Normalized defining polynomial
\( x^{25} - 5 x^{24} + 10 x^{23} - 5 x^{22} - 25 x^{21} + 80 x^{20} - 120 x^{19} + 95 x^{18} - 155 x^{16} + \cdots + 1 \)
Invariants
| Degree: | $25$ |
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| Signature: | $(1, 12)$ |
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| Discriminant: |
\(136202515664062500000000000000000000\)
\(\medspace = 2^{20}\cdot 3^{20}\cdot 5^{28}\)
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| |
| Root discriminant: | \(25.43\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5}a^{20}-\frac{1}{5}a^{15}+\frac{1}{5}a^{10}-\frac{1}{5}a^{5}+\frac{1}{5}$, $\frac{1}{5}a^{21}-\frac{1}{5}a^{16}+\frac{1}{5}a^{11}-\frac{1}{5}a^{6}+\frac{1}{5}a$, $\frac{1}{5}a^{22}-\frac{1}{5}a^{17}+\frac{1}{5}a^{12}-\frac{1}{5}a^{7}+\frac{1}{5}a^{2}$, $\frac{1}{95}a^{23}+\frac{3}{95}a^{22}-\frac{9}{95}a^{21}+\frac{3}{95}a^{20}+\frac{5}{19}a^{19}-\frac{1}{95}a^{18}+\frac{32}{95}a^{17}+\frac{14}{95}a^{16}+\frac{47}{95}a^{15}-\frac{4}{19}a^{14}+\frac{6}{95}a^{13}-\frac{47}{95}a^{12}-\frac{1}{5}a^{11}-\frac{37}{95}a^{10}-\frac{3}{19}a^{9}-\frac{26}{95}a^{8}+\frac{2}{95}a^{7}+\frac{4}{95}a^{6}+\frac{37}{95}a^{5}-\frac{1}{19}a^{4}-\frac{29}{95}a^{3}-\frac{22}{95}a^{2}+\frac{31}{95}a+\frac{23}{95}$, $\frac{1}{64\cdots 45}a^{24}+\frac{65\cdots 87}{64\cdots 45}a^{23}-\frac{32\cdots 14}{64\cdots 45}a^{22}-\frac{39\cdots 88}{64\cdots 45}a^{21}+\frac{82\cdots 41}{64\cdots 45}a^{20}+\frac{78\cdots 19}{64\cdots 45}a^{19}+\frac{21\cdots 48}{64\cdots 45}a^{18}+\frac{17\cdots 99}{64\cdots 45}a^{17}-\frac{30\cdots 02}{64\cdots 45}a^{16}-\frac{31\cdots 91}{64\cdots 45}a^{15}+\frac{98\cdots 36}{64\cdots 45}a^{14}-\frac{63\cdots 03}{64\cdots 45}a^{13}+\frac{13\cdots 56}{64\cdots 45}a^{12}-\frac{88\cdots 18}{64\cdots 45}a^{11}+\frac{30\cdots 91}{64\cdots 45}a^{10}-\frac{13\cdots 56}{64\cdots 45}a^{9}-\frac{68\cdots 22}{64\cdots 45}a^{8}+\frac{14\cdots 44}{64\cdots 45}a^{7}-\frac{20\cdots 47}{64\cdots 45}a^{6}-\frac{32\cdots 01}{64\cdots 45}a^{5}-\frac{14\cdots 59}{64\cdots 45}a^{4}+\frac{16\cdots 17}{64\cdots 45}a^{3}-\frac{22\cdots 09}{64\cdots 45}a^{2}-\frac{12\cdots 68}{64\cdots 45}a-\frac{10\cdots 74}{64\cdots 45}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
|
Unit group
| Rank: | $12$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$a^{24}-5a^{23}+10a^{22}-5a^{21}-25a^{20}+80a^{19}-120a^{18}+95a^{17}-155a^{15}+420a^{14}-860a^{13}+1375a^{12}-1545a^{11}+1060a^{10}-205a^{9}-150a^{8}-275a^{7}+870a^{6}-870a^{5}+405a^{4}-5a^{3}-90a^{2}+45a-10$, $\frac{12\cdots 73}{34\cdots 55}a^{24}-\frac{10\cdots 16}{64\cdots 45}a^{23}+\frac{37\cdots 81}{12\cdots 89}a^{22}-\frac{35\cdots 76}{64\cdots 45}a^{21}-\frac{59\cdots 44}{64\cdots 45}a^{20}+\frac{16\cdots 83}{64\cdots 45}a^{19}-\frac{21\cdots 19}{64\cdots 45}a^{18}+\frac{25\cdots 34}{12\cdots 89}a^{17}+\frac{57\cdots 71}{64\cdots 45}a^{16}-\frac{33\cdots 81}{64\cdots 45}a^{15}+\frac{83\cdots 57}{64\cdots 45}a^{14}-\frac{16\cdots 06}{64\cdots 45}a^{13}+\frac{49\cdots 38}{12\cdots 89}a^{12}-\frac{13\cdots 89}{34\cdots 55}a^{11}+\frac{13\cdots 11}{64\cdots 45}a^{10}+\frac{13\cdots 28}{64\cdots 45}a^{9}-\frac{30\cdots 54}{64\cdots 45}a^{8}-\frac{15\cdots 26}{12\cdots 89}a^{7}+\frac{17\cdots 66}{64\cdots 45}a^{6}-\frac{12\cdots 81}{64\cdots 45}a^{5}+\frac{37\cdots 17}{64\cdots 45}a^{4}+\frac{15\cdots 24}{64\cdots 45}a^{3}-\frac{28\cdots 41}{12\cdots 89}a^{2}+\frac{40\cdots 39}{64\cdots 45}a-\frac{48\cdots 74}{64\cdots 45}$, $\frac{10\cdots 43}{12\cdots 89}a^{24}-\frac{24\cdots 99}{64\cdots 45}a^{23}+\frac{46\cdots 27}{64\cdots 45}a^{22}-\frac{17\cdots 58}{64\cdots 45}a^{21}-\frac{12\cdots 96}{64\cdots 45}a^{20}+\frac{76\cdots 68}{12\cdots 89}a^{19}-\frac{54\cdots 61}{64\cdots 45}a^{18}+\frac{40\cdots 98}{64\cdots 45}a^{17}+\frac{45\cdots 43}{64\cdots 45}a^{16}-\frac{77\cdots 04}{64\cdots 45}a^{15}+\frac{40\cdots 52}{12\cdots 89}a^{14}-\frac{40\cdots 74}{64\cdots 45}a^{13}+\frac{63\cdots 87}{64\cdots 45}a^{12}-\frac{68\cdots 33}{64\cdots 45}a^{11}+\frac{44\cdots 14}{64\cdots 45}a^{10}-\frac{11\cdots 75}{12\cdots 89}a^{9}-\frac{70\cdots 21}{64\cdots 45}a^{8}-\frac{14\cdots 62}{64\cdots 45}a^{7}+\frac{41\cdots 53}{64\cdots 45}a^{6}-\frac{38\cdots 34}{64\cdots 45}a^{5}+\frac{32\cdots 54}{12\cdots 89}a^{4}+\frac{15\cdots 16}{64\cdots 45}a^{3}-\frac{42\cdots 13}{64\cdots 45}a^{2}+\frac{18\cdots 92}{64\cdots 45}a-\frac{27\cdots 26}{64\cdots 45}$, $\frac{95\cdots 72}{64\cdots 45}a^{24}+\frac{93\cdots 75}{12\cdots 89}a^{23}-\frac{89\cdots 18}{64\cdots 45}a^{22}+\frac{33\cdots 23}{64\cdots 45}a^{21}+\frac{24\cdots 68}{64\cdots 45}a^{20}-\frac{73\cdots 28}{64\cdots 45}a^{19}+\frac{20\cdots 54}{12\cdots 89}a^{18}-\frac{38\cdots 78}{34\cdots 55}a^{17}-\frac{15\cdots 03}{64\cdots 45}a^{16}+\frac{14\cdots 77}{64\cdots 45}a^{15}-\frac{38\cdots 82}{64\cdots 45}a^{14}+\frac{15\cdots 62}{12\cdots 89}a^{13}-\frac{11\cdots 43}{64\cdots 45}a^{12}+\frac{12\cdots 98}{64\cdots 45}a^{11}-\frac{40\cdots 58}{34\cdots 55}a^{10}+\frac{21\cdots 32}{64\cdots 45}a^{9}+\frac{36\cdots 98}{12\cdots 89}a^{8}+\frac{30\cdots 88}{64\cdots 45}a^{7}-\frac{80\cdots 58}{64\cdots 45}a^{6}+\frac{70\cdots 07}{64\cdots 45}a^{5}-\frac{24\cdots 82}{64\cdots 45}a^{4}-\frac{12\cdots 51}{12\cdots 89}a^{3}+\frac{84\cdots 72}{64\cdots 45}a^{2}-\frac{24\cdots 17}{64\cdots 45}a+\frac{35\cdots 33}{64\cdots 45}$, $\frac{58\cdots 62}{64\cdots 45}a^{24}-\frac{28\cdots 09}{64\cdots 45}a^{23}+\frac{10\cdots 98}{12\cdots 89}a^{22}-\frac{19\cdots 71}{64\cdots 45}a^{21}-\frac{14\cdots 32}{64\cdots 45}a^{20}+\frac{44\cdots 43}{64\cdots 45}a^{19}-\frac{62\cdots 41}{64\cdots 45}a^{18}+\frac{46\cdots 85}{68\cdots 31}a^{17}+\frac{81\cdots 16}{64\cdots 45}a^{16}-\frac{89\cdots 48}{64\cdots 45}a^{15}+\frac{22\cdots 67}{64\cdots 45}a^{14}-\frac{46\cdots 34}{64\cdots 45}a^{13}+\frac{14\cdots 11}{12\cdots 89}a^{12}-\frac{77\cdots 91}{64\cdots 45}a^{11}+\frac{25\cdots 47}{34\cdots 55}a^{10}-\frac{32\cdots 47}{64\cdots 45}a^{9}-\frac{94\cdots 21}{64\cdots 45}a^{8}-\frac{36\cdots 40}{12\cdots 89}a^{7}+\frac{48\cdots 26}{64\cdots 45}a^{6}-\frac{42\cdots 83}{64\cdots 45}a^{5}+\frac{15\cdots 82}{64\cdots 45}a^{4}+\frac{27\cdots 36}{64\cdots 45}a^{3}-\frac{91\cdots 32}{12\cdots 89}a^{2}+\frac{13\cdots 04}{64\cdots 45}a-\frac{15\cdots 67}{64\cdots 45}$, $\frac{23\cdots 36}{64\cdots 45}a^{24}-\frac{96\cdots 59}{64\cdots 45}a^{23}+\frac{14\cdots 62}{64\cdots 45}a^{22}+\frac{35\cdots 56}{64\cdots 45}a^{21}-\frac{59\cdots 86}{64\cdots 45}a^{20}+\frac{13\cdots 39}{64\cdots 45}a^{19}-\frac{14\cdots 16}{64\cdots 45}a^{18}+\frac{28\cdots 02}{34\cdots 55}a^{17}+\frac{93\cdots 24}{64\cdots 45}a^{16}-\frac{30\cdots 69}{64\cdots 45}a^{15}+\frac{69\cdots 36}{64\cdots 45}a^{14}-\frac{13\cdots 64}{64\cdots 45}a^{13}+\frac{18\cdots 97}{64\cdots 45}a^{12}-\frac{15\cdots 89}{64\cdots 45}a^{11}+\frac{27\cdots 16}{34\cdots 55}a^{10}+\frac{48\cdots 84}{64\cdots 45}a^{9}-\frac{14\cdots 66}{64\cdots 45}a^{8}-\frac{85\cdots 67}{64\cdots 45}a^{7}+\frac{13\cdots 44}{64\cdots 45}a^{6}-\frac{63\cdots 74}{64\cdots 45}a^{5}-\frac{25\cdots 34}{64\cdots 45}a^{4}+\frac{20\cdots 86}{64\cdots 45}a^{3}-\frac{56\cdots 18}{64\cdots 45}a^{2}-\frac{25\cdots 44}{64\cdots 45}a-\frac{43\cdots 91}{64\cdots 45}$, $\frac{43\cdots 94}{34\cdots 55}a^{24}+\frac{41\cdots 46}{64\cdots 45}a^{23}-\frac{82\cdots 41}{64\cdots 45}a^{22}+\frac{40\cdots 17}{64\cdots 45}a^{21}+\frac{20\cdots 44}{64\cdots 45}a^{20}-\frac{66\cdots 54}{64\cdots 45}a^{19}+\frac{98\cdots 04}{64\cdots 45}a^{18}-\frac{75\cdots 54}{64\cdots 45}a^{17}-\frac{35\cdots 57}{64\cdots 45}a^{16}+\frac{13\cdots 26}{64\cdots 45}a^{15}-\frac{34\cdots 86}{64\cdots 45}a^{14}+\frac{70\cdots 56}{64\cdots 45}a^{13}-\frac{11\cdots 06}{64\cdots 45}a^{12}+\frac{65\cdots 18}{34\cdots 55}a^{11}-\frac{83\cdots 41}{64\cdots 45}a^{10}+\frac{12\cdots 16}{64\cdots 45}a^{9}+\frac{15\cdots 09}{64\cdots 45}a^{8}+\frac{22\cdots 46}{64\cdots 45}a^{7}-\frac{72\cdots 92}{64\cdots 45}a^{6}+\frac{70\cdots 31}{64\cdots 45}a^{5}-\frac{30\cdots 56}{64\cdots 45}a^{4}-\frac{19\cdots 49}{64\cdots 45}a^{3}+\frac{80\cdots 49}{64\cdots 45}a^{2}-\frac{33\cdots 13}{64\cdots 45}a+\frac{51\cdots 74}{64\cdots 45}$, $\frac{10\cdots 57}{64\cdots 45}a^{24}-\frac{85\cdots 44}{12\cdots 89}a^{23}+\frac{65\cdots 16}{64\cdots 45}a^{22}+\frac{13\cdots 74}{64\cdots 45}a^{21}-\frac{26\cdots 44}{64\cdots 45}a^{20}+\frac{60\cdots 78}{64\cdots 45}a^{19}-\frac{13\cdots 88}{12\cdots 89}a^{18}+\frac{23\cdots 04}{64\cdots 45}a^{17}+\frac{45\cdots 01}{64\cdots 45}a^{16}-\frac{13\cdots 01}{64\cdots 45}a^{15}+\frac{30\cdots 17}{64\cdots 45}a^{14}-\frac{11\cdots 21}{12\cdots 89}a^{13}+\frac{83\cdots 11}{64\cdots 45}a^{12}-\frac{70\cdots 51}{64\cdots 45}a^{11}+\frac{19\cdots 46}{64\cdots 45}a^{10}+\frac{26\cdots 88}{64\cdots 45}a^{9}-\frac{20\cdots 81}{12\cdots 89}a^{8}-\frac{40\cdots 91}{64\cdots 45}a^{7}+\frac{61\cdots 31}{64\cdots 45}a^{6}-\frac{28\cdots 91}{64\cdots 45}a^{5}-\frac{45\cdots 03}{64\cdots 45}a^{4}+\frac{23\cdots 96}{12\cdots 89}a^{3}-\frac{32\cdots 69}{64\cdots 45}a^{2}-\frac{68\cdots 06}{64\cdots 45}a+\frac{30\cdots 36}{64\cdots 45}$, $\frac{14\cdots 35}{12\cdots 89}a^{24}-\frac{69\cdots 68}{12\cdots 89}a^{23}+\frac{65\cdots 14}{64\cdots 45}a^{22}-\frac{24\cdots 77}{64\cdots 45}a^{21}-\frac{18\cdots 86}{64\cdots 45}a^{20}+\frac{10\cdots 66}{12\cdots 89}a^{19}-\frac{15\cdots 41}{12\cdots 89}a^{18}+\frac{54\cdots 86}{64\cdots 45}a^{17}+\frac{10\cdots 12}{64\cdots 45}a^{16}-\frac{11\cdots 14}{64\cdots 45}a^{15}+\frac{56\cdots 71}{12\cdots 89}a^{14}-\frac{11\cdots 98}{12\cdots 89}a^{13}+\frac{88\cdots 84}{64\cdots 45}a^{12}-\frac{95\cdots 82}{64\cdots 45}a^{11}+\frac{58\cdots 99}{64\cdots 45}a^{10}-\frac{61\cdots 18}{12\cdots 89}a^{9}-\frac{23\cdots 70}{12\cdots 89}a^{8}-\frac{22\cdots 04}{64\cdots 45}a^{7}+\frac{60\cdots 62}{64\cdots 45}a^{6}-\frac{52\cdots 89}{64\cdots 45}a^{5}+\frac{36\cdots 93}{12\cdots 89}a^{4}+\frac{79\cdots 19}{12\cdots 89}a^{3}-\frac{51\cdots 51}{64\cdots 45}a^{2}+\frac{14\cdots 53}{64\cdots 45}a-\frac{18\cdots 51}{64\cdots 45}$, $\frac{47\cdots 02}{64\cdots 45}a^{24}-\frac{44\cdots 04}{64\cdots 45}a^{23}+\frac{19\cdots 26}{64\cdots 45}a^{22}-\frac{32\cdots 07}{64\cdots 45}a^{21}+\frac{14\cdots 06}{64\cdots 45}a^{20}+\frac{11\cdots 58}{64\cdots 45}a^{19}-\frac{28\cdots 96}{64\cdots 45}a^{18}+\frac{35\cdots 49}{64\cdots 45}a^{17}-\frac{19\cdots 68}{64\cdots 45}a^{16}-\frac{14\cdots 91}{64\cdots 45}a^{15}+\frac{62\cdots 37}{64\cdots 45}a^{14}-\frac{78\cdots 51}{34\cdots 55}a^{13}+\frac{29\cdots 01}{64\cdots 45}a^{12}-\frac{42\cdots 72}{64\cdots 45}a^{11}+\frac{41\cdots 66}{64\cdots 45}a^{10}-\frac{19\cdots 72}{64\cdots 45}a^{9}-\frac{60\cdots 56}{64\cdots 45}a^{8}+\frac{54\cdots 69}{64\cdots 45}a^{7}+\frac{15\cdots 47}{64\cdots 45}a^{6}-\frac{30\cdots 36}{64\cdots 45}a^{5}+\frac{19\cdots 42}{64\cdots 45}a^{4}-\frac{34\cdots 89}{64\cdots 45}a^{3}-\frac{39\cdots 69}{64\cdots 45}a^{2}+\frac{22\cdots 68}{64\cdots 45}a-\frac{35\cdots 29}{64\cdots 45}$, $\frac{49\cdots 03}{64\cdots 45}a^{24}-\frac{25\cdots 81}{64\cdots 45}a^{23}+\frac{52\cdots 82}{64\cdots 45}a^{22}-\frac{27\cdots 79}{64\cdots 45}a^{21}-\frac{25\cdots 22}{12\cdots 89}a^{20}+\frac{41\cdots 67}{64\cdots 45}a^{19}-\frac{62\cdots 84}{64\cdots 45}a^{18}+\frac{49\cdots 13}{64\cdots 45}a^{17}-\frac{70\cdots 61}{64\cdots 45}a^{16}-\frac{15\cdots 09}{12\cdots 89}a^{15}+\frac{21\cdots 63}{64\cdots 45}a^{14}-\frac{44\cdots 16}{64\cdots 45}a^{13}+\frac{71\cdots 87}{64\cdots 45}a^{12}-\frac{80\cdots 49}{64\cdots 45}a^{11}+\frac{10\cdots 02}{12\cdots 89}a^{10}-\frac{99\cdots 63}{64\cdots 45}a^{9}-\frac{94\cdots 79}{64\cdots 45}a^{8}-\frac{13\cdots 82}{64\cdots 45}a^{7}+\frac{45\cdots 49}{64\cdots 45}a^{6}-\frac{92\cdots 76}{12\cdots 89}a^{5}+\frac{20\cdots 23}{64\cdots 45}a^{4}+\frac{45\cdots 09}{64\cdots 45}a^{3}-\frac{54\cdots 33}{64\cdots 45}a^{2}+\frac{22\cdots 01}{64\cdots 45}a-\frac{82\cdots 12}{12\cdots 89}$, $\frac{15\cdots 86}{34\cdots 55}a^{24}-\frac{73\cdots 26}{34\cdots 55}a^{23}+\frac{24\cdots 96}{68\cdots 31}a^{22}-\frac{54\cdots 76}{34\cdots 55}a^{21}-\frac{43\cdots 58}{34\cdots 55}a^{20}+\frac{10\cdots 64}{34\cdots 55}a^{19}-\frac{13\cdots 64}{34\cdots 55}a^{18}+\frac{13\cdots 44}{68\cdots 31}a^{17}+\frac{57\cdots 86}{34\cdots 55}a^{16}-\frac{22\cdots 62}{34\cdots 55}a^{15}+\frac{55\cdots 46}{34\cdots 55}a^{14}-\frac{10\cdots 86}{34\cdots 55}a^{13}+\frac{31\cdots 36}{68\cdots 31}a^{12}-\frac{15\cdots 26}{34\cdots 55}a^{11}+\frac{66\cdots 42}{34\cdots 55}a^{10}+\frac{24\cdots 49}{34\cdots 55}a^{9}-\frac{17\cdots 74}{34\cdots 55}a^{8}-\frac{13\cdots 48}{68\cdots 31}a^{7}+\frac{11\cdots 96}{34\cdots 55}a^{6}-\frac{69\cdots 62}{34\cdots 55}a^{5}+\frac{85\cdots 56}{34\cdots 55}a^{4}+\frac{11\cdots 24}{34\cdots 55}a^{3}-\frac{13\cdots 82}{68\cdots 31}a^{2}+\frac{17\cdots 04}{34\cdots 55}a-\frac{16\cdots 78}{34\cdots 55}$
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| Regulator: | \( 106797269.80793719 \) (assuming GRH) |
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| Unit signature rank: | \( 1 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 106797269.80793719 \cdot 1}{2\cdot\sqrt{136202515664062500000000000000000000}}\cr\approx \mathstrut & 1.09553436754442 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 100 |
| The 10 conjugacy class representatives for $C_5:F_5$ |
| Character table for $C_5:F_5$ |
Intermediate fields
| 5.1.253125.1, 5.1.162000.1, 5.1.4050000.3, 5.1.50000.1, 5.1.4050000.4, 5.1.4050000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 2.20.16.2 | $x^{20} + 5 x^{17} + 5 x^{16} + 10 x^{14} + 20 x^{13} + 10 x^{12} + 10 x^{11} + 30 x^{10} + 30 x^{9} + 15 x^{8} + 20 x^{7} + 30 x^{6} + 21 x^{5} + 10 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 3$ | $5$ | $4$ | $16$ | 20T5 | $$[\ ]_{5}^{4}$$ | |
|
\(3\)
| 3.5.4.1 | $x^{5} + 3$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 3.20.16.1 | $x^{20} + 10 x^{19} + 40 x^{18} + 80 x^{17} + 90 x^{16} + 112 x^{15} + 240 x^{14} + 320 x^{13} + 200 x^{12} + 240 x^{11} + 480 x^{10} + 320 x^{9} + 80 x^{8} + 320 x^{7} + 320 x^{6} + 80 x^{4} + 160 x^{3} + 35$ | $5$ | $4$ | $16$ | 20T5 | $$[\ ]_{5}^{4}$$ | |
|
\(5\)
| 5.5.5.2 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.20.23.9 | $x^{20} + 5 x^{4} + 5$ | $20$ | $1$ | $23$ | 20T5 | not computed |