N
|
Maximal Class
|
|
Second Maximal Class
|
Lower Class
|
Tot
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
1
|
4
|
|
1
|
|
|
5
|
4
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5
|
2
|
8
|
2
|
1
|
|
|
11
|
10
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
11
|
3
|
10
|
2
|
1
|
|
|
13
|
12
|
|
|
|
1
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
14
|
4
|
5
|
4
|
1
|
|
|
10
|
9
|
|
|
|
|
|
|
|
|
1
|
|
|
|
1
|
|
|
|
|
|
|
11
|
5
|
10
|
5
|
|
1
|
|
16
|
16
|
1
|
|
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
17
|
6
|
10
|
4
|
3
|
|
|
17
|
16
|
2
|
|
|
|
1
|
|
|
1
|
|
|
|
|
4
|
|
|
|
|
|
|
21
|
7
|
8
|
4
|
|
|
|
12
|
14
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
14
|
8
|
9
|
5
|
1
|
1
|
|
16
|
14
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
1
|
17
|
9
|
10
|
2
|
1
|
|
|
13
|
15
|
1
|
2
|
|
|
|
|
|
|
|
|
|
|
3
|
|
|
|
|
|
|
16
|
10
|
12
|
6
|
2
|
|
|
20
|
18
|
|
|
1
|
1
|
|
|
|
1
|
|
|
|
|
3
|
|
|
|
|
|
|
23
|
10
|
86
|
34
|
11
|
2
|
|
133
|
128
|
5
|
3
|
1
|
2
|
1
|
|
|
2
|
1
|
|
|
|
15
|
1
|
|
|
|
|
1
|
149
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
11
|
10
|
4
|
|
2
|
|
16
|
16
|
|
2
|
|
|
1
|
|
|
|
|
|
|
1
|
4
|
|
|
|
|
|
|
20
|
12
|
8
|
2
|
2
|
|
|
12
|
11
|
|
1
|
|
|
|
|
1
|
|
|
|
|
|
2
|
|
|
|
|
|
|
14
|
13
|
9
|
3
|
|
|
|
12
|
12
|
|
|
|
|
|
|
|
1
|
|
|
|
|
1
|
|
|
|
|
|
|
13
|
14
|
14
|
5
|
4
|
|
|
23
|
20
|
1
|
|
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
24
|
15
|
12
|
5
|
1
|
1
|
|
19
|
17
|
|
|
|
|
|
|
|
|
1
|
|
|
|
1
|
|
|
|
|
|
|
20
|
16
|
8
|
2
|
2
|
1
|
|
13
|
12
|
2
|
|
1
|
|
|
|
|
|
|
|
|
|
3
|
|
1
|
|
|
|
1
|
17
|
17
|
15
|
8
|
|
|
|
23
|
26
|
2
|
1
|
|
1
|
|
|
|
1
|
|
|
|
|
5
|
|
|
|
|
|
|
28
|
18
|
14
|
8
|
2
|
2
|
|
26
|
23
|
1
|
|
1
|
|
|
|
|
|
1
|
|
|
|
3
|
|
|
|
|
|
|
29
|
19
|
19
|
6
|
1
|
|
|
26
|
27
|
2
|
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
28
|
20
|
9
|
5
|
1
|
|
|
15
|
16
|
1
|
1
|
|
|
|
|
|
1
|
|
|
|
|
3
|
|
|
|
|
|
|
18
|
20
|
204
|
82
|
24
|
8
|
|
318
|
308
|
14
|
8
|
3
|
3
|
2
|
|
1
|
5
|
3
|
|
|
1
|
40
|
1
|
1
|
|
|
|
2
|
360
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
21
|
14
|
4
|
3
|
1
|
|
22
|
19
|
|
1
|
1
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
24
|
22
|
15
|
3
|
2
|
|
|
20
|
20
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
22
|
23
|
10
|
4
|
|
|
|
14
|
14
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
14
|
24
|
10
|
4
|
1
|
|
|
15
|
17
|
3
|
|
1
|
1
|
|
|
|
|
|
|
|
|
5
|
|
1
|
|
|
|
1
|
21
|
25
|
13
|
4
|
2
|
|
|
19
|
17
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
19
|
26
|
13
|
7
|
2
|
2
|
|
24
|
24
|
3
|
1
|
|
|
|
|
|
|
|
|
|
|
4
|
|
|
|
|
|
|
28
|
27
|
8
|
6
|
|
1
|
|
15
|
15
|
1
|
|
|
|
1
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
17
|
28
|
12
|
9
|
1
|
1
|
|
23
|
21
|
|
|
|
|
2
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
25
|
29
|
17
|
7
|
1
|
1
|
|
26
|
25
|
1
|
|
1
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
28
|
30
|
22
|
4
|
1
|
1
|
1
|
29
|
26
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
29
|
30
|
338
|
134
|
37
|
15
|
1
|
525
|
506
|
23
|
11
|
6
|
4
|
5
|
|
1
|
5
|
3
|
|
|
1
|
59
|
1
|
2
|
|
|
|
3
|
587
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
31
|
13
|
15
|
1
|
|
|
29
|
29
|
1
|
|
|
|
|
|
|
1
|
|
|
|
|
2
|
|
|
|
|
|
|
31
|
32
|
16
|
7
|
2
|
|
|
25
|
24
|
1
|
|
|
|
1
|
|
1
|
|
|
|
|
|
3
|
|
|
|
|
|
|
28
|
33
|
7
|
6
|
1
|
|
|
14
|
16
|
3
|
|
|
|
2
|
|
|
1
|
|
|
|
|
6
|
|
|
|
|
|
|
20
|
34
|
11
|
8
|
2
|
|
|
21
|
21
|
2
|
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
23
|
35
|
16
|
7
|
|
|
|
23
|
23
|
|
|
1
|
1
|
|
|
|
|
|
|
|
|
2
|
1
|
|
|
|
|
1
|
26
|
36
|
18
|
5
|
|
|
|
23
|
23
|
|
|
1
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
24
|
37
|
13
|
6
|
3
|
2
|
|
24
|
21
|
2
|
|
|
|
|
|
|
1
|
|
|
|
|
3
|
|
|
|
|
|
|
27
|
38
|
17
|
5
|
2
|
|
|
24
|
24
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
26
|
39
|
12
|
4
|
2
|
|
|
18
|
17
|
1
|
|
1
|
|
|
|
|
|
|
|
|
|
2
|
1
|
|
|
|
|
1
|
21
|
40
|
18
|
8
|
1
|
|
|
27
|
28
|
|
2
|
|
|
1
|
1
|
|
|
|
|
|
|
4
|
|
|
|
|
|
|
31
|
40
|
479
|
205
|
51
|
17
|
1
|
753
|
732
|
34
|
14
|
9
|
5
|
9
|
1
|
2
|
8
|
3
|
|
|
1
|
86
|
3
|
2
|
|
|
|
5
|
844
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
41
|
9
|
6
|
1
|
|
|
16
|
16
|
|
1
|
1
|
1
|
1
|
|
1
|
|
|
|
|
|
5
|
|
|
|
|
|
|
21
|
42
|
22
|
4
|
3
|
|
|
29
|
28
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
31
|
43
|
17
|
7
|
3
|
3
|
|
30
|
26
|
2
|
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
32
|
44
|
17
|
9
|
|
1
|
|
27
|
28
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
29
|
45
|
13
|
6
|
2
|
|
|
21
|
21
|
2
|
|
1
|
|
|
|
|
1
|
|
|
|
|
4
|
1
|
|
|
|
|
1
|
26
|
46
|
14
|
9
|
2
|
1
|
|
26
|
23
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
Gap
|
27
|
47
|
8
|
5
|
2
|
|
|
15
|
13
|
|
|
|
1
|
|
|
|
|
|
|
|
|
1
|
16
|
48
|
14
|
7
|
3
|
|
|
24
|
24
|
2
|
1
|
1
|
|
|
|
|
|
1
|
|
|
|
5
|
29
|
49
|
17
|
4
|
|
|
|
21
|
22
|
|
1
|
|
|
|
|
|
|
1
|
|
|
|
2
|
23
|
50
|
14
|
5
|
1
|
|
|
20
|
20
|
1
|
|
1
|
|
|
|
|
|
|
|
|
|
2
|
22
|
50
|
624
|
267
|
68
|
22
|
1
|
982
|
953
|
43
|
19
|
13
|
7
|
11
|
1
|
3
|
9
|
5
|
|
|
1
|
112
|
4
|
2
|
|
|
|
6
|
1100
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
51
|
7
|
9
|
2
|
2
|
|
20
|
20
|
3
|
1
|
1
|
|
|
|
|
|
|
|
|
|
5
|
Gap
|
25
|
52
|
16
|
8
|
3
|
|
|
27
|
24
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
28
|
53
|
13
|
4
|
1
|
1
|
|
19
|
17
|
|
|
|
|
1
|
1
|
|
|
|
|
|
|
2
|
21
|
54
|
22
|
8
|
4
|
2
|
|
36
|
33
|
3
|
|
|
|
1
|
|
|
|
|
|
|
|
4
|
40
|
55
|
11
|
5
|
2
|
|
|
18
|
16
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
19
|
56
|
13
|
8
|
|
1
|
|
22
|
21
|
|
|
1
|
|
|
|
|
2
|
|
|
|
|
3
|
25
|
57
|
14
|
7
|
|
1
|
|
22
|
22
|
1
|
|
|
|
|
|
|
|
|
|
|
|
1
|
23
|
58
|
16
|
8
|
|
|
|
24
|
25
|
1
|
|
|
|
|
|
|
1
|
|
|
|
|
2
|
26
|
59
|
10
|
7
|
2
|
|
|
19
|
19
|
2
|
|
|
1
|
|
|
|
|
|
|
|
|
3
|
22
|
60
|
16
|
7
|
2
|
|
|
25
|
25
|
2
|
|
|
|
1
|
|
|
|
1
|
|
|
|
4
|
29
|
60
|
762
|
338
|
84
|
29
|
1
|
1214
|
1175
|
55
|
20
|
15
|
8
|
16
|
2
|
3
|
12
|
6
|
|
|
1
|
138
|
4
|
2
|
|
|
|
6
|
1358
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
61
|
15
|
4
|
|
1
|
|
20
|
23
|
2
|
2
|
|
|
|
|
|
3
|
1
|
|
|
|
8
|
Gap
|
28
|
62
|
12
|
6
|
2
|
1
|
|
21
|
20
|
1
|
1
|
|
|
1
|
|
|
|
|
|
|
|
3
|
24
|
63
|
21
|
9
|
|
1
|
|
31
|
32
|
2
|
|
|
|
1
|
|
|
1
|
|
|
|
|
4
|
35
|
64
|
13
|
8
|
1
|
2
|
|
24
|
22
|
|
1
|
1
|
|
|
|
|
1
|
|
|
|
|
3
|
27
|
65
|
15
|
9
|
1
|
2
|
|
27
|
24
|
|
|
2
|
1
|
|
|
|
|
|
|
|
|
3
|
30
|
66
|
16
|
9
|
|
2
|
|
27
|
26
|
1
|
|
1
|
|
|
|
|
1
|
|
|
|
|
3
|
30
|
67
|
12
|
12
|
2
|
|
|
26
|
26
|
1
|
1
|
|
|
|
|
|
|
1
|
|
|
|
3
|
29
|
68
|
15
|
8
|
1
|
1
|
|
25
|
25
|
2
|
|
1
|
|
|
|
|
1
|
|
|
|
|
4
|
29
|
69
|
13
|
8
|
2
|
|
|
23
|
22
|
1
|
|
1
|
|
|
|
|
|
|
|
|
|
2
|
25
|
70
|
10
|
5
|
3
|
|
|
18
|
15
|
|
|
|
|
|
1
|
|
|
|
|
|
|
1
|
19
|
70
|
904
|
416
|
96
|
39
|
1
|
1456
|
1410
|
65
|
25
|
21
|
9
|
18
|
3
|
3
|
19
|
8
|
|
|
1
|
172
|
4
|
2
|
|
|
|
6
|
1634
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
71
|
19
|
7
|
3
|
2
|
|
31
|
26
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Gap
|
31
|
72
|
15
|
10
|
|
|
|
25
|
25
|
|
|
1
|
1
|
1
|
1
|
|
|
1
|
|
|
|
5
|
30
|
73
|
12
|
6
|
1
|
1
|
|
20
|
20
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
22
|
74
|
11
|
11
|
2
|
|
|
24
|
25
|
3
|
|
|
|
|
|
|
|
1
|
|
|
|
4
|
28
|
75
|
13
|
13
|
3
|
2
|
|
31
|
27
|
|
1
|
|
|
|
|
|
|
|
|
|
|
1
|
32
|
76
|
16
|
11
|
1
|
2
|
|
30
|
30
|
2
|
1
|
|
1
|
|
|
|
1
|
|
|
|
|
5
|
35
|
77
|
23
|
8
|
4
|
1
|
|
36
|
33
|
|
2
|
|
|
|
|
|
|
|
|
|
|
2
|
38
|
78
|
18
|
13
|
2
|
|
|
33
|
33
|
1
|
1
|
1
|
|
3
|
|
|
|
|
|
|
|
6
|
39
|
79
|
12
|
10
|
1
|
1
|
|
24
|
24
|
2
|
|
|
|
1
|
|
|
|
|
|
|
|
3
|
27
|
80
|
16
|
7
|
1
|
|
|
24
|
23
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
25
|
80
|
1059
|
512
|
114
|
48
|
1
|
1734
|
1676
|
74
|
31
|
23
|
11
|
24
|
4
|
3
|
20
|
10
|
|
|
1
|
201
|
4
|
2
|
|
|
|
6
|
1941
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
81
|
11
|
5
|
1
|
1
|
|
18
|
17
|
1
|
|
2
|
|
|
|
|
|
1
|
|
|
|
4
|
Gap
|
22
|
82
|
16
|
13
|
1
|
1
|
|
31
|
31
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
2
|
|
|
|
1
|
3
|
36
|
83
|
13
|
6
|
3
|
|
|
22
|
20
|
1
|
|
|
|
1
|
|
|
2
|
|
|
|
|
4
|
|
|
|
|
|
|
26
|
84
|
25
|
15
|
2
|
1
|
|
43
|
42
|
1
|
1
|
1
|
|
|
|
|
|
|
|
|
|
3
|
|
|
1
|
|
|
1
|
47
|
85
|
13
|
11
|
|
1
|
|
25
|
25
|
1
|
|
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
1
|
|
1
|
27
|
86
|
14
|
7
|
2
|
3
|
|
26
|
22
|
|
1
|
|
|
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
27
|
87
|
19
|
6
|
3
|
1
|
|
29
|
27
|
1
|
1
|
|
|
|
|
|
|
1
|
|
|
|
3
|
|
|
|
|
|
|
32
|
88
|
18
|
7
|
3
|
1
|
|
29
|
28
|
2
|
1
|
|
|
1
|
|
|
|
|
1
|
|
|
5
|
|
|
|
|
|
|
34
|
89
|
23
|
9
|
1
|
2
|
|
35
|
34
|
1
|
1
|
|
|
1
|
1
|
|
|
|
|
|
|
4
|
|
|
|
|
|
|
39
|
90
|
14
|
13
|
2
|
2
|
|
31
|
29
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
33
|
90
|
1225
|
604
|
132
|
61
|
1
|
2023
|
1951
|
84
|
38
|
26
|
11
|
27
|
5
|
3
|
22
|
12
|
1
|
|
1
|
230
|
6
|
2
|
1
|
1
|
1
|
11
|
2264
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
91
|
21
|
7
|
3
|
1
|
|
32
|
28
|
|
|
|
|
|
|
|
1
|
|
|
|
|
1
|
|
|
|
|
|
|
33
|
92
|
14
|
13
|
3
|
1
|
|
31
|
30
|
2
|
1
|
|
|
|
|
|
1
|
1
|
|
|
|
5
|
|
|
|
|
|
|
36
|
93
|
16
|
11
|
|
|
|
27
|
27
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
27
|
94
|
18
|
8
|
3
|
|
|
29
|
30
|
1
|
3
|
|
|
|
|
|
|
|
|
|
|
4
|
|
|
|
|
|
|
33
|
95
|
13
|
9
|
1
|
|
|
23
|
24
|
1
|
1
|
1
|
|
|
1
|
|
|
|
1
|
|
|
5
|
|
|
|
|
|
|
28
|
96
|
24
|
11
|
1
|
3
|
|
39
|
35
|
|
|
|
|
1
|
|
|
|
|
|
|
|
1
|
|
|
|
|
|
|
40
|
97
|
17
|
6
|
2
|
2
|
|
27
|
26
|
2
|
1
|
|
|
|
|
|
|
1
|
|
|
|
4
|
|
|
|
|
|
|
31
|
98
|
6
|
10
|
|
3
|
|
19
|
18
|
1
|
1
|
|
|
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
21
|
99
|
12
|
9
|
2
|
|
|
23
|
24
|
2
|
1
|
|
|
|
|
|
|
|
|
|
|
3
|
1
|
|
|
|
|
1
|
27
|
100
|
20
|
9
|
|
1
|
|
30
|
30
|
|
1
|
|
|
1
|
1
|
|
1
|
|
|
|
1
|
5
|
1
|
|
|
|
|
1
|
36
|
100
|
1386
|
697
|
147
|
72
|
1
|
2303
|
2223
|
93
|
47
|
27
|
11
|
29
|
7
|
3
|
25
|
14
|
2
|
|
2
|
260
|
8
|
2
|
1
|
1
|
1
|
13
|
2576
|
|
a2/3
|
a3*
|
a1
|
a2/3↑
|
a1↑
|
Max
|
Two
|
D10
|
D5
|
H4
|
G19
|
c18
|
E6/14
|
H4↑
|
c21
|
E8/9
|
G16
|
c18↑
|
c21↑
|
Sec
|
b.10
|
d
|
F
|
d*
|
F↑
|
Low
|
|
N
|
Maximal Class
|
|
Second Maximal Class
|
Lower Class
|
Tot
|
Second 3-class groups G = Gal(K2|K)
of lower than second maximal class
show an astonishing and remarkable gap of length greater than 3.6 millions
for discriminants 4.5*106 < d < 8.1*106,
followed by a real explosion between 8.1*106 and 8.5*106,
where the transfer kernel types F.13↑, F.13, and d.25* appear for the first time.
The latest of all occurrences is revealed by transfer kernel type G.16 at d = 8711453.
|
*
|
Relative frequency of second 3-class groups
and two-stage towers
A comparison of the absolute intervals
0 < d < 106 and 0 < d < 107
shows that all frequencies remain rather stable.
Groups of maximal class (e = 2) [Bl], [Mi] are clearly dominating
with 133 / 149 = 89% and 2303 / 2576 = 89.4%,
the contribution of groups of second maximal class (e = 3) [Ne] remains modest
with 15 / 149 = 10% and 260 / 2576 = 10.1%,
and the groups of lower than second maximal class (e ≥ 4) are almost negligible
with 1 / 149 = 0.7% and 13 / 2576 = 0.5%.
The broad subset of groups with two-stage towers consists of
groups of maximal class (type a.3* and the ground states of types a.2 and a.3)
and groups of second maximal class (types D.5 and D.10)
with 128 / 149 = 86% and 2223 / 2576 = 86.3%.
From the computational point of view, all groups of maximal class
and the groups with transfer kernel types D.10, D.5, H.4, G.19, c.18, c.21,
with constant frequency of 147 / 149 = 98.7% and 2537 / 2576 = 98.5%,
can be identified already by the
structure of the 3-class groups
of the four unramified cyclic cubic extension fields N1,…,N4 of K
and thus considerably faster,
since for the other types the structure of the 3-class group of the
Hilbert 3-class field of K, which is of absolute degree 18, must be determined.
The value 98.5% in comparison to 65.7% for complex fields
means that the classification of real fields is much less time consuming.
|
*
|
Relative frequency of totally real cubic fields with fundamental discriminant
and various 3-class numbers
For each of the 2576 real quadratic fields K with discriminant d
we have four totally real cubic fields L1,…,L4 with fundamental discriminant d.
The total number of these fields is therefore 10304.
By a theorem of F. Gerth III [Ge],
their 3-class groups Cl3(Li) are cyclic and non-trivial.
The clearly dominating part of 9976 / 10304 = 96.8% has 3-class number h3(Li) = 3,
a very modest part of 323 / 10304 = 3.1% has 3-class number h3(Li) = 9,
and a negligible part of 5 / 10304 = 0.1% has 3-class number h3(Li) = 27.
|
*
|
Relative frequency of totally real S3-fields
with exotic 3-class groups of type (3,3,3)
For each of the 2576 real quadratic fields K with discriminant d
we have four totally real S3-fields N1,…,N4 with discriminant d3.
The total number of these fields is therefore 10304.
1030 / 10304 = 10.0% of these fields have an exotic 3-class group of type (3,3,3).
Every other field has an almost homogeneous 3-class group of type (3q+r,3q) with 0 ≤ r ≤ 1.
|
|