX-D of discriminant
    d(P) = 4C3-27D2 > 0
    for each field with discriminant
    d(L) <= i(P)2d(L) = d(P) < B
    within the range
    0 < C <= B1/2, 0 < D < 2(C3 / 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.
  2. Second, by the fast new method of computing the delta-invariants 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 3-class rank by computing cycles of reduced indefinite binary quadratic forms. For prime conductors f >= 2, we have
    d(k) = d(L) / f2 <= d(L)/4 < B/4.
    Therefore we had to determine the delta-invariants delta(d(k),f) of all real quadratic fields k with discriminant d(k) < 50000 and with arbitrary 3-class 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 3-class 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) = f2. Therefore,
    f = d(L)1/2 < B1/2
    and conductors lie between the bounds
    7 <= f < 2000001/2 (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.J0497-PHY.

Bibliography

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