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*10^{4} < D < 0 of negative cubic discriminants with 3169 fields
and the range
0 < D < 10^{5} 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 nonradical 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*10^{4} < D < 2*10^{4}
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 10^{5} < D < 2*10^{5}
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) = X^{3}  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, 3class rank r, 3admissible conductors f, and generators of ideal cubes,
i. e., constructing the sextic Galois closures N with S_{3}group Gal(NQ)
over their quadratic subfields K
and not the cubic fields themselves.
This idea resulted in the development of my DIFFQIalgorithm
(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*f^{2} in Z 
d is a quadratic discriminant and f is a 3admissible conductor for d }.
We observe a natural disjoint partition of FCD:
FCD = union_{ d in QD } AC_{3}(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
AC_{3}(d) = { f = 3^{e}*q_{1}*...*q_{s} 
s >= 0, e in {0,1,2},
q_{i} pairwise distinct primes different from 3,
q_{i} = (d / q_{i}) (mod 3),
e != 1 for d = 1,2 (mod 3), e != 2 for d = 3 (mod 9) }.
Using the concept of 3admissible prime conductors for d, we can also write
AC_{3}(d) = { f = p_{1}*...*p_{t} 
t >= 0, p_{i} in PC_{3}(d) pairwise coprime }, where
PC_{3}(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 F_{3}vector space
V_{3} = I_{3} / (K^{x})^{3}
of
nontrivial ideal cube generators,
we map each prime p and the critical prime power p = 9 on its
3ring space V_{3}(p)
in V_{3}:
p > V_{3}(p) =
[ I_{3}(p) intersection R_{p}*K(p)^{3} ] / K(p)^{3}.
We characterize each f = p_{1}*...*p_{t} in AC_{3}(d)
by occupation numbers a(V_{3}) of the full vector space V_{3}
and a(H) for each hyperplane H < V_{3}, where
a(T) = #{ 1 <= i <= t  V_{3}(p_{i}) = T },
for any subspace T < V_{3}.
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(V_{3}) (resp. v = tu)
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(V_{3}(9)) = 2).

The Results.
In the following tables, we list the minimal occurrences of cubic discriminants
D = d*f^{2} 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 3class rank of that quadratic field.
In the column MF we indicate the
multiplicity formula,
that gives the number m of
nonisomorphic 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
a) Nonradical (nonpure) cubic fields
m

MF

r

(u,v,w)

(n,s)

#

D=d*f^{2}

PFT

1

(0.0)

1

(0,0,0)

(0,0)

3243

23 = 23*1^{2}

Alpha_{1}

1

(0.1)

0

(1,0,0)

(1,0)

873

44 = 11*2^{2}

Beta

0

(1.1)

1

(0,1,0)

(1,0)

*

[ 236 = 59*2^{2} ]

*

1

(0.1)

0

(1,0,0)

(0,1)

124

648 = 8*9^{2}
1960 = 40*7^{2}

Alpha_{2}
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
Alpha_{2}

2

(0.1)

0

(2,0,0)

(0,2)

2*0

?
?

Alpha_{2}
Beta

3

(0.1)

0

(1,0,1)

(1,0)

3*9

3159 = 39*9^{2}

Beta

3

(0.1)

1

(1,0,0)

(1,0)

3*85

1228 = 307*2^{2}
7724 = 1931*2^{2}

Alpha_{1}
Beta

3

(0.1)

1

(1,0,0)

(0,1)

3*6

2891 = 59*7^{2}
2891 = 59*7^{2}
?

Alpha_{1}
Beta
Alpha_{2}

3

(1.1)

1

(0,2,0)

(2,0)

3*5

10700 = 107*(2*5)^{2}
?

Beta
Alpha_{1}

3

(1.1)

1

(0,2,0)

(1,1)

3*4

16268 = 83*(2*7)^{2}
?
?

Beta
Alpha_{1}
Alpha_{2}

3

(1.2)

1

(0,1,1)

(1,0)

3*2

20655 = 255*9^{2}
?

Alpha_{1}
Beta

4

(0.0)

2

(0,0,0)

(0,0)

4*47

3299 = 3299*1^{2}

Alpha_{1}

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*9^{2} ]

*

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*9^{2}

?

0

(2.2)

2

(0,1,1)

(1,0)

*

[ 708831 = 8751*9^{2} ]

*

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*9^{2}

?

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*9^{2} ]

*

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*9^{2}

?

Constructed by Francisco Diaz y Diaz
and clear without further analysis:

13

(0.0)

3

(0,0,0)

(0,0)

*

3321607 = 3321607*1^{2}

Alpha_{1}

40

(0.0)

4

(0,0,0)

(0,0)

*

653329427 = 653329427 *1^{2}

Alpha_{1}

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  p_{i} = 1,8 (mod 9) }.
Further we list the smallest radicand R for each case,
taken from
my extensive 2002 table with R < 10^{6}.
m

MF

r

(u,v,w)

(n,s)

#

D=d*f^{2}

R

PFT

0

(1.1)

0

(0,1,0)

(1,0)

*

[ 12 = 3*2^{2} ]

*

*

1

(1.1)

0

(0,2,0)

(2,0)

*

108 = 3*6^{2} = 3*(2*3)^{2}

2

Beta

1

(1.1)

0

(0,3,0)

(3,0)

*

2700 = 3*30^{2} = 3*(2*3*5)^{2}

20

Beta

1

(1.2)

0

(0,1,1)

(1,0)

1

243 = 3*9^{2}

3

Gamma

1

(0.1)

0

(1,0,0)

(1,0)

*

867 = 3*17^{2}

17

Gamma

2

(1.2)

0

(0,2,1)

(2,0)

*

972 = 3*18^{2} = 3*(2*9)^{2}

6

Beta

2

(1.1)

0

(1,2,0)

(3,0)

*

31212 = 3*102^{2} = 3*(2*3*17)^{2}

34

Beta

2

(1.1)

0

(1,3,0)

(4,0)

*

780300 = 3*510^{2} = 3*(2*3*5*17)^{2}

340

Beta

2

(0.1)

0

(2,0,0)

(1,1)

*

312987 = 3*323^{2} = 3*(17*19)^{2}

323

Gamma

3

(1.1)

0

(0,4,0)

(3,1)

*

132300 = 3*210^{2} = 3*(2*3*5*7)^{2}

70

Beta

4

(1.2)

0

(0,3,1)

(3,0)

*

24300 = 3*90^{2} = 3*(2*5*9)^{2}

30

Beta

4

(1.1)

0

(2,2,0)

(3,1)

*

11267532 = 3*1938^{2} = 3*(2*3*17*19)^{2}

646

Beta

4

(1.1)

0

(2,3,0)

(4,1)

*

281688300 = 3*9690^{2} = 3*(2*3*5*17*19)^{2}

6460

Beta

4

(0.1)

0

(3,0,0)

(1,2)

*

428479203 = 3*11951^{2} = 3*(17*19*37)^{2}

11951

Beta

5

(1.1)

0

(0,5,0)

(4,1)

*

16008300 = 3*2310^{2} = 3*(2*3*5*7*11)^{2}

770

Beta

6

(1.1)

0

(1,4,0)

(4,1)

*

38234700 = 3*3570^{2} = 3*(2*3*5*7*17)^{2}

1190

Beta

8

(1.2)

0

(0,4,1)

(3,1)

*

1190700 = 3*630^{2} = 3*(2*5*7*9)^{2}

210

Beta

8

(1.1)

0

(3,2,0)

(3,2)

*

15425251308 = 3*71706^{2} = 3*(2*3*17*19*37)^{2}

23902

Beta

8

(1.1)

0

(3,3,0)

(4,2)

*

385631282700 = 3*358530^{2} = 3*(2*3*5*17*19*37)^{2}

239020

Beta

8

(0.1)

0

(4,0,0)

(2,2)

*

1203598081227 = 3*633403^{2} = 3*(17*19*37*53)^{2}

633403

Gamma

10

(1.1)

0

(1,5,0)

(5,1)

*

4626398700 = 3*39270^{2} = 3*(2*3*5*7*11*17)^{2}

13090

Beta

11

(1.1)

0

(0,6,0)

(5,1)

*

2705402700 = 3*6^{2} = 3*(2*3*5*7*11*13)^{2}

10010

Beta

12

(1.1)

0

(2,4,0)

(4,2)

*

13802726700 = 3*67830^{2} = 3*(2*3*5*7*17*19)^{2}

22610

Beta

16

(1.2)

0

(0,5,1)

(4,1)

*

144074700 = 3*6930^{2} = 3*(2*5*7*9*11)^{2}

2310

Beta


Minimal occurrences of totally real cubic fields
a) NonGalois (noncyclic) cubic fields
m

MF

r

(u,v,w)

(n,s)

#

D=d*f^{2}

PFT

1

(0.0)

1

(0,0,0)

(0,0)

6924

229 = 229*1^{2}

Delta_{1}

0

(1.1)

0

(0,1,0)

(1,0)

*

[ 20 = 5*2^{2} ]

*

1

(0.1)

0

(1,0,0)

(1,0)

2173

148 = 37*2^{2}

Epsilon

1

(0.1)

0

(1,0,0)

(0,1)

267

2597 = 53*7^{2}
9653 = 197*7^{2}
27881 = 569*7^{2}

Delta_{2}
Beta_{2}
Epsilon

1

(1.2)

0

(0,1,1)

(1,0)

68

1944 = 24*9^{2}

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}

Beta_{2}
Gamma

1

(1.1)

0

(0,2,0)

(0,2)

1

146853 = 37*(7*9)^{2}

Alpha_{3}

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*9^{2}

Epsilon

3

(0.1)

1

(1,0,0)

(1,0)

3*33

28212 = 7053*2^{2}
57588 = 14397*2^{2}
57588 = 14397*2^{2}

Delta_{1}
Beta_{1}
Epsilon

3

(0.1)

1

(1,0,0)

(0,1)

3*5

86485 = 1765*7^{2}
86485 = 1765*7^{2}
189777 = 3873*7^{2}

Beta_{1}
Delta_{1}
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}

Beta_{1}

4

(0.0)

2

(0,0,0)

(0,0)

4*16

32009 = 32009*1^{2}
32009 = 32009*1^{2}

Alpha_{1}
Delta_{1}

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 = 2^{t1},
that depends only on the number t of prime divisors
of the conductor f = p_{1}*...*p_{t},
where t >= 1 and
p_{i} = 1 (mod 3) pairwise distinct primes
or p_{i} = 9.
m

t

#

D=f^{2}

PFT

1

1

42

49 = 7^{2}

Zeta

2

2

2*14

3969 = 63^{2} = (7*9)^{2}

Zeta

And outside of my extension,
constructed by myself:

4

3

*

670761 = 819^{2} = (7*9*13)^{2}

Zeta

8

4

*

242144721 = 15561^{2} = (7*9*13*19)^{2}

Zeta

