For the 16 principal factorization types of cubic fields L in the following table we denote by

r ... the unit rank of the quadratic subfield k of the normal field N
z ... the 3-exponent of the subgroup of torsion units of k
u ... the 3-exponent of the unit norm index ( Uk : N(UN) )
a ... the 3-exponent of the group of absolute principal factors ( PLG : PQ )
b ... the 3-exponent of the group of relative principal factors ( DN|k-Ik : Ik )
c ... the 3-exponent of the capitulation kernel ( PN * Ik : Pk )
where the operator * denotes the intersection
PF-Type ... the principal factorization type (red color emphasizes Arnie's monster)


Principal Factorization Types of Cubic Fields
Nr. Signature Designation r z r + z u 1 + u AbsPF a RelPF b CapPF c PF-Type
1 (3,0) Cyclic 0 0 0 0 1 1 0 0 ZETA
2 (3,0) Totally Real 1 0 1 1 2 0 0 2 ALPHA 1
3




1
2
0 1 1 ALPHA 2
4




1
2
0 2 0 ALPHA 3
5




1
2
1 0 1 BETA 1
6




1
2
1 1 0 BETA 2
7




1
2
2 0 0 GAMMA
8




0 1 0 0 1 DELTA 1
9




0
1
0 1 0 DELTA 2
10




0
1
1 0 0 EPSILON
11 (1,1) Pure 0 1 1 1 2 1 1 0 ALPHA
12




1
2
2 0 0 BETA
13




0 1 1 0 0 GAMMA
14 (1,1) Complex 0 0 0 0 1 0 0 1 ALPHA 1
15




0
1
0 1 0 ALPHA 2
16




0
1
1 0 0 BETA