Pure Quintic Fields 3

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Pure Quintic Fields 3

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§ 7. DPF types of pure quintic fields.

We define thirteen possible differential principal factorization types
of pure quintic number fields L = Q(D1/5),
according to the generators of the group NormN|kUN
as the primary invariant
and the triplet (a,i,r) consisting of
the order a = #Pa of the group of absolute DPF of L|Q,
the index i = (Pi:Pa) in the group of intermediate DPF of M|k+, and
the index r = (Pr:Pi) in the group of relative DPF of N|k,
as the secondary invariant
(similar to but not identical with our definitions in [Ma2]).
The connection between the various quantities is given by the chain of equations
a * i * r = #Pr = (PNGal(N|k):Pk) = (EN|k:UN1 - σ) = 5 (Uk:NormN|kUN).
Note that the unit index (UN:U0) does not enter the definition of the DPF types.

NormN|kUN (Uk:NormN|kUN) (PNGal(N|k):Pk) a i r DPF type
<-1,η5> 25 125 5 1 25 α1
<-1,η5> 25 125 5 5 5 α2
<-1,η5> 25 125 5 25 1 α3
<-1,η5> 25 125 25 1 5 β1
<-1,η5> 25 125 25 5 1 β2
<-1,η5> 25 125 125 1 1 γ
<-1,η> 5 25 5 1 5 δ1
<-1,η> 5 25 5 5 1 δ2
<-1,η> 5 25 25 1 1 ε
<-1,ζ,η5> 5 25 5 1 5 ζ1
<-1,ζ,η5> 5 25 5 5 1 ζ2
<-1,ζ,η5> 5 25 25 1 1 η
<-1,ζ,η> 1 5 5 1 1 θ

§ 8. Computational results.

With the aid of PARI/GP and MAGMA we have determined the DPF type
and other invariants of the 90 pure quintic number fields L = Q(D1/5)
with normalized radicands D = q1e1 … qses
(minimal among the powers Dn, 1 ≤ n ≤ 4, with corresponding ejs reduced mod 5)
in the range 2 ≤ D ≤ 110 (date of completion: Jan 11th, 2014).
Prime factors are given for composite D only
(qj ≡ + 1 (mod 5) in red, qj ≡ - 1 (mod 5) in magenta, qj ≡ ± 7 (mod 25) in green).
The first kind of fields is refined by distinguishing 5|D (kind Ia) and (5,D) = 1 (kind Ib).
The exponent of the power in the unit index (UN:U0) = 5e is denoted by e.
The unit norm index (Uk:NormN|kUN) is abbreviated by u.
The symbol Cl briefly denotes the 5-class group Cl5 of a number field.
An asterisk (*) denotes the smallest radicand with given type and 5-class groups.

No. D factors kind Cl(L) Cl(M) Cl(N) e u DPF type *
1 2 Ib 1 1 1 5 5 ε *
2 3 Ib 1 1 1 5 5 ε
3 5 Ia 1 1 1 5 1 θ *
4 6 2,3 Ib 1 1 (5) 6 25 γ *
5 7 II 1 1 1 5 1 θ
6 10 2,5 Ia 1 1 1 5 5 ε
7 11 Ib (5) (5) (5,5) 3 25 α2 *
8 12 3,22 Ib 1 1 (5) 6 25 γ
9 13 Ib 1 1 1 5 5 ε
10 14 2,7 Ib 1 1 (5) 6 25 γ
11 15 3,5 Ia 1 1 1 5 5 ε
12 17 Ib 1 1 1 5 5 ε
13 18 2,32 II 1 1 1 5 5 ε
14 19 Ib (5) (5) (5,5) 3 5 δ2 *
15 20 5,22 Ia 1 1 1 5 5 ε
16 21 3,7 Ib 1 1 (5) 6 25 γ
17 22 2,11 Ib (5) (5) (5,5,5) 4 25 β2 *
18 23 Ib 1 1 1 5 5 ε
19 26 2,13 II 1 1 1 5 5 ε
20 28 7,22 Ib 1 1 (5) 6 25 γ
21 29 Ib (5) (5) (5,5) 3 5 δ2
22 30 2,3,5 Ia 1 1 (5) 6 25 γ
23 31 Ib (5,5) (5,5,5) (5,5,5,5,5) 2 25 α1 *
24 33 3,11 Ib (5,5) (5,5) (5,5,5,5) 1 25 α2 *
25 34 2,17 Ib 1 1 (5) 6 25 γ
26 35 5,7 Ia 1 1 (5) 6 5 η *
27 37 Ib 1 1 1 5 5 ε
28 38 2,19 Ib (5) (5) (5,5,5) 4 25 β2
29 39 3,13 Ib 1 1 (5) 6 25 γ
30 40 5,23 Ia 1 1 1 5 5 ε
31 41 Ib (5) (5) (5,5) 3 25 α2
32 42 2,3,7 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ *
33 43 II 1 1 1 5 1 θ
34 44 11,22 Ib (5) (5) (5,5,5) 4 25 β2
35 45 5,32 Ia 1 1 1 5 5 ε
36 46 2,23 Ib 1 1 (5) 6 25 γ
37 47 Ib 1 1 1 5 5 ε
38 48 3,24 Ib 1 1 (5) 6 25 γ
39 51 3,17 II 1 1 1 5 5 ε
40 52 13,22 Ib 1 1 (5) 6 25 γ
41 53 Ib 1 1 1 5 5 ε
42 55 5,11 Ia (5) (5) (5,5) 3 25 α2
43 56 7,23 Ib 1 1 (5) 6 25 γ
44 57 3,19 II (5) (5) (5,5) 3 5 δ2
45 58 2,29 Ib (5) (5) (5,5,5) 4 25 β2
46 59 Ib (5) (5) (5,5) 3 5 δ2
47 60 3,5,22 Ia 1 1 (5) 6 25 γ
48 61 Ib (5) (5) (5,5) 3 25 α2
49 62 2,31 Ib (5) (5) (5,5,5) 4 25 β2
50 63 7,32 Ib 1 1 (5) 6 25 γ
51 65 5,13 Ia 1 1 1 5 5 ε
52 66 2,3,11 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ
53 67 Ib 1 1 1 5 5 ε
54 68 17,22 II 1 1 1 5 5 ε
55 69 3,23 Ib 1 1 (5) 6 25 γ
56 70 2,5,7 Ia 1 1 (5) 6 25 γ
57 71 Ib (5) (5) (5,5) 3 25 α2
58 73 Ib 1 1 1 5 5 ε
59 74 2,37 II 1 1 1 5 5 ε
60 75 3,52 Ia 1 1 1 5 5 ε
61 76 19,22 II (5) (5) (5,5) 3 5 δ2
62 77 7,11 Ib (5) (5) (5,5,5) 4 25 β2
63 78 2,3,13 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ
64 79 Ib (5) (5) (5,5) 3 5 δ2
65 80 5,24 Ia 1 1 1 5 5 ε
66 82 2,41 II (5) (5) (5,5) 3 25 α2
67 83 Ib 1 1 1 5 5 ε
68 84 3,7,22 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ
69 85 5,17 Ia 1 1 1 5 5 ε
70 86 2,43 Ib 1 1 (5) 6 25 γ
71 87 3,29 Ib (5) (5) (5,5,5) 4 25 β2
72 88 11,23 Ib (5,5) (5,5) (5,5,5,5) 1 25 α2
73 89 Ib (5) (5) (5,5) 3 5 δ2
74 90 2,5,32 Ia 1 1 (5) 6 25 γ
75 91 7,13 Ib 1 1 (5) 6 25 γ
76 92 23,22 Ib 1 1 (5) 6 25 γ
77 93 3,31 II (5) (5) (5,5) 3 25 α2
78 94 2,47 Ib 1 1 (5) 6 25 γ
79 95 5,19 Ia (5) (5) (5,5) 3 5 δ2
80 97 Ib 1 1 1 5 5 ε
81 99 11,32 II (5) (5) (5,5) 3 25 α2
82 101 II (5) (5,5) (5,5,5,5) 5 5 ζ1 *
83 102 2,3,17 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ
84 103 Ib 1 1 1 5 5 ε
85 104 13,23 Ib 1 1 (5) 6 25 γ
86 105 3,5,7 Ia 1 1 (5) 6 25 γ
87 106 2,53 Ib 1 1 (5) 6 25 γ
88 107 II 1 1 1 5 1 θ
89 109 Ib (5) (5) (5,5) 3 5 δ2
90 110 2,5,11 Ia (5) (5) (5,5,5) 4 25 β2

On Jan 13th, 2014, we additionally determined the DPF type
and other invariants of the 35 pure quintic number fields L = Q(D1/5)
with normalized radicands in the range 111 ≤ D ≤ 150.

No. D factors kind Cl(L) Cl(M) Cl(N) e u DPF type *
91 111 3,37 Ib 1 1 (5) 6 25 γ
92 112 7,24 Ib 1 1 (5) 6 25 γ
93 113 Ib 1 1 1 5 5 ε
94 114 2,3,19 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ
95 115 5,23 Ia 1 1 1 5 5 ε
96 116 29,22 Ib (5) (5) (5,5,5) 4 25 β2
97 117 13,32 Ib 1 1 (5) 6 25 γ
98 118 2,59 II (5) (5) (5,5) 3 5 δ2
99 119 7,17 Ib 1 1 (5) 6 25 γ
100 120 3,5,23 Ia 1 1 (5) 6 25 γ
101 122 2,61 Ib (5) (5) (5,5,5) 4 25 β2
102 123 3,41 Ib (5,5) (5,5,5) (5,5,5,5,5,5) 3 25 α2 *
103 124 31,22 II (5) (5) (5,5) 3 25 α2
104 126 2,7,32 II 1 1 (5) 6 25 γ
105 127 Ib 1 1 1 5 5 ε
106 129 3,43 Ib 1 1 (5) 6 25 γ
107 130 2,5,13 Ia 1 1 (5) 6 25 γ
108 131 Ib (5,5) (5,5) (5,5,5,5) 1 25 α2
109 132 3,11,22 II (5) (5) (5,5,5) 4 25 β2
110 133 7,19 Ib (5) (5) (5,5,5) 4 25 β2
111 134 2,67 Ib 1 1 (5) 6 25 γ
112 136 17,23 Ib 1 1 (5) 6 25 γ
113 137 Ib 1 1 1 5 5 ε
114 138 2,3,23 Ib (5) (5,5) (5,5,5,5,5) 6 25 γ
115 139 Ib (5) (5) (5,5,5) 4 25 β2
116 140 5,7,22 Ia (5) (5,5) (5,5,5,5) 5 5 ε *
117 141 3,47 Ib (5) (5,5) (5,5,5,5) 5 5 ε
118 142 2,71 Ib (5) (5) (5,5,5) 4 25 β2
119 143 11,13 II (5) (5) (5,5) 3 25 α2
120 145 5,29 Ia (5) (5) (5,5) 3 5 δ2
121 146 2,73 Ib 1 1 (5) 6 25 γ
122 147 3,72 Ib 1 1 (5) 6 25 γ
123 148 37,22 Ib 1 1 (5) 6 25 γ
124 149 II (5) (5) (5,5) 3 5 δ2
125 150 2,3,52 Ia 1 1 (5) 6 25 γ

Summarized, we finally obtain the following Statistics of DPF types,
expressed by absolute frequencies and relative frequencies in percent (%).
Arrows indicate monotonic tendencies (↑ … increasing, ↓ … descending).

D ≤ tot ε γ α1 α2 β2 δ2 θ η ζ1 Cl(N)=1 Cl(N)=(5)
50 38 14 11 1 3 3 2 3 1 0 17 11
37 29 8 8 5 8 44 29
100 81 26 25 1 10 7 8 3 1 0 29 22
32 31 12 9 10 4 36 27
150 125 33 45 1 14 14 12 4 1 1 35 39
26↓ 36↑ 11 11↑ 10 3↓ 28↓ 31

The missing types have been found by us later in 2018:
type β1 occurs for D = 186 = 2*3*31,
type δ1 occurs for D = 211 (prime),
type α3 occurs for D = 319 = 11*29, and
type ζ2 did not appear until D = 505 = 5*101.

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