Tables of Complex and Totally Real Cubic Fields

with Discriminants -3*104 < D < -2*104 and 105 < D < 2*105

The Inspiration.

The tables of Ian O. Angell, constructed in 1972 and 1975 as part of the requirements for a Ph. D. thesis written under the supervision of Harold J. Godwin, have been "a veritable tour de force of computing", as Harvey Cohn described these first extensive compilations of machine calculated discriminants, fundamental units, and ideal class numbers of complex and totally real cubic number fields. Angell covered the range -2*104 < D < 0 of negative cubic discriminants with 3169 fields and the range 0 < D < 105 of positive cubic discriminants with 4804 fields. The software, written for this purpose, was the first computer implementation of G. F. Voronoi's algorithms that included the determination of ideal class numbers. In 1970, H. C. Williams and C. R. Zarnke had already implemented the algorithms of Voronoi to compute fundamental systems of units for the first time. However, since Angell only mentions Delaunay and Faddeev as reference for the number geometric methods of Voronoi, he probably didn't use the recent work of Williams and Zarnke.

The Enterprise.

Impressed by Angell's tables, a rewarding challenge to demonstrate my intellectual "nuclear weapons" potential seemed to be an extension of these tables in two directions. First by a continuation of the discriminantal ranges and second by adding new arithmetical invariants like principal factorization types (PFT's). Having gained experience with the intricate Voronoi algorithm for series of pure cubic fields ordered by ascending radicands between 1986 and 1988, I dared to attack the project of extending Angell's first table in 1989. A complication that arises for non-radical extensions is that they must be ordered by ascending discriminants without omitting any of them, though they are constructed from generating polynomials in a completely different order. As a first valuable check I reproduced and confirmed Angell's own range. Then I carried out the continuation -3*104 < D < -2*104 yielding a total of 4885 fields with an interesting population of PFT's. Since the complexity of Voronoi's algorithm for totally real cubic fields exeeds the difficulties of the algorithm for complex fields considerably, it was not earlier than 1991 that I was able to extend Angell's second table. Again, I first reproduced and confirmed Angell's own range, which had been corrected meanwhile by Llorente / Oneto and Ennola / Turunen, and then doubled the range by a continuation 105 < D < 2*105 obtaining a total of 10015 fields with a fascinating population of PFT's that includes Arnie's monster.

Deeper Analysis of the Minimal Occurrences of Formal Cubic Discriminants with Certain Types of Conductors.

My extensions of Angell's tables were constructed by two independent methods.
First by the classical procedure of finding generating polynomials P(X) = X3 - C*X - D in a sufficiently large range for the coefficients C,D in Z. Unfortunately, this method has repeatedly turned out to be error prone, as the missing fields in the first versions of the tables of Angell (10 of 4804, or 0.21%) and Fung / Williams (669 of 182417, or 0.37%) have proved. Failures can be due to an incorrect determination of the polynomial index i(P) that connects the field discriminant D with the polynomial discriminant d(P) by the relation d(P) = D*i(P)2, to errors in deciding whether fields with coinciding discriminant are isomorphic, and finally simply to selecting a too small range for the polynomial coefficients C,D.
Therefore, my revolutionary idea in 1989 was to synthesize cubic discriminants "artificially", completely forgetting the cubic polynomials, and only working class field theoretically with quadratic fields and their invariants, like discriminant d, 3-class rank r, 3-admissible conductors f, and generators of ideal cubes, i. e., constructing the sextic Galois closures N with S3-group Gal(N|Q) over their quadratic subfields K and not the cubic fields themselves. This idea resulted in the development of my DIFFQI-algorithm (DIhedral Fields From Quadratic Invariants), the second independent and modern tool for synthesizing cubic (and replacing p = 3 by an arbitrary odd prime p, even dihedral) discriminants. DIFFQI consists of two layers:

The low level layer of DIFFQI. We start by filtering out all Formal Cubic Discriminants (FCD's) from the set of rational integers, Z:
FCD = { d*f2 in Z | d is a quadratic discriminant and f is a 3-admissible conductor for d }.
We observe a natural disjoint partition of FCD:
FCD = union_{ d in QD } AC3(d), where
(denoting by SF(r,m) the set { n in Z | n square free and n = r (mod m) })
QD = [ SF(1,4) - {1} ] union 4*SF(2,4) union 4*SF(3,4),
and for each d in QD
AC3(d) = { f = 3e*q1*...*qs | s >= 0, e in {0,1,2},
qi pairwise distinct primes different from 3, qi = (d / qi) (mod 3),
e != 1 for d = 1,2 (mod 3), e != 2 for d = 3 (mod 9) }.
Using the concept of 3-admissible prime conductors for d, we can also write
AC3(d) = { f = p1*...*pt | t >= 0, pi in PC3(d) pairwise coprime }, where
PC3(d) = { p prime or p = 9 | either p prime different from 3, p = (d / p) (mod 3)
or p = 9 if d = 1,2 (mod 3) or p = 3 if d = 3 (mod 9) or p in {3,9} if d = -3 (mod 9) }.

The high level layer of DIFFQI. Given a fixed quadratic field K with discriminant d in QD and with associated F3-vector space V3 = I3 / (Kx)3 of non-trivial ideal cube generators, we map each prime p and the critical prime power p = 9 on its 3-ring space V3(p) in V3: p --> V3(p) = [ I3(p) intersection Rp*K(p)3 ] / K(p)3. We characterize each f = p1*...*pt in AC3(d) by occupation numbers a(V3) of the full vector space V3 and a(H) for each hyperplane H < V3, where a(T) = #{ 1 <= i <= t | V3(pi) = T }, for any subspace T < V3. The irregular case, where 9 divides f and d = -3(mod 9), will be characterized by an indicator w = 1 that takes the value w = 0 otherwise. We adopt the following abbreviations and concepts: u = a(V3) (resp. v = t-u) is the number of free (resp. restrictive) prime conductors dividing f. The family (u,(a(H))H,w) is called the type of the conductor f, which degenerates to the triplet (u,v,w), if there is essentially only a single hyperplane H, whence v = a(H) (or v = 1 + a(H) in the exceptional situation, where codim(V3(9)) = 2).

The Results.

In the following tables, we list the minimal occurrences of cubic discriminants D = d*f2 in FCD with conductors f of all possible types (u,v,w), the number # of associated cubic fields in my extensions of Angell's tables, and the principal factorization type PFT, denoting by n (resp. s) the number of prime divisors of f that do not split (resp. split) in the quadratic field with fundamental discriminant d in QD, and by r the 3-class rank of that quadratic field. In the column MF we indicate the multiplicity formula, that gives the number m of non-isomorphic fields sharing the common conductor f, for each case. If m >= 1, then D is an actual cubic discriminant. For m = 0, however, D is only a formal cubic discriminant (indicated by brackets, [ ]).
Red color emphasizes discriminants whose multiplicity cannot be determined by Hasse's theory, i. e., formula 0.0 and 0.1. In these cases my own formulae 1.1, 1.2,... must be applied. A question mark ? means either that the corresponding case didn't occur up to now or that a PFT has not been determined yet. An asterisk * indicates either that a case cannot occur or that a count is difficult.

Minimal occurrences of complex cubic fields

m MF r (u,v,w) (n,s) # D=d*f2 PFT
1 (0.0) 1 (0,0,0) (0,0) 3243 -23 = -23*12 Alpha1
1 (0.1) 0 (1,0,0) (1,0) 873 -44 = -11*22 Beta
0 (1.1) 1 (0,1,0) (1,0) * [ -236 = -59*22 ] *
1 (0.1) 0 (1,0,0) (0,1) 124 -648 = -8*92
-1960 = -40*72
Alpha2
Beta
2 (0.1) 0 (2,0,0) (2,0) 2*32 -1836 = -51*(2*3)2 Beta
2 (0.1) 0 (2,0,0) (1,1) 2*9 -3564 = -11*(2*9)2
?
Beta
Alpha2
2 (0.1) 0 (2,0,0) (0,2) 2*0 ?
?
Alpha2
Beta
3 (0.1) 0 (1,0,1) (1,0) 3*9 -3159 = -39*92 Beta
3 (0.1) 1 (1,0,0) (1,0) 3*85 -1228 = -307*22
-7724 = -1931*22
Alpha1
Beta
3 (0.1) 1 (1,0,0) (0,1) 3*6 -2891 = -59*72
-2891 = -59*72
?
Alpha1
Beta
Alpha2
3 (1.1) 1 (0,2,0) (2,0) 3*5 -10700 = -107*(2*5)2
?
Beta
Alpha1
3 (1.1) 1 (0,2,0) (1,1) 3*4 -16268 = -83*(2*7)2
?
?
Beta
Alpha1
Alpha2
3 (1.2) 1 (0,1,1) (1,0) 3*2 -20655 = -255*92
?
Alpha1
Beta
4 (0.0) 2 (0,0,0) (0,0) 4*47 -3299 = -3299*12 Alpha1
And outside of my extension,
constructed by my special friends G. W. Fung and H. C. Williams,
but analyzed by myself:
0 (1.1) 1 (0,1,1) (1,0) * [ -55647 = -687*92 ] *
6 (0.1) 0 (2,0,1) (2,0) * -70956 = -219*(2*9)2 Beta
9 (0.1) 1 (1,0,1) (1,0) * -274347 = -3387*92 ?
0 (2.2) 2 (0,1,1) (1,0) * [ -708831 = -8751*92 ] *
3 (1.1) 1 (0,3,0) (2,1) * -725004 = -411*(2*3*7)2 ?
Constructed by Karim Belabas
and analyzed by myself:
9 (1.2) 2 (0,1,1) (1,0) * -3449871 = -42591*92 ?
12 (0.1) 0 (3,0,1) (3,0) * -5856300 = -723*(2*5*9)2 Beta
0 (1.1) 2 (0,1,1) (1,0) * [ -10404531 = -128451*92 ] *
18 (1.1) 1 (1,2,1) (3,0) * -27434700 = -3387*(2*5*9)2 ?
27 (0.1) 2 (1,0,1) (1,0) * -167644728 = -2069688*92 ?
Constructed by Francisco Diaz y Diaz
and clear without further analysis:
13 (0.0) 3 (0,0,0) (0,0) * -3321607 = -3321607*12 Alpha1
40 (0.0) 4 (0,0,0) (0,0) * -653329427 = -653329427 *12 Alpha1

b) Pure cubic fields

Here we can give a general expression for the number u of free prime divisors of the conductor f:
u = # { 1 <= i <= t | pi = 1,8 (mod 9) }.
Further we list the smallest radicand R for each case,
taken from my extensive 2002 table with R < 106.

m MF r (u,v,w) (n,s) # D=d*f2 R PFT
0 (1.1) 0 (0,1,0) (1,0) * [ -12 = -3*22 ] * *
1 (1.1) 0 (0,2,0) (2,0) * -108 = -3*62 = -3*(2*3)2 2 Beta
1 (1.1) 0 (0,3,0) (3,0) * -2700 = -3*302 = -3*(2*3*5)2 20 Beta
1 (1.2) 0 (0,1,1) (1,0) 1 -243 = -3*92 3 Gamma
1 (0.1) 0 (1,0,0) (1,0) * -867 = -3*172 17 Gamma
2 (1.2) 0 (0,2,1) (2,0) * -972 = -3*182 = -3*(2*9)2 6 Beta
2 (1.1) 0 (1,2,0) (3,0) * -31212 = -3*1022 = -3*(2*3*17)2 34 Beta
2 (1.1) 0 (1,3,0) (4,0) * -780300 = -3*5102 = -3*(2*3*5*17)2 340 Beta
2 (0.1) 0 (2,0,0) (1,1) * -312987 = -3*3232 = -3*(17*19)2 323 Gamma
3 (1.1) 0 (0,4,0) (3,1) * -132300 = -3*2102 = -3*(2*3*5*7)2 70 Beta
4 (1.2) 0 (0,3,1) (3,0) * -24300 = -3*902 = -3*(2*5*9)2 30 Beta
4 (1.1) 0 (2,2,0) (3,1) * -11267532 = -3*19382 = -3*(2*3*17*19)2 646 Beta
4 (1.1) 0 (2,3,0) (4,1) * -281688300 = -3*96902 = -3*(2*3*5*17*19)2 6460 Beta
4 (0.1) 0 (3,0,0) (1,2) * -428479203 = -3*119512 = -3*(17*19*37)2 11951 Beta
5 (1.1) 0 (0,5,0) (4,1) * -16008300 = -3*23102 = -3*(2*3*5*7*11)2 770 Beta
6 (1.1) 0 (1,4,0) (4,1) * -38234700 = -3*35702 = -3*(2*3*5*7*17)2 1190 Beta
8 (1.2) 0 (0,4,1) (3,1) * -1190700 = -3*6302 = -3*(2*5*7*9)2 210 Beta
8 (1.1) 0 (3,2,0) (3,2) * -15425251308 = -3*717062 = -3*(2*3*17*19*37)2 23902 Beta
8 (1.1) 0 (3,3,0) (4,2) * -385631282700 = -3*3585302 = -3*(2*3*5*17*19*37)2 239020 Beta
8 (0.1) 0 (4,0,0) (2,2) * -1203598081227 = -3*6334032 = -3*(17*19*37*53)2 633403 Gamma
10 (1.1) 0 (1,5,0) (5,1) * -4626398700 = -3*392702 = -3*(2*3*5*7*11*17)2 13090 Beta
11 (1.1) 0 (0,6,0) (5,1) * -2705402700 = -3*62 = -3*(2*3*5*7*11*13)2 10010 Beta
12 (1.1) 0 (2,4,0) (4,2) * -13802726700 = -3*678302 = -3*(2*3*5*7*17*19)2 22610 Beta
16 (1.2) 0 (0,5,1) (4,1) * -144074700 = -3*69302 = -3*(2*5*7*9*11)2 2310 Beta

Minimal occurrences of totally real cubic fields

a) Non-Galois (non-cyclic) cubic fields

m MF r (u,v,w) (n,s) # D=d*f2 PFT
1 (0.0) 1 (0,0,0) (0,0) 6924 229 = 229*12 Delta1
0 (1.1) 0 (0,1,0) (1,0) * [ 20 = 5*22 ] *
1 (0.1) 0 (1,0,0) (1,0) 2173 148 = 37*22 Epsilon
1 (0.1) 0 (1,0,0) (0,1) 267 2597 = 53*72
9653 = 197*72
27881 = 569*72
Delta2
Beta2
Epsilon
1 (1.2) 0 (0,1,1) (1,0) 68 1944 = 24*92 Epsilon
1 (1.1) 0 (0,2,0) (2,0) 173 756 = 21*(2*3)2 Gamma
1 (1.1) 0 (0,2,0) (1,1) 94 4212 = 13*(2*9)2
155316 = 21*(2*43)2
Beta2
Gamma
1 (1.1) 0 (0,2,0) (0,2) 1 146853 = 37*(7*9)2 Alpha3
1 (1.1) 0 (0,3,0) (3,0) 1 91476 = 21*(2*3*11)2 Gamma
1 (1.1) 0 (0,3,0) (2,1) 1 105300 = 13*(2*5*9)2 Gamma
2 (0.1) 0 (2,0,0) (2,0) 2*9 37300 = 373*(2*5)2 Epsilon
2 (0.1) 0 (2,0,0) (1,1) 2*3 38612 = 197*(2*7)2 Epsilon
2 (1.2) 0 (1,1,1) (2,0) 2*2 45684 = 141*(2*9)2 Epsilon
2 (1.2) 0 (0,2,1) (2,0) 2*6 66825 = 33*(5*9)2 Gamma
2 (1.1) 0 (1,2,0) (3,0) 2*3 83700 = 93*(2*3*5)2 Gamma
2 (1.1) 0 (1,2,0) (2,1) 2*2 164052 = 93*(2*3*7)2 Gamma
3 (0.1) 0 (1,0,1) (1,0) 3*3 58077 = 717*92 Epsilon
3 (0.1) 1 (1,0,0) (1,0) 3*33 28212 = 7053*22
57588 = 14397*22
57588 = 14397*22
Delta1
Beta1
Epsilon
3 (0.1) 1 (1,0,0) (0,1) 3*5 86485 = 1765*72
86485 = 1765*72
189777 = 3873*72
Beta1
Delta1
Epsilon
3 (1.1) 0 (0,2,1) (2,0) 3*1 22356 = 69*(2*9)2 Gamma
3 (1.1) 1 (0,2,0) (2,0) 3*1 54324 = 1509*(2*3)2 Beta1
4 (0.0) 2 (0,0,0) (0,0) 4*16 32009 = 32009*12
32009 = 32009*12
Alpha1
Delta1
And outside of my extension,
constructed by Llorente / Quer,
but analyzed by myself:
4 (0.1) 0 (3,0,0) (2,1) * 8250228 = 4677*(2*3*7)2 ?
4 (1.2) 0 (1,2,1) (3,0) * 1725300 = 213*(2*5*9)2 ?
6 (0.1) 1 (2,0,0) (2,0) * 3054132 = 84837*(2*3)2 ?
6 (0.1) 0 (2,0,1) (3,0) * 9796788 = 30237*(2*9)2 ?
6 (1.2) 1 (0,2,1) (2,0) * 6367572 = 19653*(2*9)2 ?
6 (1.1) 0 (1,2,1) (3,0) * 5807700 = 717*(2*5*9)2 ?
Constructed by Karim Belabas
and analyzed by myself:
5 (1.1) 0 (0,5,0) (3,2) * 13302897300 = 277*(2*5*7*9*11)2 ?

b) Cyclic cubic fields

Here we have a particularly simple multiplicity formula,
m = 2t-1,
that depends only on the number t of prime divisors of the conductor f = p1*...*pt,
where t >= 1 and pi = 1 (mod 3) pairwise distinct primes or pi = 9.

m t # D=f2 PFT
1 1 42 49 = 72 Zeta
2 2 2*14 3969 = 632 = (7*9)2 Zeta
And outside of my extension,
constructed by myself:
4 3 * 670761 = 8192 = (7*9*13)2 Zeta
8 4 * 242144721 = 155612 = (7*9*13*19)2 Zeta

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