Normalized defining polynomial
\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} - 32 x^{16} + 201 x^{15} - 718 x^{14} + \cdots + 19264 \)
Invariants
| Degree: | $21$ |
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| Signature: | $[3, 9]$ |
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| Discriminant: |
\(-26160559119874379648562947375700703\)
\(\medspace = -\,7^{17}\cdot 13^{18}\)
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| |
| Root discriminant: | \(43.54\) |
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| Galois root discriminant: | $7^{5/6}13^{6/7}\approx 45.60967255955715$ | ||
| Ramified primes: |
\(7\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{172}a^{16}+\frac{19}{172}a^{15}-\frac{5}{86}a^{14}+\frac{15}{172}a^{13}-\frac{3}{43}a^{12}-\frac{11}{172}a^{11}-\frac{7}{43}a^{10}-\frac{13}{172}a^{9}+\frac{8}{43}a^{8}-\frac{3}{172}a^{7}+\frac{14}{43}a^{6}+\frac{11}{172}a^{5}+\frac{33}{86}a^{4}-\frac{13}{172}a^{3}+\frac{15}{172}a^{2}+\frac{18}{43}a$, $\frac{1}{172}a^{17}+\frac{4}{43}a^{15}-\frac{5}{86}a^{14}+\frac{1}{43}a^{13}+\frac{1}{86}a^{12}-\frac{17}{86}a^{11}-\frac{10}{43}a^{10}-\frac{11}{86}a^{9}+\frac{17}{86}a^{8}-\frac{4}{43}a^{7}+\frac{11}{86}a^{6}+\frac{18}{43}a^{5}-\frac{5}{43}a^{4}-\frac{39}{172}a^{3}-\frac{21}{43}a^{2}+\frac{2}{43}a$, $\frac{1}{62608}a^{18}-\frac{127}{62608}a^{17}-\frac{95}{62608}a^{16}+\frac{569}{31304}a^{15}-\frac{6431}{62608}a^{14}+\frac{426}{3913}a^{13}+\frac{2549}{62608}a^{12}+\frac{875}{4472}a^{11}+\frac{1357}{8944}a^{10}+\frac{627}{15652}a^{9}-\frac{901}{4816}a^{8}-\frac{825}{7826}a^{7}-\frac{2703}{62608}a^{6}+\frac{807}{2236}a^{5}+\frac{887}{4472}a^{4}+\frac{2935}{8944}a^{3}-\frac{165}{2236}a^{2}-\frac{205}{2236}a-\frac{5}{26}$, $\frac{1}{125216}a^{19}-\frac{1}{125216}a^{18}-\frac{81}{125216}a^{17}+\frac{11}{15652}a^{16}+\frac{5917}{125216}a^{15}-\frac{99}{1456}a^{14}-\frac{6411}{125216}a^{13}-\frac{11}{86}a^{12}-\frac{2783}{17888}a^{11}+\frac{1275}{62608}a^{10}+\frac{677}{2912}a^{9}-\frac{12491}{62608}a^{8}-\frac{27679}{125216}a^{7}-\frac{3057}{8944}a^{6}+\frac{2283}{8944}a^{5}-\frac{2549}{17888}a^{4}-\frac{2105}{8944}a^{3}-\frac{45}{344}a^{2}-\frac{3}{43}a-\frac{3}{26}$, $\frac{1}{20\cdots 88}a^{20}+\frac{39\cdots 11}{20\cdots 88}a^{19}+\frac{63\cdots 27}{20\cdots 88}a^{18}+\frac{15\cdots 17}{25\cdots 36}a^{17}-\frac{33\cdots 03}{20\cdots 88}a^{16}+\frac{30\cdots 95}{10\cdots 44}a^{15}-\frac{24\cdots 59}{20\cdots 88}a^{14}-\frac{84\cdots 09}{25\cdots 36}a^{13}+\frac{32\cdots 95}{20\cdots 88}a^{12}+\frac{12\cdots 83}{10\cdots 44}a^{11}-\frac{23\cdots 17}{20\cdots 88}a^{10}+\frac{15\cdots 77}{10\cdots 44}a^{9}+\frac{27\cdots 97}{20\cdots 88}a^{8}-\frac{22\cdots 83}{10\cdots 44}a^{7}-\frac{42\cdots 79}{10\cdots 44}a^{6}+\frac{13\cdots 67}{29\cdots 84}a^{5}+\frac{41\cdots 23}{14\cdots 92}a^{4}-\frac{19\cdots 31}{74\cdots 96}a^{3}+\frac{31\cdots 73}{92\cdots 62}a^{2}+\frac{86\cdots 89}{18\cdots 24}a-\frac{20\cdots 48}{10\cdots 17}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{17\cdots 73}{19\cdots 72}a^{20}+\frac{23\cdots 69}{39\cdots 44}a^{19}-\frac{79\cdots 61}{57\cdots 92}a^{18}+\frac{15\cdots 15}{39\cdots 44}a^{17}-\frac{17\cdots 64}{35\cdots 87}a^{16}+\frac{46\cdots 83}{39\cdots 44}a^{15}+\frac{53\cdots 43}{19\cdots 72}a^{14}+\frac{16\cdots 67}{39\cdots 44}a^{13}-\frac{46\cdots 94}{24\cdots 09}a^{12}+\frac{44\cdots 41}{39\cdots 44}a^{11}+\frac{11\cdots 35}{19\cdots 72}a^{10}-\frac{57\cdots 51}{39\cdots 44}a^{9}-\frac{40\cdots 77}{19\cdots 72}a^{8}+\frac{13\cdots 15}{39\cdots 44}a^{7}-\frac{35\cdots 51}{99\cdots 36}a^{6}-\frac{10\cdots 97}{71\cdots 74}a^{5}+\frac{18\cdots 77}{57\cdots 92}a^{4}+\frac{14\cdots 71}{71\cdots 74}a^{3}-\frac{34\cdots 91}{71\cdots 74}a^{2}-\frac{61\cdots 34}{35\cdots 87}a+\frac{18\cdots 37}{83\cdots 09}$, $\frac{88\cdots 11}{49\cdots 18}a^{20}-\frac{13\cdots 43}{39\cdots 44}a^{19}+\frac{73\cdots 45}{39\cdots 44}a^{18}-\frac{22\cdots 21}{57\cdots 92}a^{17}+\frac{83\cdots 69}{19\cdots 72}a^{16}-\frac{52\cdots 13}{57\cdots 92}a^{15}+\frac{29\cdots 47}{49\cdots 18}a^{14}-\frac{30\cdots 45}{57\cdots 92}a^{13}+\frac{29\cdots 25}{19\cdots 72}a^{12}-\frac{54\cdots 65}{39\cdots 44}a^{11}-\frac{44\cdots 39}{99\cdots 36}a^{10}-\frac{40\cdots 73}{39\cdots 44}a^{9}+\frac{99\cdots 25}{49\cdots 18}a^{8}-\frac{11\cdots 23}{39\cdots 44}a^{7}-\frac{24\cdots 51}{99\cdots 36}a^{6}+\frac{48\cdots 43}{28\cdots 96}a^{5}-\frac{14\cdots 61}{57\cdots 92}a^{4}-\frac{64\cdots 61}{35\cdots 87}a^{3}+\frac{62\cdots 13}{14\cdots 48}a^{2}-\frac{11\cdots 03}{35\cdots 87}a-\frac{11\cdots 73}{83\cdots 09}$, $\frac{16\cdots 25}{14\cdots 92}a^{20}-\frac{11\cdots 45}{14\cdots 92}a^{19}+\frac{15\cdots 05}{79\cdots 88}a^{18}-\frac{15\cdots 09}{64\cdots 34}a^{17}+\frac{10\cdots 43}{10\cdots 44}a^{16}-\frac{10\cdots 43}{51\cdots 72}a^{15}+\frac{22\cdots 63}{10\cdots 44}a^{14}-\frac{22\cdots 30}{32\cdots 17}a^{13}+\frac{34\cdots 53}{79\cdots 88}a^{12}+\frac{13\cdots 87}{74\cdots 96}a^{11}-\frac{12\cdots 05}{14\cdots 92}a^{10}-\frac{36\cdots 77}{39\cdots 44}a^{9}+\frac{15\cdots 95}{10\cdots 44}a^{8}-\frac{21\cdots 93}{51\cdots 72}a^{7}-\frac{32\cdots 09}{51\cdots 72}a^{6}+\frac{16\cdots 33}{11\cdots 84}a^{5}+\frac{30\cdots 33}{74\cdots 96}a^{4}-\frac{73\cdots 73}{28\cdots 96}a^{3}+\frac{10\cdots 22}{46\cdots 31}a^{2}+\frac{15\cdots 97}{92\cdots 62}a-\frac{18\cdots 25}{10\cdots 17}$, $\frac{15\cdots 37}{20\cdots 88}a^{20}-\frac{69\cdots 71}{15\cdots 76}a^{19}+\frac{31\cdots 63}{29\cdots 84}a^{18}-\frac{19\cdots 13}{14\cdots 92}a^{17}+\frac{18\cdots 95}{29\cdots 84}a^{16}-\frac{12\cdots 93}{74\cdots 96}a^{15}+\frac{27\cdots 97}{20\cdots 88}a^{14}-\frac{39\cdots 41}{10\cdots 44}a^{13}+\frac{37\cdots 91}{20\cdots 88}a^{12}+\frac{24\cdots 41}{25\cdots 36}a^{11}-\frac{30\cdots 29}{20\cdots 88}a^{10}-\frac{25\cdots 61}{51\cdots 72}a^{9}+\frac{17\cdots 61}{20\cdots 88}a^{8}-\frac{11\cdots 33}{64\cdots 34}a^{7}-\frac{37\cdots 05}{10\cdots 44}a^{6}+\frac{20\cdots 67}{29\cdots 84}a^{5}+\frac{77\cdots 93}{37\cdots 48}a^{4}-\frac{35\cdots 07}{28\cdots 96}a^{3}+\frac{44\cdots 93}{37\cdots 48}a^{2}+\frac{13\cdots 71}{18\cdots 24}a-\frac{27\cdots 97}{21\cdots 34}$, $\frac{13\cdots 75}{10\cdots 44}a^{20}-\frac{36\cdots 89}{51\cdots 72}a^{19}+\frac{13\cdots 57}{74\cdots 96}a^{18}-\frac{27\cdots 63}{10\cdots 44}a^{17}+\frac{30\cdots 39}{11\cdots 84}a^{16}-\frac{53\cdots 33}{10\cdots 44}a^{15}+\frac{24\cdots 63}{10\cdots 44}a^{14}-\frac{67\cdots 75}{10\cdots 44}a^{13}+\frac{46\cdots 75}{10\cdots 44}a^{12}+\frac{10\cdots 85}{10\cdots 44}a^{11}+\frac{34\cdots 45}{10\cdots 44}a^{10}-\frac{69\cdots 91}{10\cdots 44}a^{9}+\frac{13\cdots 51}{10\cdots 44}a^{8}-\frac{88\cdots 81}{10\cdots 44}a^{7}-\frac{24\cdots 15}{51\cdots 72}a^{6}+\frac{15\cdots 43}{14\cdots 92}a^{5}-\frac{10\cdots 15}{14\cdots 92}a^{4}-\frac{16\cdots 47}{18\cdots 24}a^{3}+\frac{85\cdots 67}{37\cdots 48}a^{2}-\frac{21\cdots 79}{18\cdots 24}a-\frac{10\cdots 07}{10\cdots 17}$, $\frac{46\cdots 75}{20\cdots 88}a^{20}-\frac{35\cdots 99}{22\cdots 68}a^{19}+\frac{94\cdots 47}{20\cdots 88}a^{18}-\frac{73\cdots 67}{10\cdots 44}a^{17}+\frac{11\cdots 03}{20\cdots 88}a^{16}-\frac{32\cdots 17}{51\cdots 72}a^{15}+\frac{92\cdots 59}{20\cdots 88}a^{14}-\frac{16\cdots 75}{10\cdots 44}a^{13}+\frac{35\cdots 53}{20\cdots 88}a^{12}+\frac{65\cdots 41}{25\cdots 36}a^{11}-\frac{10\cdots 33}{29\cdots 84}a^{10}-\frac{75\cdots 85}{51\cdots 72}a^{9}+\frac{83\cdots 51}{20\cdots 88}a^{8}-\frac{58\cdots 01}{19\cdots 72}a^{7}-\frac{11\cdots 49}{10\cdots 44}a^{6}+\frac{97\cdots 61}{29\cdots 84}a^{5}-\frac{53\cdots 79}{37\cdots 48}a^{4}-\frac{34\cdots 83}{74\cdots 96}a^{3}+\frac{15\cdots 47}{37\cdots 48}a^{2}+\frac{97\cdots 38}{46\cdots 31}a-\frac{30\cdots 79}{10\cdots 17}$, $\frac{51\cdots 01}{14\cdots 92}a^{20}-\frac{11\cdots 27}{74\cdots 96}a^{19}+\frac{68\cdots 05}{19\cdots 72}a^{18}-\frac{60\cdots 41}{10\cdots 44}a^{17}+\frac{98\cdots 79}{10\cdots 44}a^{16}-\frac{26\cdots 81}{10\cdots 44}a^{15}+\frac{59\cdots 21}{79\cdots 88}a^{14}-\frac{16\cdots 23}{10\cdots 44}a^{13}+\frac{10\cdots 61}{10\cdots 44}a^{12}-\frac{11\cdots 49}{14\cdots 92}a^{11}+\frac{12\cdots 65}{14\cdots 92}a^{10}-\frac{17\cdots 59}{10\cdots 44}a^{9}+\frac{28\cdots 41}{10\cdots 44}a^{8}-\frac{27\cdots 57}{10\cdots 44}a^{7}-\frac{30\cdots 37}{49\cdots 18}a^{6}+\frac{15\cdots 83}{14\cdots 92}a^{5}+\frac{10\cdots 45}{14\cdots 92}a^{4}+\frac{67\cdots 61}{74\cdots 96}a^{3}-\frac{21\cdots 51}{37\cdots 48}a^{2}-\frac{10\cdots 97}{92\cdots 62}a+\frac{57\cdots 67}{21\cdots 34}$, $\frac{28\cdots 31}{20\cdots 88}a^{20}-\frac{19\cdots 75}{20\cdots 88}a^{19}+\frac{54\cdots 65}{20\cdots 88}a^{18}-\frac{46\cdots 05}{12\cdots 68}a^{17}+\frac{35\cdots 27}{20\cdots 88}a^{16}-\frac{23\cdots 63}{10\cdots 44}a^{15}+\frac{43\cdots 47}{15\cdots 76}a^{14}-\frac{12\cdots 17}{12\cdots 68}a^{13}+\frac{17\cdots 97}{20\cdots 88}a^{12}+\frac{20\cdots 61}{10\cdots 44}a^{11}-\frac{43\cdots 79}{20\cdots 88}a^{10}-\frac{11\cdots 25}{10\cdots 44}a^{9}+\frac{53\cdots 99}{20\cdots 88}a^{8}-\frac{11\cdots 53}{10\cdots 44}a^{7}-\frac{80\cdots 29}{10\cdots 44}a^{6}+\frac{58\cdots 41}{29\cdots 84}a^{5}-\frac{22\cdots 99}{14\cdots 92}a^{4}-\frac{30\cdots 69}{74\cdots 96}a^{3}+\frac{42\cdots 21}{18\cdots 24}a^{2}+\frac{66\cdots 09}{18\cdots 24}a-\frac{32\cdots 69}{10\cdots 17}$, $\frac{68\cdots 93}{20\cdots 88}a^{20}-\frac{36\cdots 25}{20\cdots 88}a^{19}+\frac{71\cdots 47}{20\cdots 88}a^{18}-\frac{11\cdots 13}{51\cdots 72}a^{17}-\frac{31\cdots 83}{20\cdots 88}a^{16}-\frac{66\cdots 03}{10\cdots 44}a^{15}+\frac{11\cdots 69}{20\cdots 88}a^{14}-\frac{67\cdots 97}{51\cdots 72}a^{13}-\frac{92\cdots 89}{20\cdots 88}a^{12}+\frac{53\cdots 61}{10\cdots 44}a^{11}+\frac{55\cdots 51}{20\cdots 88}a^{10}-\frac{25\cdots 41}{10\cdots 44}a^{9}+\frac{39\cdots 17}{20\cdots 88}a^{8}+\frac{26\cdots 79}{10\cdots 44}a^{7}-\frac{16\cdots 95}{10\cdots 44}a^{6}+\frac{57\cdots 43}{29\cdots 84}a^{5}+\frac{43\cdots 97}{14\cdots 92}a^{4}-\frac{36\cdots 83}{74\cdots 96}a^{3}-\frac{62\cdots 49}{18\cdots 24}a^{2}+\frac{63\cdots 97}{18\cdots 24}a+\frac{21\cdots 35}{10\cdots 17}$, $\frac{37\cdots 99}{20\cdots 88}a^{20}-\frac{49\cdots 15}{20\cdots 88}a^{19}-\frac{71\cdots 75}{20\cdots 88}a^{18}+\frac{82\cdots 09}{51\cdots 72}a^{17}-\frac{71\cdots 53}{20\cdots 88}a^{16}+\frac{42\cdots 91}{10\cdots 44}a^{15}-\frac{60\cdots 53}{20\cdots 88}a^{14}+\frac{60\cdots 39}{51\cdots 72}a^{13}-\frac{13\cdots 15}{20\cdots 88}a^{12}+\frac{12\cdots 35}{10\cdots 44}a^{11}+\frac{30\cdots 97}{20\cdots 88}a^{10}-\frac{20\cdots 43}{10\cdots 44}a^{9}-\frac{61\cdots 29}{20\cdots 88}a^{8}+\frac{15\cdots 85}{10\cdots 44}a^{7}-\frac{28\cdots 57}{10\cdots 44}a^{6}-\frac{15\cdots 23}{29\cdots 84}a^{5}+\frac{18\cdots 99}{14\cdots 92}a^{4}-\frac{99\cdots 31}{74\cdots 96}a^{3}-\frac{10\cdots 91}{18\cdots 24}a^{2}+\frac{20\cdots 41}{18\cdots 24}a-\frac{14\cdots 54}{10\cdots 17}$, $\frac{19\cdots 13}{15\cdots 76}a^{20}-\frac{18\cdots 03}{20\cdots 88}a^{19}+\frac{54\cdots 29}{20\cdots 88}a^{18}-\frac{44\cdots 53}{10\cdots 44}a^{17}+\frac{83\cdots 69}{20\cdots 88}a^{16}-\frac{32\cdots 15}{64\cdots 34}a^{15}+\frac{55\cdots 61}{20\cdots 88}a^{14}-\frac{94\cdots 63}{10\cdots 44}a^{13}+\frac{22\cdots 59}{20\cdots 88}a^{12}+\frac{42\cdots 35}{39\cdots 44}a^{11}-\frac{36\cdots 61}{20\cdots 88}a^{10}-\frac{10\cdots 11}{12\cdots 68}a^{9}+\frac{47\cdots 61}{20\cdots 88}a^{8}-\frac{10\cdots 53}{51\cdots 72}a^{7}-\frac{55\cdots 85}{10\cdots 44}a^{6}+\frac{53\cdots 43}{29\cdots 84}a^{5}-\frac{75\cdots 01}{74\cdots 96}a^{4}-\frac{62\cdots 55}{28\cdots 96}a^{3}+\frac{81\cdots 49}{37\cdots 48}a^{2}+\frac{19\cdots 51}{18\cdots 24}a-\frac{28\cdots 69}{21\cdots 34}$
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| Regulator: | \( 3123124290.4668374 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 3123124290.4668374 \cdot 1}{2\cdot\sqrt{26160559119874379648562947375700703}}\cr\approx \mathstrut & 1.17881204229849 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 42 |
| The 7 conjugacy class representatives for $F_7$ |
| Character table for $F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.1.81124178863.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 42 |
| Degree 7 sibling: | 7.1.81124178863.1 |
| Degree 14 sibling: | deg 14 |
| Minimal sibling: | 7.1.81124178863.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{7}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.1.0.1}{1} }^{21}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
|
\(13\)
| 13.1.7.6a1.1 | $x^{7} + 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $$[\ ]_{7}^{2}$$ |
| 13.1.7.6a1.1 | $x^{7} + 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $$[\ ]_{7}^{2}$$ | |
| 13.1.7.6a1.1 | $x^{7} + 13$ | $7$ | $1$ | $6$ | $D_{7}$ | $$[\ ]_{7}^{2}$$ |