By making use of our generalization of Barrucand and Cohn’s theory of principal factorizations in pure cubic fields and their Galois closures with 3 possible types to pure quintic fields and their pure metacyclic normal fields with 13 possible types, we compile an extensive database with arithmetical invariants of the 900 pairwise non-isomorphic fields N having normalized radicands in the range 2≤D<103 . Our classification is based on the Galois cohomology of the unit group UN, viewed as a module over the automorphism group Gal(N/K) of N over the cyclotomic field K=Q(ξ5) , by employing theorems of Hasse and Iwasawa on the Herbrand quotient of the unit norm index (Uk:NN/K(UN)) by the number #(PN/K/PK) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different DN/K . The precise structure of the F5 -vector space of differential principal factors is expressed in terms of norm kernels and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units (U<SUB>N</SUB>:U<SUB>0</SUB>). The statistical distribution of the 13 principal factorization types and their refined splitting into similarity classes with representative prototypes is discussed thoroughly.
At the end of his 1975 article on class numbers of pure quintic fields, Parry suggested verbatim: In conclusion, the author would like to say that he believes a numerical study of pure quintic fields would be most interesting ( [
Even in 1991, when we generalized Barrucand and Cohn’s theory [
All these conjectures have been proven by our most recent numerical investigations. Our classification is based on the Hasse-Iwasawa theorem about the Herbrand quotient of the unit group
We begin with a collection of explicit multiplicity formulas in §2 which are required for understanding the subsequent extensive presentation of our computational results in twenty tables of crucial invariants in §3. This information admits the classification of all 900 pure quintic fields
We draw the attention to remaining open questions in §3.3, and we collect theoretical consequences of our experimental results in §4.3. The exposition is concluded with a retrospective final §5.
For the convenience of the reader, we provide a summary of formulas for calculating invariants of pure quintic fields
Let f be the class field theoretic conductor of the relatively quintic Kummer extension N/K over the cyclotomic field
We adapt the general multiplicity formulas in ( [
If L is a field of species 1b ( [
If L is a field of species 2 ( [
X j : = 1 5 ( 4 j − ( − 1 ) j ) , that is
The following twenty Tables 6-25 establish a complete classification of all 900 pure metacyclic fields
The possible DPF types are listed in dependence on
t | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|
m | 1 | 4 | 16 | 64 | 256 | 1024 |
u | v | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|
0 | m | 0 | 1 | 3 | 13 | 51 | 205 |
1 | 0 | 4 | 12 | 52 | 204 | 820 | |
2 | 0 | 16 | 48 | 208 | 816 | ||
3 | 0 | 64 | 192 | 832 | |||
4 | 0 | 256 | 768 |
u | v | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|
0 | m | 0 | 0 | 1 | 3 | 13 | 51 |
1 | 1 | 0 | 4 | 12 | 52 | 204 | |
2 | 4 | 0 | 16 | 48 | 208 | 816 | |
3 | 16 | 0 | 64 | 192 | 832 | ||
4 | 64 | 0 | 256 | 768 |
T | U | A | I | R | ||
---|---|---|---|---|---|---|
2 | − | − | 1 | 0 | 2 | |
2 | − | − | 1 | 1 | 1 | |
2 | − | − | 1 | 2 | 0 | |
2 | − | − | 2 | 0 | 1 | |
2 | − | − | 2 | 1 | 0 | |
2 | − | − | 3 | 0 | 0 | |
1 | × | − | 1 | 0 | 1 | |
1 | × | − | 1 | 1 | 0 | |
1 | × | − | 2 | 0 | 0 | |
1 | − | × | 1 | 0 | 1 | |
1 | − | × | 1 | 1 | 0 | |
1 | − | × | 2 | 0 | 0 | |
0 | × | × | 1 | 0 | 0 |
The steps of the following classification algorithm are ordered by increasing requirements of CPU time. To avoid unnecessary time consumption, the algorithm stops at early stages already, as soon as the DPF type is determined unambiguously. The illustrating subfield lattice of N is drawn in
Algorithm 3.1 (Classification into 13 DPF types.)
Input: a normalized fifth power free radicand
Step 1: By purely rational methods, without any number field constructions, the prime factorization of the radicand D (including the counters
Step 2: The field L of degree 5 is constructed. The primes
the elements of
Step 3: If
Step 4: If
Step 5: If the type of the field N is not yet determined uniquely, then
Output: the DPF type of the field
Proof. The claims of Step 1 concerning the types
For Step 2, the formulas (4.1) and (4.2) in ( [
For Step 3, the formulas (4.5) and (4.6) in ( [
For Step 4, the formulas (4.9) and (4.10) in ( [
Concerning Step 5, the signature of N is
Remark 3.1 Whereas the execution of Step 1 and 2 in Algorithm 3.1, implemented as a Magma program [
We conjecture that considerable amounts of CPU time can be saved in our Algorithm 3.1 by computing the logarithmic 5-class numbers
However, first there would be required rigorous proofs of the heuristic connections between
The normalized radicand
Prime factors are given for composite radicands D only. Dedekind’s species, S, of radicands is refined by distinguishing
T | E | E+ | or | E | E+ |
---|---|---|---|---|---|
2 | 1 | ||||
1 | 0 | 3 | 1 | ||
2 | 0 | ||||
3 | 1 | 5 | 2 | ||
4 | 1 | ||||
6 | 2 | ||||
2 | 1 | 4 | 2 | ||
3 | 1 | ||||
5 | 2 | ||||
5 | 2 | ||||
4 | 1 | ||||
6 | 2 | ||||
5 | 2 |
An asterisk denotes the smallest radicand with given Dedekind kind, DPF type and 5-class groups
Principal factors, P, are listed when their constitution is not a consequence of the other information. According to ( [
The quartet
A symbol × for the component 2 emphasizes a prime divisor
If D has only prime divisors
The complete statistical evaluation of the following twenty Tables 6-25 is given in
Among our 13 differential principal factorization types, type
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | *2 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
2 | 3 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
3 | *5 | 1a | 1 | 0 | 0 | 0 | 5 | |||||
4 | *6 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
5 | *7 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
6 | *10 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
7 | *11 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
8 | 12 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
9 | 13 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
10 | *14 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
11 | 15 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
12 | 17 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
13 | *18 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
14 | *19 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
15 | 20 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
16 | 21 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
17 | *22 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
18 | 23 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
19 | 26 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
20 | 28 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
21 | 29 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
22 | *30 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
23 | *31 | 1b | 1 | 2 | 3 | 5 | 2 | |||||
24 | *33 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
25 | 34 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
26 | *35 | 1a | 4 | 0 | 0 | 1 | 6 | |||||
27 | 37 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
28 | *38 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
29 | 39 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
30 | 40 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
31 | 41 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
32 | *42 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
33 | 43 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
34 | 44 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
35 | 45 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
36 | 46 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
37 | 47 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
38 | 48 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
39 | 51 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
40 | 52 | 1b | 3 | 0 | 0 | 1 | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | (1,2,4,5) | T | P |
---|---|---|---|---|---|---|---|---|---|---|---|---|
41 | 53 | 1b | 1 | 0 | 0 | 0 | 2 | |||||
42 | *55 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
43 | 56 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
44 | *57 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
45 | 58 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
46 | 59 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
47 | 60 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
48 | 61 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
49 | 62 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
50 | 63 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
51 | 65 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
52 | *66 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
53 | 67 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
54 | 68 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
55 | 69 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
56 | *70 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
57 | 71 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
58 | 73 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
59 | 74 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
60 | 75 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
61 | 76 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
62 | *77 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
63 | *78 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
64 | 79 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
65 | 80 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
66 | *82 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
67 | 83 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
68 | 84 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
69 | 85 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
70 | 86 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
71 | 87 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
72 | 88 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
73 | 89 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
74 | 90 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
75 | 91 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
76 | 92 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
77 | 93 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
78 | 94 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
79 | *95 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
80 | 97 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
81 | 99 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
82 | *101 | 2 | 1 | 1 | 2 | 4 | 5 | |||||
83 | 102 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
84 | 103 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
85 | 104 | 1b | 3 | 0 | 0 | 1 | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
86 | 105 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
87 | 106 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
88 | 107 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
89 | 109 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
90 | *110 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
91 | 111 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
92 | 112 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
93 | 113 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
94 | *114 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
95 | 115 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
96 | 116 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
97 | 117 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
98 | 118 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
99 | 119 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
100 | 120 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
101 | 122 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
102 | *123 | 1b | 3 | 2 | 3 | 6 | 3 | |||||
103 | 124 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
104 | *126 | 2 | 4 | 0 | 0 | 1 | 6 | |||||
105 | 127 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
106 | 129 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
107 | 130 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
108 | *131 | 1b | 1 | 2 | 2 | 4 | 1 | |||||
109 | *132 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
110 | *133 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
111 | 134 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
112 | 136 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
113 | 137 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
114 | 138 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
115 | *139 | 1b | 1 | 1 | 1 | 3 | 4 | |||||
116 | *140 | 1a | 16 | 1 | 2 | 4 | 5 | 7 | ||||
117 | *141 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
118 | 142 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
119 | 143 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
120 | 145 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
121 | 146 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
122 | 147 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
123 | 148 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
124 | *149 | 2 | 1 | 1 | 1 | 2 | 3 |
125 | 150 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
126 | *151 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
127 | 152 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
128 | 153 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
129 | *154 | 1b | 12 | 2 | 3 | 7 | 4 | |||||
130 | *155 | 1a | 4 | 2 | 3 | 5 | 2 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
131 | 156 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
132 | 157 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
133 | 158 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
134 | 159 | 1b | 3 | 1 | 2 | 4 | 5 | 5 | ||||
135 | 161 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
136 | 163 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
137 | 164 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
138 | 165 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
139 | 166 | 1b | 3 | 1 | 2 | 4 | 5 | 5 | ||||
140 | 167 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
141 | 168 | 2 | 4 | 0 | 0 | 1 | 6 | |||||
142 | 170 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
143 | *171 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
144 | 172 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
145 | 173 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
146 | *174 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
147 | 175 | 1a | 4 | 0 | 0 | 1 | 6 | |||||
148 | 176 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
149 | 177 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
150 | 178 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
151 | 179 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
152 | *180 | 1a | 16 | 1 | 2 | 4 | 5 | 3 | ||||
153 | 181 | 1b | 1 | 2 | 2 | 4 | 1 | |||||
154 | *182 | 2 | 4 | 1 | 2 | 4 | 5 | 7 | ||||
155 | 183 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
156 | 184 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
157 | 185 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
158 | *186 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
159 | 187 | 1b | 3 | 1 | 1 | 3 | 4 |
160 | 188 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
161 | *190 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
162 | *191 | 1b | 1 | 1 | 2 | 4 | 5 | |||||
163 | 193 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
164 | 194 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
165 | 195 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
166 | 197 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
167 | 198 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
168 | 199 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
169 | 201 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
170 | *202 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
171 | *203 | 1b | 4 | 1 | 2 | 4 | 5 | |||||
172 | 204 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
173 | 205 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
174 | 206 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
175 | 207 | 2 | 1 | 0 | 0 | 0 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
176 | 208 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
177 | *209 | 1b | 3 | 2 | 3 | 7 | 4 | |||||
178 | *210 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
179 | *211 | 1b | 1 | 3 | 5 | 9 | 2 | |||||
180 | 212 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
181 | 213 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
182 | 214 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
183 | 215 | 1a | 4 | 0 | 0 | 1 | 6 | |||||
184 | 217 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
185 | *218 | 2 | 1 | 1 | 2 | 4 | 5 | 2 | ||||
186 | 219 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
187 | 220 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
188 | 221 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
189 | 222 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
190 | 223 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
191 | 226 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
192 | 227 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
193 | 228 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
194 | 229 | 1b | 1 | 1 | 1 | 2 | 3 |
195 | 230 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
196 | *231 | 1b | 12 | 1 | 2 | 2 | 6 | |||||
197 | 232 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
198 | 233 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
199 | 234 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
200 | 235 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
201 | 236 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
202 | 237 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
203 | 238 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
204 | 239 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
205 | 240 | 1a | 16 | 1 | 2 | 4 | 5 | 3 | ||||
206 | 241 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
207 | 244 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
208 | 245 | 1a | 4 | 0 | 0 | 1 | 6 | |||||
209 | 246 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
210 | *247 | 1b | 3 | 1 | 2 | 5 | 6 | |||||
211 | 248 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
212 | 249 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
213 | 251 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
214 | 252 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
215 | *253 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
216 | 254 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
217 | 255 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
218 | 257 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
219 | 258 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
220 | *259 | 1b | 4 | 1 | 2 | 5* | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
221 | 260 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
222 | 261 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
223 | 262 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
224 | 263 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
225 | 264 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
226 | 265 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
227 | *266 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
228 | 267 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
229 | 268 | 2 | 1 | 0 | 0 | 0 | 5 |
230 | 269 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
231 | 270 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
232 | 271 | 1b | 1 | 1 | 2 | 4 | 5 | |||||
233 | 272 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
234 | *273 | 1b | 12 | 2 | 4 | 8 | 5 | |||||
235 | 274 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
236 | *275 | 1a | 4 | 2 | 2 | 4 | 1 | |||||
237 | *276 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
238 | 277 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
239 | 278 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
240 | 279 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
241 | 280 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
242 | *281 | 1b | 1 | 3 | 5* | 9* | 2 | |||||
243 | 282 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
244 | 283 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
245 | 284 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
246 | *285 | 1a | 16 | 1 | 2 | 5 | 6 | |||||
247 | *286 | 1b | 13 | 2 | 3 | 7 | 4 | |||||
248 | *287 | 1b | 4 | 2 | 2 | 4 | 1 | |||||
249 | *290 | 1a | 16 | 1 | 2 | 4 | 5 | 29 | ||||
250 | 291 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
251 | 292 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
252 | 293 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
253 | 294 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
254 | 295 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
255 | 296 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
256 | 297 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
257 | *298 | 1b | 4 | 1 | 2 | 4 | 5 | 5 | ||||
258 | 299 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
259 | *301 | 2 | 4 | 0 | 0 | 1 | 6 | |||||
260 | *302 | 1b | 4 | 1 | 2 | 4 | 5 | |||||
261 | 303 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
262 | 304 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
263 | 305 | 1a | 4 | 2 | 2 | 4 | 1 | |||||
264 | 306 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
265 | 307 | 2 | 1 | 0 | 0 | 0 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
266 | 308 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
267 | 309 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
268 | 310 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
269 | 311 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
270 | 312 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
271 | 313 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
272 | 314 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
273 | 315 | 1a | 16 | 1 | 2 | 4 | 5 | 7 | ||||
274 | 316 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
275 | 317 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
276 | 318 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
277 | *319 | 1b | 3 | 2 | 2 | 5 | 2 | |||||
278 | 321 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
279 | 322 | 1b | 12 | 2 | 4 | 8 | 5 | |||||
280 | 323 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
281 | 325 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
282 | 326 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
283 | 327 | 1b | 3 | 1 | 2 | 5 | 6 | |||||
284 | 328 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
285 | *329 | 1b | 4 | 1 | 2 | 4 | 5 | 47 | ||||
286 | *330 | 1a | 64 | 2 | 4 | 9 | 6 | |||||
287 | 331 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
288 | 332 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
289 | 333 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
290 | 334 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
291 | 335 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
292 | 336 | 1b | 12 | 2 | 4 | 8 | 5 | |||||
293 | 337 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
294 | 339 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
295 | 340 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
296 | *341 | 1b | 3 | 3 | 5 | 9 | 2 | |||||
297 | 342 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
298 | 344 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
299 | 345 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
300 | 346 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
301 | 347 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
302 | *348 | 1b | 13 | 2 | 4 | 8 | 5 | |||||
303 | 349 | 2 | 1 | 1 | 1 | 2 | 3 |
304 | 350 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
305 | 351 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
306 | 353 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
307 | 354 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
308 | 355 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
309 | 356 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
310 | 357 | 2 | 4 | 0 | 0 | 1 | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
311 | 358 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
312 | 359 | 1b | 1 | 1 | 1 | 3 | 4 | |||||
313 | 360 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
314 | 362 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
315 | 364 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
316 | 365 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
317 | 366 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
318 | 367 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
319 | 368 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
320 | 369 | 1b | 3 | 2 | 3 | 6 | 3 | |||||
321 | 370 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
322 | 371 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
323 | 372 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
324 | 373 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
325 | 374 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
326 | 376 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
327 | *377 | 1b | 3 | 2 | 3 | 6 | 3 | |||||
328 | 378 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
329 | *379 | 1b | 1 | 1 | 2 | 4 | 5 | 5 | ||||
330 | 380 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
331 | 381 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
332 | 382 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
333 | 383 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
334 | *385 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
335 | 386 | 1b | 4 | 1 | 2 | 4 | 5 | 2 |
336 | 387 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
337 | 388 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
338 | 389 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
339 | *390 | 1a | 64 | 1 | 2 | 5 | 6 | 2, 5 | ||||
340 | 391 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
341 | 393 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
342 | 394 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
343 | 395 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
344 | 396 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
345 | 397 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
346 | *398 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
347 | *399 | 2 | 4 | 1 | 1 | 3 | 4 | |||||
348 | *401 | 2 | 1 | 2 | 3 | 5 | 2 | |||||
349 | 402 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
350 | 403 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
351 | 404 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
352 | 405 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
353 | 406 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
354 | 407 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
355 | 408 | 1b | 13 | 1 | 2 | 5 | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
356 | 409 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
357 | 410 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
358 | 411 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
359 | 412 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
360 | 413 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
361 | 414 | 1b | 13 | 1 | 2 | 5 | 6 | 2, 3 | ||||
362 | 415 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
363 | 417 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
364 | *418 | 2 | 3 | 2 | 3 | 6 | 3 | |||||
365 | 419 | 1b | 1 | 1 | 1 | 3 | 4 | |||||
366 | 420 | 1a | 64 | 1 | 2 | 5 | 6 |
367 | *421 | 1b | 1 | 1 | 2 | 3 | 4 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
368 | *422 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
369 | 423 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
370 | 424 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
371 | 425 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
372 | 426 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
373 | 427 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
374 | 428 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
375 | 429 | 1b | 13 | 2 | 3 | 7 | 4 | |||||
376 | 430 | 1a | 16 | 1 | 2 | 4 | 5 | 43 | ||||
377 | 431 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
378 | 433 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
379 | 434 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
380 | 435 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
381 | 436 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
382 | 437 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
383 | 438 | 1b | 13 | 1 | 2 | 5 | 6 | 3, 5 | ||||
384 | 439 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
385 | 440 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
386 | 442 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
387 | 443 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
388 | 444 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
389 | 445 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
390 | 446 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
391 | 447 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
392 | 449 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
393 | *451 | 2 | 1 | 2 | 3 | 6 | 3 | K ( 11 ) K ( 41 ) 4 , K ( 11 ) ,1 K ( 41 ) ,2 3 | ||||
394 | 452 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
395 | 453 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
396 | 454 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
397 | 455 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
398 | 456 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
399 | 457 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
400 | 458 | 1b | 3 | 1 | 1 | 3 | 4 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
401 | 459 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
402 | 460 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
403 | 461 | 1b | 1 | 1 | 2 | 3 | 4 | |||||
404 | *462 | 1b | 52 | 2 | 4 | 9 | 6 | 5 2 × 7, 3 × 5 × 11 2 | ||||
405 | 463 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
406 | 464 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
407 | *465 | 1a | 16 | 2 | 3 | 7* | 4 | |||||
408 | 466 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
409 | 467 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
410 | 468 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
411 | 469 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
412 | 470 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
413 | 471 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
414 | 472 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
415 | *473 | 1b | 4 | 2 | 3 | 7* | 4 | |||||
416 | 474 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
417 | 475 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
418 | 476 | 2 | 4 | 0 | 0 | 1 | 6 | |||||
419 | 477 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
420 | 478 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
421 | 479 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
422 | 481 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
423 | *482 | 2 | 1 | 1 | 2 | 4 | 5 | |||||
424 | 483 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
425 | 485 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
426 | 487 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
427 | 488 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
428 | 489 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
429 | 490 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
430 | 491 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
431 | 492 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
432 | 493 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
433 | 494 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
434 | 495 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
435 | 496 | 1b | 3 | 1 | 1 | 3 | 4 |
436 | 497 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
437 | 498 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
438 | 499 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
439 | 501 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
440 | *502 | 1b | 4 | 2* | 4* | 8* | 5 | |||||
441 | 503 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
442 | 504 | 1b | 15 | 1 | 2 | 5 | 6 | |||||
443 | *505 | 1a | 4 | 1 | 1 | 3 | 4 | |||||
444 | 506 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
445 | 508 | 1b | 3 | 0 | 0 | 1 | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
446 | 509 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
447 | 510 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
448 | 511 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
449 | 513 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
450 | 514 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
451 | 515 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
452 | 516 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
453 | *517 | 1b | 3 | 2 | 3 | 6 | 3 | |||||
454 | 518 | 2 | 4 | 0 | 0 | 1 | 6 | |||||
455 | 519 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
456 | 520 | 1a | 16 | 1 | 2 | 4 | 5 | 2 | ||||
457 | 521 | 1b | 1 | 1 | 2 | 3 | 4 | |||||
458 | 522 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
459 | 523 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
460 | 524 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
461 | 525 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
462 | 526 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
463 | 527 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
464 | 528 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
465 | 530 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
466 | 531 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
467 | *532 | 2 | 4 | 1 | 2 | 4 | 5 | |||||
468 | 533 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
469 | 534 | 1b | 13 | 1 | 2 | 5 | 6 |
470 | 535 | 1a | 4 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
471 | 536 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
472 | 537 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
473 | 538 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
474 | 539 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
475 | 540 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
476 | 541 | 1b | 1 | 2 | 2 | 4 | 1 | |||||
477 | 542 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
478 | 543 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
479 | 545 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
480 | *546 | 1b | 52 | 2 | 4 | 9 | 6 | |||||
481 | 547 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
482 | 548 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
483 | 549 | 2 | 3 | 2 | 2 | 4 | 1 | |||||
484 | *550 | 1a | 16 | 2 | 3 | 6 | 3 | |||||
485 | *551 | 2 | 1 | 2 | 2 | 5 | 2 | |||||
486 | 552 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
487 | 553 | 1b | 4 | 1 | 2 | 4 | 5 | 5 | ||||
488 | 554 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
489 | 555 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
490 | 556 | 1b | 3 | 1 | 1 | 3 | 4 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
491 | 557 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
492 | 558 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
493 | 559 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
494 | 560 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
495 | 561 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
496 | 562 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
497 | 563 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
498 | 564 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
499 | 565 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
500 | 566 | 1b | 3 | 0 | 0 | 1 | 6 |
501 | 567 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
502 | 568 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
503 | 569 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
504 | *570 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
505 | 571 | 1b | 1 | 2 | 2 | 4 | 1 | |||||
506 | 572 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
507 | 573 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
508 | *574 | 2 | 4 | 1 | 1 | 3 | 4 | |||||
509 | 575 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
510 | 577 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
511 | 579 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
512 | 580 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
513 | 581 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
514 | 582 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
515 | 583 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
516 | 584 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
517 | 585 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
518 | 586 | 1b | 4 | 1 | 2 | 4 | 5 | 2 | ||||
519 | 587 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
520 | 589 | 1b | 3 | 2 | 2 | 5 | 2 | |||||
521 | *590 | 1a | 16 | 2 | 3 | 7* | 4 | |||||
522 | 591 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
523 | 592 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
524 | 593 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
525 | 594 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
526 | 595 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
527 | 596 | 1b | 4 | 1 | 2 | 4 | 5 | |||||
528 | 597 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
529 | 598 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
530 | 599 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
531 | 600 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
532 | 601 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
533 | *062 | 1b | 16 | 1 | 2 | 5 | 6 | 2, 7 | ||||
534 | 603 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
535 | 604 | 1b | 4 | 1 | 2 | 4 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
536 | 605 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
537 | *606 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
538 | 607 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
539 | *609 | 1b | 12 | 2 | 3 | 7 | 4 | |||||
540 | 610 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
541 | 611 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
542 | 612 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
543 | 613 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
544 | 614 | 1b | 4 | 1 | 2 | 4 | 5 | 2 | ||||
545 | 615 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
546 | 616 | 1b | 12 | 2 | 3 | 7 | 4 | |||||
547 | 617 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
548 | 618 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
549 | 619 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
550 | *620 | 1a | 16 | 2 | 4* | 8* | 5 | |||||
551 | 621 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
552 | 622 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
553 | 623 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
554 | 624 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
555 | 626 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
556 | *627 | 1b | 13 | 3 | 4 | 9 | 2 | |||||
557 | 628 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
558 | 629 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
559 | 630 | 1a | 64 | 1 | 2 | 5 | 6 | 3, 5 | ||||
560 | 631 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
561 | 632 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
562 | 633 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
563 | 634 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
564 | 635 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
565 | 636 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
566 | 637 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
567 | *638 | 1b | 13 | 2 | 3 | 7 | 4 | |||||
568 | 639 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
569 | 641 | 1b | 1 | 1 | 2 | 4 | 5 | |||||
570 | 642 | 1b | 12 | 1 | 2 | 5 | 6 |
571 | 643 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
572 | 644 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
573 | 645 | 1a | 16 | 1 | 2 | 4 | 5 | 43 | ||||
574 | 646 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
575 | 647 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
576 | *649 | 2 | 1 | 2 | 2 | 5 | 2 | |||||
577 | 650 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
578 | 651 | 2 | 4 | 1 | 1 | 3 | 4 | |||||
579 | 652 | 1b | 3 | 1 | 2 | 4 | 5 | 5 | ||||
580 | 653 | 1b | 1 | 0 | 0 | 0 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
581 | 654 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
582 | 655 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
583 | 656 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
584 | 657 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
585 | 658 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
586 | 659 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
587 | *660 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
588 | 661 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
589 | 662 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
590 | 663 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
591 | 664 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
592 | *665 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
593 | 666 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
594 | 667 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
595 | 668 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
596 | 669 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
597 | 670 | 1a | 16 | 1 | 2 | 4 | 5 | 2 | ||||
598 | *671 | 1b | 3 | 3 | 4 | 8 | 1 | |||||
599 | 673 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
600 | 674 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
601 | 677 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
602 | 678 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
603 | 679 | 1b | 4 | 0 | 0 | 1 | 6 |
604 | 680 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
605 | 681 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
606 | *682 | 2 | 3 | 2 | 3 | 6 | 3 | |||||
607 | 683 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
608 | 684 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
609 | 685 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
610 | 687 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
611 | 688 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
612 | 689 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
613 | 690 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
614 | *691 | 1b | 1 | 3 | 4 | 8 | 1 | |||||
615 | 692 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
616 | *693 | 2 | 4 | 2 | 3 | 6 | 3 | |||||
617 | 694 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
618 | *695 | 1a | 4 | 1 | 2 | 4 | 5 | 139 | ||||
619 | 696 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
620 | 697 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
621 | 698 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
622 | 699 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
623 | 700 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
624 | 701 | 2 | 1 | 2 | 3 | 5 | 2 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
625 | *702 | 1b | 13 | 2 | 4 | 8 | 5 | |||||
626 | 703 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
627 | 705 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
628 | 706 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
629 | *707 | 2 | 4 | 1 | 1 | 3 | 4 | |||||
630 | 708 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
631 | 709 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
632 | *710 | 1a | 16 | 2 | 2 | 4 | 1 | |||||
633 | 711 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
634 | 712 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
635 | 713 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
636 | 714 | 1b | 52 | 2 | 4 | 9 | 6 |
637 | 715 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
638 | 716 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
639 | 717 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
640 | 718 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
641 | 719 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
642 | 720 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
643 | 721 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
644 | 723 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
645 | 724 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
646 | 725 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
647 | 726 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
648 | 727 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
649 | 728 | 1b | 12 | 1 | 2 | 5 | 6 | 5, 7 | ||||
650 | 730 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
651 | 731 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
652 | 732 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
653 | 733 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
654 | 734 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
655 | 735 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
656 | 737 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
657 | 738 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
658 | 739 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
659 | 740 | 1a | 16 | 1 | 2 | 4 | 5 | 37 | ||||
660 | 741 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
661 | 742 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
662 | 743 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
663 | 744 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
664 | *745 | 1a | 4 | 1 | 1 | 3 | 4 | |||||
665 | 746 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
666 | 747 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
667 | 748 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
668 | *749 | 2 | 4 | 1 | 2 | 4 | 5 | |||||
669 | *751 | 2 | 1 | 2 | 2 | 4 | 1 | |||||
670 | 752 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
671 | 753 | 1b | 4 | 1 | 1 | 3 | 4 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
672 | 754 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
673 | 755 | 1a | 4 | 1 | 1 | 3 | 4 | |||||
674 | 756 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
675 | 757 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
676 | 758 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
677 | 759 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
678 | 760 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
679 | 761 | 1b | 1 | 2 | 3 | 5 | 2 | |||||
680 | 762 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
681 | 763 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
682 | 764 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
683 | 765 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
684 | 766 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
685 | 767 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
686 | 769 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
687 | *770 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
688 | 771 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
689 | 772 | 1b | 4 | 1 | 2 | 4 | 5 | 2 | ||||
690 | 773 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
691 | 774 | 2 | 4 | 0 | 0 | 1 | 6 | |||||
692 | 775 | 1a | 4 | 2 | 3 | 5 | 2 | |||||
693 | 776 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
694 | 777 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
695 | 778 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
696 | *779 | 1b | 3 | 3 | 4 | 8 | 1 | |||||
697 | 780 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
698 | 781 | 1b | 3 | 3 | 5 | 9 | 2 | |||||
699 | 782 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
700 | 783 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
701 | *785 | 1a | 4 | 1 | 2 | 4 | 5 | |||||
702 | 786 | 1b | 13 | 2 | 3 | 7 | 4 | |||||
703 | 787 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
704 | 788 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
705 | 789 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
706 | 790 | 1a | 16 | 1 | 2 | 4 | 5 | 5 | ||||
707 | 791 | 1b | 4 | 0 | 0 | 1 | 6 |
708 | 792 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
709 | 793 | 2 | 1 | 1 | 2 | 4 | 5 | |||||
710 | 794 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
711 | 795 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
712 | 796 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
713 | 797 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
714 | *798 | 1b | 52 | 2 | 4 | 9 | 6 | |||||
715 | 799 | 2 | 1 | 0 | 0 | 0 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
716 | 801 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
717 | 802 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
718 | 803 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
719 | 804 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
720 | 805 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
721 | 806 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
722 | 807 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
723 | *808 | 1b | 4 | 2 | 2 | 4 | 1 | |||||
724 | 809 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
725 | 810 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
726 | 811 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
727 | 812 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
728 | 813 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
729 | 814 | 1b | 13 | 2 | 3 | 6 | 3 | |||||
730 | 815 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
731 | 816 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
732 | 817 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
733 | 818 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
734 | 819 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
735 | 820 | 1a | 16 | 2 | 2 | 4 | 1 | |||||
736 | 821 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
737 | 822 | 1b | 13 | 2 | 4 | 8 | 5 | |||||
738 | 823 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
739 | 824 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
740 | *825 | 1a | 16 | 1 | 2 | 4 | 5 | |||||
741 | 826 | 2 | 4 | 1 | 1 | 3 | 4 |
742 | 827 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
743 | 828 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
744 | 829 | 1b | 1 | 1 | 1 | 3 | 4 | |||||
745 | 830 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
746 | 831 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
747 | 833 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
748 | 834 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
749 | 835 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
750 | 836 | 1b | 13 | 2 | 3 | 7 | 4 | |||||
751 | 837 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
752 | 838 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
753 | 839 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
754 | 840 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
755 | 842 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
756 | *843 | 2 | 1 | 1 | 2 | 3 | 4 | |||||
757 | 844 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
758 | 845 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
759 | 846 | 1b | 13 | 1 | 2 | 5 | 6 | 2, 3 | ||||
760 | 847 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
761 | 848 | 1b | 3 | 0 | 0 | 1 | 6 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
762 | 849 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
763 | 850 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
764 | 851 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
765 | 852 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
766 | 853 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
767 | 854 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
768 | 855 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
769 | 856 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
770 | 857 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
771 | *858 | 1b | 51 | 2 | 4 | 9 | 6 | |||||
772 | 859 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
773 | 860 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
774 | *861 | 1b | 12 | 3 | 4 | 8 | 1 |
775 | 862 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
776 | 863 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
777 | 865 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
778 | 866 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
779 | 868 | 2 | 4 | 1 | 1 | 3 | 4 | |||||
780 | 869 | 1b | 3 | 2 | 2 | 5 | 2 | |||||
781 | 870 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
782 | 871 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
783 | 872 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
784 | 873 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
785 | 874 | 2 | 3 | 1 | 1 | 3 | 4 | |||||
786 | 876 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
787 | 877 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
788 | 878 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
789 | 879 | 1b | 4 | 1 | 2 | 4 | 5 | 3 | ||||
790 | 880 | 1a | 16 | 2 | 2 | 4 | 1 | |||||
791 | 881 | 1b | 1 | 1 | 2 | 3 | 4 | |||||
792 | 883 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
793 | 884 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
794 | 885 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
795 | 886 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
796 | 887 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
797 | 888 | 1b | 13 | 1 | 2 | 5 | 6 | 2, 3 | ||||
798 | 889 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
799 | 890 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
800 | 891 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
801 | 892 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
802 | *893 | 2 | 1 | 1 | 1 | 3 | 4 | |||||
803 | *894 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
804 | 895 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
805 | 897 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
806 | 898 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
807 | 899 | 2 | 1 | 2 | 2 | 5 | 2 | |||||
808 | 901 | 2 | 1 | 0 | 0 | 0 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
809 | *902 | 1b | 13 | 2 | 3 | 7 | 4 | |||||
810 | 903 | 1b | 16 | 1 | 2 | 5 | 6 | |||||
811 | 904 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
812 | 905 | 1a | 4 | 1 | 1 | 2 | 3 | |||||
813 | 906 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
814 | 907 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
815 | 908 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
816 | 909 | 1b | 4 | 2 | 2 | 4 | 1 | |||||
817 | 910 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
818 | 911 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
819 | 912 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
820 | 913 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
821 | 914 | 1b | 4 | 1 | 2 | 4 | 5 | 2 | ||||
822 | 915 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
823 | 916 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
824 | 917 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
825 | 918 | 2 | 3 | 0 | 0 | 1 | 6 | |||||
826 | 919 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
827 | 920 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
828 | 921 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
829 | 922 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
830 | 923 | 1b | 3 | 2 | 2 | 4 | 1 | |||||
831 | *924 | 2 | 12 | 1 | 2 | 5 | 6 | |||||
832 | 925 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
833 | 926 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
834 | 927 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
835 | 929 | 1b | 1 | 1 | 1 | 2 | 3 | |||||
836 | 930 | 1a | 64 | 2 | 4 | 9 | 6 | 5, 31 | ||||
837 | 931 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
838 | 932 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
839 | 933 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
840 | 934 | 1b | 3 | 0 | 0 | 1 | 6 | |||||
841 | 935 | 1a | 16 | 2 | 2 | 4 | 1 | |||||
842 | 936 | 1b | 13 | 1 | 2 | 5 | 6 | 3, 5 | ||||
843 | 937 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
844 | 938 | 1b | 12 | 1 | 2 | 5 | 6 |
845 | 939 | 1b | 3 | 1 | 2 | 4 | 5 | 5 | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
846 | 940 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
847 | 941 | 1b | 1 | 2 | 2 | 4 | 1 | |||||
848 | 942 | 1b | 12 | 2 | 4 | 8 | 5 | 157 | ||||
849 | 943 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
850 | 944 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
851 | 945 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
852 | 946 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
853 | 947 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
854 | 948 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
855 | 949 | 2 | 1 | 0 | 0 | 0 | 5 |
No. | D | Factors | S | f4 | m | VL | VM | VN | E | T | P | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
856 | 950 | 1a | 16 | 1 | 1 | 3 | 4 | |||||
857 | 951 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
858 | 952 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
859 | 953 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
860 | 954 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
861 | *955 | 1a | 4 | 2* | 3* | 6* | 3 | |||||
862 | 956 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
863 | *957 | 2 | 3 | 2 | 2 | 5 | 2 | |||||
864 | 958 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
865 | 959 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
866 | 962 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
867 | 963 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
868 | 964 | 1b | 3 | 1 | 2 | 4 | 5 | |||||
869 | 965 | 1a | 4 | 0 | 0 | 1 | 6 | |||||
870 | 966 | 1b | 52 | 2 | 4 | 9 | 6 | |||||
871 | 967 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
872 | 969 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
873 | 970 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
874 | 971 | 1b | 1 | 2 | 2 | 4 | 1 | |||||
875 | 973 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
876 | 974 | 2 | 1 | 0 | 0 | 0 | 5 | |||||
877 | 975 | 1a | 16 | 0 | 0 | 1 | 6 | |||||
878 | 976 | 2 | 1 | 1 | 1 | 2 | 3 |
879 | 977 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
880 | 978 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
881 | 979 | 1b | 3 | 2 | 3 | 7 | 4 | |||||
882 | 980 | 1a | 16 | 1 | 2 | 4 | 5 | 7 | ||||
883 | 981 | 1b | 3 | 1 | 1 | 3 | 4 | |||||
884 | *982 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
885 | 983 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
886 | 984 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
887 | 985 | 1a | 4 | 0 | 0 | 0 | 5 | |||||
888 | 986 | 1b | 13 | 2 | 4 | 8 | 5 | |||||
889 | 987 | 1b | 12 | 1 | 2 | 5 | 6 | |||||
890 | 988 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
891 | 989 | 1b | 4 | 0 | 0 | 1 | 6 | |||||
892 | 990 | 1a | 64 | 1 | 2 | 5 | 6 | |||||
893 | 991 | 1b | 1 | 1 | 2 | 3 | 4 | |||||
894 | 993 | 2 | 1 | 1 | 1 | 2 | 3 | |||||
895 | 994 | 1b | 12 | 2 | 3 | 7 | 4 | |||||
896 | 995 | 1a | 4 | 1 | 1 | 3 | 4 | |||||
897 | 996 | 1b | 13 | 1 | 2 | 5 | 6 | |||||
898 | 997 | 1b | 1 | 0 | 0 | 0 | 5 | |||||
899 | 998 | 1b | 4 | 1 | 1 | 3 | 4 | |||||
900 | 999 | 2 | 1 | 0 | 0 | 0 | 5 |
Type | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 | % |
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 5 | 5 | 9 | 9 | 9 | ||
10 | 17 | 23 | 30 | 35 | 42 | 52 | 57 | 63 | 75 | 8.3 | |
0 | 0 | 0 | 1 | 1 | 3 | 5 | 5 | 7 | 8 | ||
0 | 2 | 4 | 7 | 8 | 11 | 15 | 18 | 22 | 23 | ||
7 | 24 | 40 | 54 | 80 | 94 | 108 | 126 | 146 | 161 | 17.9 | |
25 | 55 | 88 | 117 | 148 | 187 | 222 | 259 | 290 | 324 | 36.0 | |
0 | 0 | 1 | 1 | 3 | 4 | 4 | 4 | 6 | 7 | ||
8 | 14 | 19 | 23 | 31 | 35 | 38 | 44 | 51 | 53 | 5.9 | |
26 | 45 | 67 | 95 | 110 | 128 | 150 | 165 | 184 | 208 | 23.4 | |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||
0 | 0 | 0 | 0 | 0 | 1 | 1 | 4 | 4 | 5 | ||
1 | 2 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | ||
3 | 6 | 8 | 9 | 11 | 13 | 15 | 17 | 18 | 19 | ||
Total | 81 | 168 | 258 | 347 | 438 | 530 | 622 | 715 | 807 | 900 | 100.0 |
It is striking that type
The appearance of the four types
In [
Let t be the number of primes
of
Definition 4.1 A set of normalized fifth power free radicands
• the refined Dedekind species
• the differential principal factorization type
• the structure of the 5-class groups
Warning 4.1 To reduce the number of invariants, we abstain from defining additional counters
We also emphasize that in the rare cases of non-elementary 5-class groups, the actual structures (abelian type invariants) of the 5-class groups will be taken into account, and not only the 5-valuations
Definition 4.2 The minimal element
The remaining elements of a similarity class, which are bigger than the prototype, only reproduce the arithmetical invariants of the prototype and do not provide any additional information, exept possibly about other primary components of the class groups, that is the structure of ℓ-class groups
Whereas there are only 13 DPF types of pure quintic fields, the number of similarity classes is obviously infinite, since firstly the number t of primes dividing the conductor is unbounded and secondly the number of states, defined by the triplet
Given a fixed refined Dedekind species
The 134 prototypes
There is only a single finite similarity class [
No. | Factors | S | m | VL | VM | VN | E | T | P | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 1b | 1 | 0 | 0 | 0 | 5 | 71 | |||||
2 | 5 | 1a | 1 | 0 | 0 | 0 | 5 | 1 | |||||
3 | 6 | 1b | 3 | 0 | 0 | 1 | 6 | 77 | |||||
4 | 7 | 2 | 1 | 0 | 0 | 0 | 5 | 18 | |||||
5 | 10 | 1a | 4 | 0 | 0 | 0 | 5 | 31 | |||||
6 | 11 | 1b | 1 | 1 | 1 | 2 | 3 | 14 | |||||
7 | 14 | 1b | 4 | 0 | 0 | 1 | 6 | 44 | |||||
8 | 18 | 2 | 1 | 0 | 0 | 0 | 5 | 37 | |||||
9 | 19 | 1b | 1 | 1 | 1 | 2 | 3 | 27 | |||||
10 | 22 | 1b | 3 | 1 | 1 | 3 | 4 | 35 | |||||
11 | 30 | 1a | 16 | 0 | 0 | 1 | 6 | 37 | |||||
12 | 31 | 1b | 1 | 2 | 3 | 5 | 2 | 2 | |||||
13 | 33 | 1b | 3 | 2 | 2 | 4 | 1 | 8 | |||||
14 | 35 | 1a | 4 | 0 | 0 | 1 | 6 | 6 | |||||
15 | 38 | 1b | 3 | 1 | 1 | 3 | 4 | 44 | |||||
16 | 42 | 1b | 12 | 1 | 2 | 5 | 6 | 22 | |||||
17 | 55 | 1a | 4 | 1 | 1 | 2 | 3 | 6 | |||||
18 | 57 | 2 | 1 | 1 | 1 | 2 | 3 | 10 | |||||
19 | 66 | 1b | 13 | 1 | 2 | 5 | 6 | 17 | |||||
20 | 70 | 1a | 16 | 0 | 0 | 1 | 6 | 14 | |||||
21 | 77 | 1b | 4 | 1 | 1 | 3 | 4 | 7 | |||||
22 | 78 | 1b | 13 | 1 | 2 | 5 | 6 | 37 | |||||
23 | 82 | 2 | 1 | 1 | 1 | 2 | 3 | 15 | |||||
24 | 95 | 1a | 4 | 1 | 1 | 2 | 3 | 9 | |||||
25 | 101 | 2 | 1 | 1 | 2 | 4 | 5 | 1 | |||||
26 | 110 | 1a | 16 | 1 | 1 | 3 | 4 | 11 | |||||
27 | 114 | 1b | 13 | 1 | 2 | 5 | 6 | 20 | |||||
28 | 123 | 1b | 3 | 2 | 3 | 6 | 3 | 2 | |||||
29 | 126 | 2 | 4 | 0 | 0 | 1 | 6 | 6 | |||||
30 | 131 | 1b | 1 | 2 | 2 | 4 | 1 | 6 | |||||
31 | 132 | 2 | 3 | 1 | 1 | 3 | 4 | 5 | |||||
32 | 133 | 1b | 4 | 1 | 1 | 3 | 4 | 7 | |||||
33 | 139 | 1b | 1 | 1 | 1 | 3 | 4 | 4 | |||||
34 | 140 | 1a | 16 | 1 | 2 | 4 | 5 | 7 | 5 | ||||
35 | 141 | 1b | 3 | 1 | 2 | 4 | 5 | 19 | |||||
36 | 149 | 2 | 1 | 1 | 1 | 2 | 3 | 6 |
37 | 151 | 2 | 1 | 1 | 1 | 2 | 3 | 3 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
38 | 154 | 1b | 12 | 2 | 3 | 7 | 4 | 3 | |||||
39 | 155 | 1a | 4 | 2 | 3 | 5 | 2 | 2 | |||||
40 | 171 | 1b | 3 | 1 | 2 | 4 | 5 | 6 | |||||
41 | 174 | 2 | 3 | 1 | 1 | 3 | 4 | 3 | |||||
42 | 180 | 1a | 16 | 1 | 2 | 4 | 5 | 3 | 5 | ||||
43 | 182 | 2 | 4 | 1 | 2 | 4 | 5 | 7 | 1 | ||||
44 | 186 | 1b | 13 | 2 | 3 | 6 | 3 | 7 | |||||
45 | 190 | 1a | 16 | 1 | 1 | 3 | 4 | 9 | |||||
46 | 191 | 1b | 1 | 1 | 2 | 4 | 5 | 3 |
No. | Factors | S | m | VL | VM | VN | E | T | P | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47 | 202 | 1b | 4 | 1 | 1 | 3 | 4 | 6 | |||||
48 | 203 | 1b | 4 | 1 | 2 | 4 | 5 | 2 | |||||
49 | 209 | 1b | 3 | 2 | 3 | 7 | 4 | 2 | |||||
50 | 210 | 1a | 64 | 1 | 2 | 5 | 6 | 5 | |||||
51 | 211 | 1b | 1 | 3 | 5 | 9 | 2 | 1 | |||||
52 | 218 | 2 | 1 | 1 | 2 | 4 | 5 | 2 | 1 | ||||
53 | 231 | 1b | 12 | 1 | 2 | 5 | 6 | 5 | |||||
54 | 247 | 1b | 3 | 1 | 2 | 5 | 6 | 2 | |||||
55 | 253 | 1b | 3 | 1 | 2 | 4 | 5 | 4 | |||||
56 | 259 | 1b | 4 | 1 | 2 | 5* | 6 | 1 | |||||
57 | 266 | 1b | 12 | 1 | 2 | 5 | 6 | 3 | |||||
58 | 273 | 1b | 12 | 2 | 4 | 8 | 5 | 4 | |||||
59 | 275 | 1a | 4 | 2 | 2 | 4 | 1 | 2 | |||||
60 | 276 | 2 | 3 | 0 | 0 | 1 | 6 | 10 | |||||
61 | 281 | 1b | 1 | 3 | 5* | 9* | 2 | 1 | |||||
62 | 285 | 1a | 16 | 1 | 2 | 5 | 6 | 1 | |||||
63 | 286 | 1b | 13 | 2 | 3 | 7 | 4 | 3 | |||||
64 | 287 | 1b | 4 | 2 | 2 | 4 | 1 | 1 | |||||
65 | 290 | 1a | 16 | 1 | 2 | 4 | 5 | 29 | 2 | ||||
66 | 298 | 1b | 4 | 1 | 2 | 4 | 5 | 5 | 2 | ||||
67 | 301 | 2 | 4 | 0 | 0 | 1 | 6 | 1 | |||||
68 | 302 | 1b | 4 | 1 | 2 | 4 | 5 | 2 | |||||
69 | 319 | 1b | 3 | 2 | 2 | 5 | 2 | 3 | |||||
70 | 329 | 1b | 4 | 1 | 2 | 4 | 5 | 47 | 7 |
71 | 330 | 1a | 64 | 2 | 4 | 9 | 6 | 2 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
72 | 341 | 1b | 3 | 3 | 5 | 9 | 2 | 2 | |||||
73 | 348 | 1b | 13 | 2 | 4 | 8 | 5 | 2 | |||||
74 | 377 | 1b | 3 | 2 | 3 | 6 | 3 | 1 | |||||
75 | 379 | 1b | 1 | 1 | 2 | 4 | 5 | 5 | 1 | ||||
76 | 385 | 1a | 16 | 1 | 1 | 3 | 4 | 1 | |||||
77 | 390 | 1a | 64 | 1 | 2 | 5 | 6 | 4 | |||||
78 | 398 | 1b | 4 | 1 | 1 | 3 | 4 | 7 | |||||
79 | 399 | 2 | 4 | 1 | 1 | 3 | 4 | 2 | |||||
80 | 401 | 2 | 1 | 2 | 3 | 5 | 2 | 2 | |||||
81 | 418 | 2 | 3 | 2 | 3 | 6 | 3 | 1 | |||||
82 | 421 | 1b | 1 | 1 | 2 | 3 | 4 | 5 | |||||
83 | 422 | 1b | 3 | 1 | 2 | 4 | 5 | 6 | |||||
84 | 451 | 2 | 1 | 2 | 3 | 6 | 3 | 1 | |||||
85 | 462 | 1b | 52 | 2 | 4 | 9 | 6 | 1 | |||||
86 | 465 | 1a | 16 | 2 | 3 | 7* | 4 | 1 | |||||
87 | 473 | 1b | 4 | 2 | 3 | 7* | 4 | 1 | |||||
88 | 482 | 2 | 1 | 1 | 2 | 4 | 5 | 2 | |||||
89 | 502 | 1b | 4 | *2 | *4 | *8 | 5 | 1 | |||||
90 | 505 | 1a | 4 | 1 | 1 | 3 | 4 | 2 |
No. | Factors | S | m | VL | VM | VN | E | T | P | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
91 | 517 | 1b | 3 | 2 | 3 | 6 | 3 | 1 | |||||
92 | 532 | 2 | 4 | 1 | 2 | 4 | 5 | 1 | |||||
93 | 546 | 1b | 52 | 2 | 4 | 9 | 6 | 3 | |||||
94 | 550 | 1a | 16 | 2 | 3 | 6 | 3 | 1 | |||||
95 | 551 | 2 | 1 | 2 | 2 | 5 | 2 | 1 | |||||
96 | 570 | 1a | 64 | 1 | 2 | 5 | 6 | 2 | |||||
97 | 574 | 2 | 4 | 1 | 1 | 3 | 4 | 3 | |||||
98 | 590 | 1a | 16 | 2 | 3 | *7 | 4 | 1 | |||||
99 | 602 | 1b | 16 | 1 | 2 | 5 | 6 | 2 | |||||
100 | 606 | 1b | 12 | 1 | 2 | 5 | 6 | 2 | |||||
101 | 609 | 1b | 12 | 2 | 3 | 7 | 4 | 1 | |||||
102 | 620 | 1a | 16 | 2 | 4* | 8* | 5 | 1 |
103 | 627 | 1b | 13 | 3 | 4 | 9 | 2 | 1 | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
104 | 638 | 1b | 13 | 2 | 3 | 7 | 4 | 2 | |||||
105 | 649 | 2 | 1 | 2 | 2 | 5 | 2 | 2 | |||||
106 | 660 | 1a | 64 | 1 | 2 | 5 | 6 | 2 | |||||
107 | 665 | 1a | 16 | 1 | 1 | 3 | 4 | 1 | |||||
108 | 671 | 1b | 3 | 3 | 4 | 8 | 1 | 1 | |||||
109 | 682 | 2 | 3 | 2 | 3 | 6 | 3 | 1 | |||||
110 | 691 | 1b | 1 | 3 | 4 | 8 | 1 | 1 | |||||
111 | 693 | 2 | 4 | 2 | 3 | 6 | 3 | 1 | |||||
112 | 695 | 1a | 4 | 1 | 2 | 4 | 5 | 139 | 1 | ||||
113 | 702 | 1b | 13 | 2 | 4 | 8 | 5 | 2 | |||||
114 | 707 | 2 | 4 | 1 | 1 | 3 | 4 | 1 | |||||
115 | 710 | 1a | 16 | 2 | 2 | 4 | 1 | 4 | |||||
116 | 745 | 1a | 4 | 1 | 1 | 3 | 4 | 2 | |||||
117 | 749 | 2 | 4 | 1 | 2 | 4 | 5 | 1 | |||||
118 | 751 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | |||||
119 | 770 | 1a | 64 | 1 | 2 | 5 | 6 | 1 | |||||
120 | 779 | 1b | 3 | 3 | 4 | 8 | 1 | 1 | |||||
121 | 785 | 1a | 4 | 1 | 2 | 4 | 5 | 1 | |||||
122 | 798 | 1b | 52 | 2 | 4 | 9 | 6 | 1 | |||||
123 | 808 | 1b | 4 | 2 | 2 | 4 | 1 | 2 | |||||
124 | 825 | 1a | 16 | 1 | 2 | 4 | 5 | 1 | |||||
125 | 843 | 2 | 1 | 1 | 2 | 3 | 4 | 1 | |||||
126 | 858 | 1b | 51 | 2 | 4 | 9 | 6 | 1 | |||||
127 | 861 | 1b | 13 | 3 | 4 | 8 | 1 | 1 | |||||
128 | 893 | 2 | 1 | 1 | 1 | 3 | 4 | 1 | |||||
129 | 894 | 1b | 12 | 1 | 2 | 5 | 6 | 1 |
exceptional number
[
type
We conjecture that all the other similarity classes are infinite. Precisely four of them can actually be given by parametrized infinite sequences in a deterministic way aside from the intrinsic probabilistic nature of the occurrence of primes in
No. | M | Factors | S | f 4 | m | VL | VM | VN | E | ( 1,2,4,5 ) | T | P | | M | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
130 | 902 | 2 × 11 × 41 | 1b | 5 2 2 4 11 4 41 4 | 13 | 2 | 3 | 7 | 4 | ( − , − , ( × ) , − ) | β 2 | 11 × 5, K ( 11 ) × K ( 41 ) | 1 |
131 | 924 | 2 2 × 3 × 7 × 11 | 2 | 2 4 3 4 7 4 11 4 | 12 | 1 | 2 | 5 | 6 | ( − , − , × , − ) | γ | 3 × 7,7 × 11 | 1 |
132 | 955 | 5 × 191 | 1a | 5 6 191 4 | 4 | 2* | 3* | 6* | 3 | ( − , − , ⊗ , − ) | α 2 | K , K 1 | 1 |
133 | 957 | 3 × 11 × 29 | 2 | 3 4 11 4 29 4 | 3 | 2 | 2 | 5 | 2 | ( − , ⊗ , ( × ) , − ) | α 3 | K ( 11 ) , K ( 29 ) | 1 |
134 | 982 | 2 × 491 | 2 | 2 4 491 4 | 1 | 1 | 1 | 2 | 3 | ( − , − , ⊗ , − ) | α 2 | K , K 1 | 2 |
residue classes and of composite integers with assigned shape of prime decomposition. This was proved in ( [
Theorem 4.1 Each of the following infinite sequences of conductors
1)
2)
3)
4)
In fact, the shape of the conductors in Theorem 4.1 does not only determine the refined Dedekind species and the DPF type, but also the structure of the 5-class groups of the fields L, M and N.
Corollary 4.1 The invariants of the similarity classes defined by the four infinite sequences of conductors in Theorem 4.1 are given as follows, in the same order:
[
type
[
type
[
type
[
type
The pure metacyclic fields N associated with these four similarity classes are Polya fields.
Remark 4.1 The statements concerning 5-class groups in Corollary 4.1 were proved by Parry in ( [
Proof. (of Theorem 4.1 and Corollary 4.1) It only remains to show the claims for the composite radicands associated with conductors
For similarity classes distinct from the four infinite classes in Theorem 4.1 we cannot provide deterministic criteria for the DPF type and for the homogeneity of multiplets with
Theorem 4.2 Each of the following infinite sequences of conductors
1)
2)
Example 4.1 It is quite easy to find complete quartets, whose members are spread rather widely. The smallest quartet
Corollary 4.2 The invariants of the similarity classes defined by the two infinite sequences of conductors in Theorem 4.2 are given as follows, in the same order. The statements concerning 5-class groups are only conjectural. Each sequence splits in two similarity classes.
The classes for
[
type
[
type
The classes for
[
type
[
type
All pure metacyclic fields N associated with these four similarity classes are Polya fields.
Proof. (of Theorem 4.2 and Corollary 4.2) We use
Remark 4.2. The statements on 5-class groups in Corollary 4.2 have been verified for all examples with
Theorem 4.3 Each of the following infinite sequences of conductors
1)
2)
Example 4.2 It is not difficult to find complete hexadecuplets, whose members are spread rather widely. The smallest hexadecuplet
belonging to the first infinite sequence contains the members
Corollary 4.3 The invariants of the similarity classes defined by the two infinite sequences of conductors in Theorem 4.3 are given as follows, in the same order. The statements concerning 5-class groups are only conjectural. Each sequence splits into two similarity classes.
The classes for
[
type
[
type
The classes for
[
type
[
type
Only the pure metacyclic fields N of type
Proof. (of Theorem 4.3 and Corollary 4.3) See ( [
Theorem 4.4 A pure metacyclic field
Proof. This is an immediate consequence of ( [
Theorem 4.5 A pure metacyclic field
1) N possesses the Polya property.
2)
3) The prime ideal
4) N is of DPF type either
Proof. This is a consequence of ( [
Inspired by the last two theorems, it is worth ones while to summarize, for each kind of prime radicands, what is known about the possibilities for differential principal factorizations.
Theorem 4.6 Let
1) If
2) If
3) If
4) If
5) If
6) If
A pure metacyclic field with prime radicand can never be of any of the types
Proof. By making use of the bounds [
we can determine the possible DPF types of pure metacyclic fields
Firstly, if
Secondly, for a prime radicand
1) If
2) If
3) If
4) If
5) If
6) If
Example 4.3 Concerning numerical realizations of Theorem 4.6, we refer to Corollary 4.1 for the parametrized infinite sequences [
The similarity class [
The similarity classes [
The similarity classes [
The similarity classes [
Although most of the 5-class groups of pure metacyclic fields N, maximal real subfields M and pure quintic subfields L are elementary abelian, there occur sparse examples with non-elementary structure. For instance, we have only 8 occurrences within the range
1)
2)
3)
4)
5)
6)
7)
8)
However, outside the range of systematic computations, we additionally found:
a)
b)
c)
d)
We point out that in all of the last four examples, the normal field N is a Polya field, since the radicands D are primes
Based on the definition of similarity classes and prototypes in §4.2, on the explicit listing of all prototypes in the range between 2 and 103 in Tables 27-30, and on theoretical foundations in §4.3, we are now in the position to establish the intended refinement of our 13 differential principal factorization types into similarity classes in Tables 31-43, as far as the range of our computations for normalized radicands
DPF types are characterized by the multiplet
DPF type
The logarithmic subfield unit index of DPF type
DPF type
The logarithmic subfield unit index of DPF type
DPF type
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 2 | 3 | 5 | 2 | 31 | 2 |
2 | 1a | 6 | 1 | 0 | 1 | 4 | 0 | 0 | 1 | 2 | 3 | 5 | 2 | 155 | 2 |
3 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 3 | 5* | 9* | 2 | 281 | 1 |
4 | 1b | 2 | 2 | 0 | 2 | 3 | 0 | 0 | 2 | 3 | 5 | 9 | 2 | 341 | 2 |
5 | 2 | 0 | 1 | 1' | 0 | 1 | 0 | 0 | 1' | 2 | 3 | 5 | 2 | 401 | 2 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 2 | 3 | 11 | 14 |
2 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 2 | 2 | 4 | 1 | 33 | 8 |
3 | 1a | 6 | 1 | 0 | 1 | 4 | 0 | 0 | 1 | 1 | 1 | 2 | 3 | 55 | 6 |
4 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 1 | 1 | 2 | 3 | 82 | 15 |
5 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 2 | 3 | 6 | 3 | 123 | 2 |
6 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 2 | 2 | 4 | 1 | 131 | 6 |
7 | 2 | 0 | 1 | 1' | 0 | 1 | 0 | 0 | 1' | 1 | 1 | 2 | 3 | 151 | 3 |
8 | 1a | 6 | 1 | 0 | 1 | 4 | 0 | 0 | 1 | 2 | 2 | 4 | 1 | 275 | 2 |
9 | 1b | 2 | 2 | 1 | 1 | 4 | 1 | 0 | 1 | 2 | 2 | 4 | 1 | 287 | 1 |
10 | 2 | 0 | 3 | 0 | 3 | 3 | 1 | 1 | 1 | 2 | 3 | 6 | 3 | 418 | 1 |
11 | 2 | 0 | 2 | 0 | 2 | 1 | 0 | 0 | 2 | 2 | 3 | 6 | 3 | 451 | 1 |
12 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 0 | 1 | 2 | 3 | 6 | 3 | 550 | 1 |
13 | 1b | 2 | 2 | 0 | 2 | 3 | 0 | 0 | 2 | 3 | 4 | 8 | 1 | 671 | 1 |
14 | 2 | 0 | 3 | 0 | 3 | 3 | 1 | 0 | 2 | 2 | 3 | 6 | 3 | 682 | 1 |
15 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 3 | 4 | 8 | 1 | 691 | 1 |
16 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 0 | 1 | 2 | 2 | 4 | 1 | 710 | 4 |
17 | 2 | 0 | 1 | 1' | 0 | 1 | 0 | 0 | 1' | 2 | 2 | 4 | 1 | 751 | 1 |
18 | 1b | 2 | 2 | 0 | 2 | 3 | 0 | 1 | 1 | 3 | 4 | 8 | 1 | 779 | 1 |
19 | 1b | 2 | 2 | 1' | 1 | 4 | 1 | 0 | 1' | 2 | 2 | 4 | 1 | 808 | 2 |
20 | 1b | 2 | 3 | 1 | 2 | 12 | 2 | 0 | 1 | 3 | 4 | 8 | 1 | 861 | 1 |
21 | 1a | 6 | 1 | 0 | 1 | 4 | 0 | 0 | 1 | 2* | 3* | 6* | 3 | 955 | 1 |
22 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 0 | 1 | 1 | 1 | 2 | 3 | 982 | 2 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 2 | 0 | 2 | 3 | 0 | 1 | 1 | 2 | 2 | 5 | 2 | 319 | 3 |
2 | 2 | 0 | 2 | 0 | 2 | 1 | 0 | 2 | 0 | 2 | 2 | 5 | 2 | 551 | 1 |
3 | 1b | 2 | 3 | 0 | 3 | 13 | 1 | 1 | 1 | 3 | 4 | 9 | 2 | 627 | 1 |
4 | 2 | 0 | 2 | 0 | 2 | 1 | 0 | 1 | 1 | 2 | 2 | 5 | 2 | 649 | 2 |
5 | 2 | 0 | 3 | 0 | 3 | 3 | 1 | 1 | 1 | 2 | 2 | 5 | 2 | 957 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 3 | 0 | 3 | 13 | 2 | 0 | 1 | 2 | 3 | 6 | 3 | 186 | 7 |
2 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 2 | 4 | 5 | 191 | 3 |
3 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 1 | 2 | 4 | 5 | 253 | 4 |
4 | 1b | 2 | 2 | 1' | 1 | 4 | 1 | 0 | 1' | 1 | 2 | 4 | 5 | 302 | 2 |
5 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 0 | 1 | 1 | 2 | 4 | 5 | 482 | 2 |
6 | 1b | 2 | 2 | 1' | 1 | 4 | 1 | 0 | 1' | 2* | 4* | 8* | 5 | 502 | 1 |
7 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 2 | 3 | 6 | 3 | 517 | 1 |
8 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 0 | 1 | 2 | 4* | 8* | 5 | 620 | 1 |
9 | 2 | 0 | 3 | 1 | 2 | 4 | 2 | 0 | 1 | 2 | 3 | 6 | 3 | 693 | 1 |
10 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 0 | 1 | 1 | 2 | 4 | 5 | 825 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 1 | 1 | 3 | 4 | 22 | 35 |
2 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 1 | 0 | 1 | 1 | 3 | 4 | 38 | 44 |
3 | 1b | 2 | 2 | 1 | 1 | 4 | 1 | 0 | 1 | 1 | 1 | 3 | 4 | 77 | 7 |
4 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 0 | 1 | 1 | 1 | 3 | 4 | 110 | 11 |
5 | 2 | 0 | 3 | 0 | 3 | 3 | 2 | 0 | 1 | 1 | 1 | 3 | 4 | 132 | 5 |
6 | 1b | 2 | 2 | 1 | 1 | 4 | 1 | 1 | 0 | 1 | 1 | 3 | 4 | 133 | 7 |
7 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 3 | 4 | 139 | 4 |
8 | 1b | 2 | 3 | 1 | 2 | 12 | 2 | 0 | 1 | 2 | 3 | 7 | 4 | 154 | 3 |
9 | 2 | 0 | 3 | 0 | 3 | 3 | 2 | 1 | 0 | 1 | 1 | 3 | 4 | 174 | 3 |
10 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 1 | 0 | 1 | 1 | 3 | 4 | 190 | 9 |
11 | 1b | 2 | 2 | 1' | 1 | 4 | 1 | 0 | 1' | 1 | 1 | 3 | 4 | 202 | 6 |
12 | 1b | 2 | 2 | 0 | 2 | 3 | 0 | 1 | 1 | 2 | 3 | 7 | 4 | 209 | 2 |
13 | 1b | 2 | 3 | 0 | 3 | 13 | 2 | 0 | 1 | 2 | 3 | 7 | 4 | 286 | 3 |
14 | 1a | 6 | 2 | 1 | 1 | 16 | 1 | 0 | 1 | 1 | 1 | 3 | 4 | 385 | 1 |
15 | 1b | 2 | 2 | 1' | 1 | 4 | 1 | 1' | 0 | 1 | 1 | 3 | 4 | 398 | 7 |
16 | 2 | 0 | 3 | 1 | 2 | 4 | 2 | 1 | 0 | 1 | 1 | 3 | 4 | 399 | 2 |
17 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 0 | 1 | 2 | 3 | 7* | 4 | 465 | 1 |
18 | 1b | 2 | 2 | 1 | 1 | 4 | 1 | 0 | 1 | 2 | 3 | 7* | 4 | 473 | 1 |
19 | 2 | 0 | 3 | 1 | 2 | 4 | 2 | 0 | 1 | 1 | 1 | 3 | 4 | 574 | 3 |
20 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 1 | 0 | 2 | 3 | 7* | 4 | 590 | 1 |
21 | 1b | 2 | 3 | 1 | 2 | 12 | 2 | 1 | 0 | 2 | 3 | 7 | 4 | 609 | 1 |
22 | 1b | 2 | 3 | 0 | 3 | 13 | 1 | 1 | 1 | 2 | 3 | 7 | 4 | 638 | 2 |
23 | 1a | 6 | 2 | 1 | 1 | 16 | 1 | 1 | 0 | 1 | 1 | 3 | 4 | 665 | 1 |
24 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 0 | 1 | 1 | 3 | 4 | 893 | 1 |
25 | 1b | 2 | 3 | 0 | 3 | 1. | 1 | 0 | 2 | 2 | 3 | 7 | 4 | 902 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 2 | 0 | 2 | 3 | 2 | 0 | 0 | 0 | 0 | 1 | 6 | 6 | 77 |
2 | 1b | 2 | 2 | 1 | 1 | 4 | 2 | 0 | 0 | 0 | 0 | 1 | 6 | 14 | 44 |
3 | 1a | 6 | 2 | 0 | 2 | 16 | 2 | 0 | 0 | 0 | 0 | 1 | 6 | 30 | 37 |
4 | 1b | 2 | 3 | 1 | 2 | 12 | 3 | 0 | 0 | 1 | 2 | 5 | 6 | 42 | 22 |
5 | 1b | 2 | 3 | 0 | 3 | 13 | 2 | 0 | 1 | 1 | 2 | 5 | 6 | 66 | 17 |
6 | 1a | 6 | 2 | 1 | 1 | 16 | 2 | 0 | 0 | 0 | 0 | 1 | 6 | 70 | 14 |
7 | 1b | 2 | 3 | 0 | 3 | 13 | 3 | 0 | 0 | 1 | 2 | 5 | 6 | 78 | 37 |
8 | 1b | 2 | 3 | 0 | 3 | 13 | 2 | 1 | 0 | 1 | 2 | 5 | 6 | 114 | 20 |
9 | 2 | 0 | 3 | 1 | 2 | 4 | 3 | 0 | 0 | 0 | 0 | 1 | 6 | 126 | 6 |
10 | 1a | 6 | 3 | 1 | 2 | 64 | 3 | 0 | 0 | 1 | 2 | 5 | 6 | 210 | 5 |
11 | 1b | 2 | 3 | 1 | 2 | 12 | 2 | 0 | 1 | 1 | 2 | 5 | 6 | 231 | 5 |
12 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 1 | 0 | 1 | 2 | 5 | 6 | 247 | 2 |
13 | 1b | 2 | 2 | 1 | 1 | 4 | 2 | 0 | 0 | 1 | 2 | 5 | 6 | 259 | 1 |
14 | 1b | 2 | 3 | 1 | 2 | 12 | 2 | 1 | 0 | 1 | 2 | 5 | 6 | 266 | 3 |
15 | 2 | 0 | 3 | 0 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 1 | 6 | 276 | 10 |
16 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 1 | 0 | 1 | 2 | 5 | 6 | 285 | 1 |
17 | 1a | 6 | 3 | 0 | 3 | 64 | 2 | 0 | 1 | 2 | 4 | 9 | 6 | 330 | 2 |
18 | 1a | 6 | 3 | 0 | 3 | 64 | 3 | 0 | 0 | 1 | 2 | 5 | 6 | 390 | 4 |
19 | 1b | 2 | 4 | 1 | 3 | 52 | 3 | 0 | 1 | 2 | 4 | 9 | 6 | 462 | 1 |
20 | 1b | 2 | 4 | 1 | 3 | 25 | 4 | 0 | 0 | 2 | 4 | 9 | 6 | 546 | 3 |
21 | 1a | 6 | 3 | 0 | 3 | 64 | 2 | 1 | 0 | 1 | 2 | 5 | 6 | 570 | 2 |
22 | 1b | 2 | 3 | 2 | 1 | 16 | 3 | 0 | 0 | 1 | 2 | 5 | 6 | 602 | 2 |
23 | 1b | 2 | 3 | 1' | 2 | 12 | 2 | 0 | 1' | 1 | 2 | 5 | 6 | 606 | 2 |
24 | 1a | 6 | 3 | 0 | 3 | 64 | 2 | 0 | 1 | 1 | 2 | 5 | 6 | 660 | 2 |
25 | 1a | 6 | 3 | 1 | 2 | 64 | 2 | 0 | 1 | 1 | 2 | 5 | 6 | 770 | 1 |
26 | 1b | 2 | 4 | 1 | 3 | 52 | 3 | 1 | 0 | 2 | 4 | 9 | 6 | 798 | 1 |
27 | 1b | 2 | 4 | 0 | 4 | 51 | 3 | 0 | 1 | 2 | 4 | 9 | 6 | 858 | 1 |
28 | 1b | 2 | 3 | 1' | 2 | 12 | 2 | 1' | 0 | 1 | 2 | 5 | 6 | 894 | 1 |
29 | 2 | 0 | 4 | 1 | 3 | 12 | 3 | 0 | 1 | 1 | 2 | 5 | 6 | 924 | 1 |
DPF type
The logarithmic subfield unit index of DPF type
DPF type
DPF type
The logarithmic subfield unit index of DPF type
DPF type
unit index
DPF type
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 3 | 5 | 9 | 2 | 211 | 1 |
2 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 2 | 3 | 4 | 421 | 5 |
3 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 0 | 1 | 1 | 2 | 3 | 4 | 843 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 2 | 3 | 19 | 27 |
2 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 0 | 1 | 1 | 2 | 3 | 57 | 10 |
3 | 1a | 6 | 1 | 0 | 1 | 4 | 0 | 1 | 0 | 1 | 1 | 2 | 3 | 95 | 9 |
4 | 2 | 0 | 1 | 1' | 0 | 1 | 0 | 1' | 0 | 1 | 1 | 2 | 3 | 149 | 6 |
5 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 1 | 0 | 2 | 3 | 6 | 3 | 377 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1b | 2 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 5 | 2 | 71 |
2 | 1a | 6 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 5 | 10 | 31 |
3 | 2 | 0 | 2 | 0 | 2 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 5 | 18 | 37 |
4 | 1a | 6 | 2 | 1 | 1 | 16 | 2 | 0 | 0 | 1 | 2 | 4 | 5 | 140 | 5 |
5 | 1b | 2 | 2 | 0 | 2 | 3 | 2 | 0 | 0 | 1 | 2 | 4 | 5 | 141 | 19 |
6 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 1 | 0 | 1 | 2 | 4 | 5 | 171 | 6 |
7 | 1a | 6 | 2 | 0 | 2 | 16 | 2 | 0 | 0 | 1 | 2 | 4 | 5 | 180 | 5 |
8 | 2 | 0 | 3 | 1 | 2 | 4 | 2 | 0 | 0 | 1 | 2 | 4 | 5 | 182 | 1 |
9 | 1b | 2 | 2 | 1 | 1 | 4 | 1 | 1 | 0 | 1 | 2 | 4 | 5 | 203 | 2 |
10 | 2 | 0 | 2 | 0 | 2 | 1 | 1 | 1 | 0 | 1 | 2 | 4 | 5 | 218 | 1 |
11 | 1b | 2 | 3 | 1 | 2 | 12 | 3 | 0 | 0 | 2 | 4 | 8 | 5 | 273 | 4 |
12 | 1a | 6 | 2 | 0 | 2 | 16 | 1 | 1 | 0 | 1 | 2 | 4 | 5 | 290 | 2 |
13 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 1 | 0 | 1 | 2 | 4 | 5 | 298 | 2 |
14 | 1b | 2 | 2 | 1 | 1 | 4 | 2 | 0 | 0 | 1 | 2 | 4 | 5 | 329 | 7 |
15 | 1b | 2 | 3 | 0 | 3 | 13 | 2 | 1 | 0 | 2 | 4 | 8 | 5 | 348 | 2 |
16 | 1b | 2 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 2 | 4 | 5 | 379 | 1 |
17 | 1b | 2 | 2 | 0 | 2 | 3 | 1 | 0 | 1 | 1 | 2 | 4 | 5 | 422 | 6 |
18 | 2 | 0 | 3 | 1 | 2 | 4 | 2 | 1 | 0 | 1 | 2 | 4 | 5 | 532 | 1 |
19 | 1a | 6 | 1 | 0 | 1 | 4 | 0 | 1 | 0 | 1 | 2 | 4 | 5 | 695 | 1 |
20 | 1b | 2 | 3 | 0 | 3 | 13 | 3 | 0 | 0 | 2 | 4 | 8 | 5 | 702 | 2 |
21 | 2 | 0 | 2 | 2 | 0 | 4 | 2 | 0 | 0 | 1 | 2 | 4 | 5 | 749 | 1 |
22 | 1a | 6 | 1 | 1 | 0 | 4 | 1 | 0 | 0 | 1 | 2 | 4 | 5 | 785 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 0 | 1 | 1' | 0 | 1 | 0 | 0 | 1' | 1 | 2 | 4 | 5 | 101 | 1 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1a | 6 | 1 | 1' | 0 | 4 | 0 | 0 | 1' | 1 | 1 | 3 | 4 | 505 | 2 |
2 | 2 | 0 | 2 | 2' | 0 | 4 | 1 | 0 | 1' | 1 | 1 | 3 | 4 | 707 | 1 |
3 | 1a | 6 | 1 | 1 | 0 | 4 | 0 | 1 | 0 | 1 | 1 | 3 | 4 | 745 | 2 |
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1a | 6 | 1 | 1 | 0 | 4 | 1 | 0 | 0 | 0 | 0 | 1 | 6 | 35 | 6 |
2 | 2 | 0 | 2 | 2 | 0 | 4 | 2 | 0 | 0 | 0 | 0 | 1 | 6 | 301 | 1 |
DPF type
In this final section, we want to show that the careful book keeping of similarity classes with representative prototypes in Tables 31-43 is useful for the quantitative illumination of many other phenomena. For an explanation, we select the phenomenon of absolute principal factorizations.
The statistical distribution of DPF types in
For the following investigation, we have to recall that the number T of all prime factors of
Conductors f with
No. | S | e0 | t | u | v | m | n | s2 | s4 | VL | VM | VN | E | M | |M| |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1a | 6 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 5 | 1 |
2 | 2 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 5 | 7 | 18 |
1 case of type
1 case of type
7 cases of type
10 cases of type
126 cases of type
8 cases of type
that is, a total of 153 cases, with respect to the complete range
with respect to the entire database, to
The feature is even aggravated for conductors f with
In this paper, it was our intention to realize Parry’s suggestion ( [
On the one hand, this enabled us to use the Galois cohomology of the unit group UN with respect to the relative automorphism group
On the other hand, our theory of the relatively cyclic quintic Kummer extension N/K as a 5-ring class field modulo the conductor
Equipped with this theoretical background, we were able to develop our Classification Algorithm 3.1 in §3.2, and to prove that it determines the DPF type of L and N in finitely many steps. It also decides whether the normal field N is a Polya field or not. (It is known that L is a (trivial!) Polya field if and only if it possesses class number
The algorithm was implemented as a Magma program script [
Nevertheless, after the completion of the statistics in §4, we came to the conviction that for deeper insight into the arithmetical structure of pure metacyclic fields N, the prime factorization of the class field theoretic conductor f of the abelian extension N/K (invariants
We gratefully acknowledge that our research was supported by the Austrian Science Fund (FWF): project P 26008-N25. This work is dedicated to the memory of Charles J. Parry († 25 December 2010) who suggested a numerical investigation of pure quintic number fields. We are indebted to the anonymous referees for valuable suggestions concerning an improvement of the paper’s layout.
The author declares no conflicts of interest regarding the publication of this paper.
Mayer, D.C. (2019) Tables of Pure Quintic Fields. Advances in Pure Mathematics, 9, 347-403. https://doi.org/10.4236/apm.2019.94017