Normalized defining polynomial
\( x^{18} - 7 x^{17} - 32 x^{16} + 294 x^{15} + 275 x^{14} - 4779 x^{13} + 1076 x^{12} + 37663 x^{11} + \cdots - 191 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[18, 0]$ |
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| Discriminant: |
\(34205654728777159191037355893457\)
\(\medspace = 17^{9}\cdot 19^{16}\)
|
| |
| Root discriminant: | \(56.48\) |
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| Galois root discriminant: | $17^{1/2}19^{8/9}\approx 56.47995320269716$ | ||
| Ramified primes: |
\(17\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{18}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(323=17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{323}(256,·)$, $\chi_{323}(1,·)$, $\chi_{323}(137,·)$, $\chi_{323}(271,·)$, $\chi_{323}(16,·)$, $\chi_{323}(273,·)$, $\chi_{323}(220,·)$, $\chi_{323}(290,·)$, $\chi_{323}(35,·)$, $\chi_{323}(101,·)$, $\chi_{323}(169,·)$, $\chi_{323}(237,·)$, $\chi_{323}(239,·)$, $\chi_{323}(305,·)$, $\chi_{323}(118,·)$, $\chi_{323}(120,·)$, $\chi_{323}(188,·)$, $\chi_{323}(254,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{26\cdots 07}a^{17}-\frac{28\cdots 81}{26\cdots 07}a^{16}+\frac{71\cdots 08}{26\cdots 07}a^{15}+\frac{81\cdots 61}{26\cdots 07}a^{14}+\frac{12\cdots 23}{26\cdots 07}a^{13}-\frac{10\cdots 70}{26\cdots 07}a^{12}+\frac{10\cdots 51}{26\cdots 07}a^{11}-\frac{33\cdots 14}{26\cdots 07}a^{10}+\frac{12\cdots 72}{26\cdots 07}a^{9}-\frac{54\cdots 74}{26\cdots 07}a^{8}+\frac{35\cdots 51}{26\cdots 07}a^{7}+\frac{77\cdots 73}{26\cdots 07}a^{6}-\frac{12\cdots 59}{26\cdots 07}a^{5}-\frac{57\cdots 48}{26\cdots 07}a^{4}+\frac{30\cdots 22}{26\cdots 07}a^{3}+\frac{84\cdots 98}{17\cdots 57}a^{2}-\frac{75\cdots 94}{26\cdots 07}a+\frac{66\cdots 08}{26\cdots 07}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $17$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{12\cdots 08}{26\cdots 07}a^{17}-\frac{73\cdots 06}{26\cdots 07}a^{16}-\frac{45\cdots 36}{26\cdots 07}a^{15}+\frac{31\cdots 57}{26\cdots 07}a^{14}+\frac{62\cdots 23}{26\cdots 07}a^{13}-\frac{51\cdots 94}{26\cdots 07}a^{12}-\frac{36\cdots 17}{26\cdots 07}a^{11}+\frac{41\cdots 85}{26\cdots 07}a^{10}+\frac{53\cdots 82}{26\cdots 07}a^{9}-\frac{17\cdots 19}{26\cdots 07}a^{8}+\frac{25\cdots 57}{26\cdots 07}a^{7}+\frac{32\cdots 64}{26\cdots 07}a^{6}-\frac{87\cdots 03}{26\cdots 07}a^{5}-\frac{23\cdots 35}{26\cdots 07}a^{4}+\frac{58\cdots 34}{26\cdots 07}a^{3}+\frac{23\cdots 60}{17\cdots 57}a^{2}-\frac{20\cdots 40}{26\cdots 07}a+\frac{43\cdots 05}{26\cdots 07}$, $\frac{23\cdots 80}{26\cdots 07}a^{17}-\frac{15\cdots 53}{26\cdots 07}a^{16}-\frac{84\cdots 06}{26\cdots 07}a^{15}+\frac{65\cdots 13}{26\cdots 07}a^{14}+\frac{99\cdots 03}{26\cdots 07}a^{13}-\frac{10\cdots 88}{26\cdots 07}a^{12}-\frac{30\cdots 41}{26\cdots 07}a^{11}+\frac{88\cdots 00}{26\cdots 07}a^{10}-\frac{23\cdots 73}{26\cdots 07}a^{9}-\frac{35\cdots 69}{26\cdots 07}a^{8}+\frac{19\cdots 29}{26\cdots 07}a^{7}+\frac{68\cdots 69}{26\cdots 07}a^{6}-\frac{47\cdots 42}{26\cdots 07}a^{5}-\frac{48\cdots 11}{26\cdots 07}a^{4}+\frac{36\cdots 40}{26\cdots 07}a^{3}+\frac{35\cdots 03}{17\cdots 57}a^{2}-\frac{54\cdots 06}{26\cdots 07}a+\frac{62\cdots 75}{26\cdots 07}$, $\frac{13\cdots 00}{26\cdots 07}a^{17}-\frac{97\cdots 91}{26\cdots 07}a^{16}-\frac{43\cdots 70}{26\cdots 07}a^{15}+\frac{40\cdots 23}{26\cdots 07}a^{14}+\frac{37\cdots 09}{26\cdots 07}a^{13}-\frac{66\cdots 26}{26\cdots 07}a^{12}+\frac{16\cdots 13}{26\cdots 07}a^{11}+\frac{52\cdots 77}{26\cdots 07}a^{10}-\frac{40\cdots 76}{26\cdots 07}a^{9}-\frac{20\cdots 16}{26\cdots 07}a^{8}+\frac{22\cdots 87}{26\cdots 07}a^{7}+\frac{36\cdots 55}{26\cdots 07}a^{6}-\frac{48\cdots 52}{26\cdots 07}a^{5}-\frac{20\cdots 18}{26\cdots 07}a^{4}+\frac{36\cdots 63}{26\cdots 07}a^{3}-\frac{10\cdots 86}{17\cdots 57}a^{2}-\frac{57\cdots 16}{26\cdots 07}a+\frac{60\cdots 60}{26\cdots 07}$, $\frac{19\cdots 62}{26\cdots 07}a^{17}-\frac{12\cdots 14}{26\cdots 07}a^{16}-\frac{69\cdots 56}{26\cdots 07}a^{15}+\frac{51\cdots 90}{26\cdots 07}a^{14}+\frac{87\cdots 44}{26\cdots 07}a^{13}-\frac{85\cdots 36}{26\cdots 07}a^{12}-\frac{37\cdots 60}{26\cdots 07}a^{11}+\frac{69\cdots 45}{26\cdots 07}a^{10}-\frac{87\cdots 67}{26\cdots 07}a^{9}-\frac{29\cdots 51}{26\cdots 07}a^{8}+\frac{11\cdots 46}{26\cdots 07}a^{7}+\frac{57\cdots 68}{26\cdots 07}a^{6}-\frac{31\cdots 08}{26\cdots 07}a^{5}-\frac{44\cdots 30}{26\cdots 07}a^{4}+\frac{26\cdots 79}{26\cdots 07}a^{3}+\frac{55\cdots 43}{17\cdots 57}a^{2}-\frac{45\cdots 13}{26\cdots 07}a-\frac{12\cdots 52}{26\cdots 07}$, $\frac{88\cdots 32}{26\cdots 07}a^{17}-\frac{51\cdots 57}{26\cdots 07}a^{16}-\frac{35\cdots 44}{26\cdots 07}a^{15}+\frac{22\cdots 69}{26\cdots 07}a^{14}+\frac{53\cdots 55}{26\cdots 07}a^{13}-\frac{37\cdots 95}{26\cdots 07}a^{12}-\frac{38\cdots 59}{26\cdots 07}a^{11}+\frac{31\cdots 76}{26\cdots 07}a^{10}+\frac{12\cdots 10}{26\cdots 07}a^{9}-\frac{13\cdots 29}{26\cdots 07}a^{8}-\frac{14\cdots 49}{26\cdots 07}a^{7}+\frac{30\cdots 69}{26\cdots 07}a^{6}-\frac{12\cdots 43}{26\cdots 07}a^{5}-\frac{29\cdots 10}{26\cdots 07}a^{4}+\frac{25\cdots 76}{26\cdots 07}a^{3}+\frac{65\cdots 25}{17\cdots 57}a^{2}-\frac{10\cdots 32}{26\cdots 07}a-\frac{27\cdots 69}{26\cdots 07}$, $\frac{23\cdots 80}{26\cdots 07}a^{17}-\frac{15\cdots 53}{26\cdots 07}a^{16}-\frac{84\cdots 06}{26\cdots 07}a^{15}+\frac{65\cdots 13}{26\cdots 07}a^{14}+\frac{99\cdots 03}{26\cdots 07}a^{13}-\frac{10\cdots 88}{26\cdots 07}a^{12}-\frac{30\cdots 41}{26\cdots 07}a^{11}+\frac{88\cdots 00}{26\cdots 07}a^{10}-\frac{23\cdots 73}{26\cdots 07}a^{9}-\frac{35\cdots 69}{26\cdots 07}a^{8}+\frac{19\cdots 29}{26\cdots 07}a^{7}+\frac{68\cdots 69}{26\cdots 07}a^{6}-\frac{47\cdots 42}{26\cdots 07}a^{5}-\frac{48\cdots 11}{26\cdots 07}a^{4}+\frac{36\cdots 40}{26\cdots 07}a^{3}+\frac{35\cdots 03}{17\cdots 57}a^{2}-\frac{54\cdots 06}{26\cdots 07}a+\frac{35\cdots 68}{26\cdots 07}$, $\frac{13\cdots 00}{26\cdots 07}a^{17}-\frac{97\cdots 91}{26\cdots 07}a^{16}-\frac{43\cdots 70}{26\cdots 07}a^{15}+\frac{40\cdots 23}{26\cdots 07}a^{14}+\frac{37\cdots 09}{26\cdots 07}a^{13}-\frac{66\cdots 26}{26\cdots 07}a^{12}+\frac{16\cdots 13}{26\cdots 07}a^{11}+\frac{52\cdots 77}{26\cdots 07}a^{10}-\frac{40\cdots 76}{26\cdots 07}a^{9}-\frac{20\cdots 16}{26\cdots 07}a^{8}+\frac{22\cdots 87}{26\cdots 07}a^{7}+\frac{36\cdots 55}{26\cdots 07}a^{6}-\frac{48\cdots 52}{26\cdots 07}a^{5}-\frac{20\cdots 18}{26\cdots 07}a^{4}+\frac{36\cdots 63}{26\cdots 07}a^{3}-\frac{10\cdots 86}{17\cdots 57}a^{2}-\frac{57\cdots 16}{26\cdots 07}a+\frac{87\cdots 67}{26\cdots 07}$, $\frac{86\cdots 38}{26\cdots 07}a^{17}-\frac{47\cdots 52}{26\cdots 07}a^{16}-\frac{35\cdots 30}{26\cdots 07}a^{15}+\frac{20\cdots 04}{26\cdots 07}a^{14}+\frac{58\cdots 42}{26\cdots 07}a^{13}-\frac{35\cdots 52}{26\cdots 07}a^{12}-\frac{48\cdots 66}{26\cdots 07}a^{11}+\frac{30\cdots 08}{26\cdots 07}a^{10}+\frac{22\cdots 76}{26\cdots 07}a^{9}-\frac{13\cdots 94}{26\cdots 07}a^{8}-\frac{55\cdots 72}{26\cdots 07}a^{7}+\frac{30\cdots 95}{26\cdots 07}a^{6}+\frac{79\cdots 37}{26\cdots 07}a^{5}-\frac{31\cdots 83}{26\cdots 07}a^{4}-\frac{59\cdots 57}{26\cdots 07}a^{3}+\frac{88\cdots 15}{17\cdots 57}a^{2}+\frac{10\cdots 91}{26\cdots 07}a-\frac{13\cdots 96}{26\cdots 07}$, $\frac{25\cdots 78}{26\cdots 07}a^{17}-\frac{16\cdots 35}{26\cdots 07}a^{16}-\frac{88\cdots 48}{26\cdots 07}a^{15}+\frac{69\cdots 25}{26\cdots 07}a^{14}+\frac{10\cdots 49}{26\cdots 07}a^{13}-\frac{11\cdots 23}{26\cdots 07}a^{12}-\frac{28\cdots 09}{26\cdots 07}a^{11}+\frac{92\cdots 39}{26\cdots 07}a^{10}-\frac{28\cdots 17}{26\cdots 07}a^{9}-\frac{37\cdots 00}{26\cdots 07}a^{8}+\frac{22\cdots 27}{26\cdots 07}a^{7}+\frac{71\cdots 71}{26\cdots 07}a^{6}-\frac{52\cdots 35}{26\cdots 07}a^{5}-\frac{50\cdots 81}{26\cdots 07}a^{4}+\frac{41\cdots 63}{26\cdots 07}a^{3}+\frac{28\cdots 03}{17\cdots 57}a^{2}-\frac{81\cdots 92}{26\cdots 07}a+\frac{14\cdots 45}{26\cdots 07}$, $\frac{71\cdots 78}{26\cdots 07}a^{17}-\frac{46\cdots 94}{26\cdots 07}a^{16}-\frac{25\cdots 81}{26\cdots 07}a^{15}+\frac{19\cdots 36}{26\cdots 07}a^{14}+\frac{30\cdots 21}{26\cdots 07}a^{13}-\frac{32\cdots 42}{26\cdots 07}a^{12}-\frac{99\cdots 98}{26\cdots 07}a^{11}+\frac{26\cdots 58}{26\cdots 07}a^{10}-\frac{66\cdots 91}{26\cdots 07}a^{9}-\frac{10\cdots 29}{26\cdots 07}a^{8}+\frac{57\cdots 83}{26\cdots 07}a^{7}+\frac{20\cdots 57}{26\cdots 07}a^{6}-\frac{14\cdots 43}{26\cdots 07}a^{5}-\frac{15\cdots 28}{26\cdots 07}a^{4}+\frac{11\cdots 77}{26\cdots 07}a^{3}+\frac{16\cdots 12}{17\cdots 57}a^{2}-\frac{18\cdots 51}{26\cdots 07}a+\frac{55\cdots 90}{26\cdots 07}$, $\frac{21\cdots 39}{26\cdots 07}a^{17}-\frac{99\cdots 97}{26\cdots 07}a^{16}-\frac{10\cdots 48}{26\cdots 07}a^{15}+\frac{43\cdots 06}{26\cdots 07}a^{14}+\frac{18\cdots 09}{26\cdots 07}a^{13}-\frac{73\cdots 66}{26\cdots 07}a^{12}-\frac{18\cdots 45}{26\cdots 07}a^{11}+\frac{61\cdots 05}{26\cdots 07}a^{10}+\frac{10\cdots 97}{26\cdots 07}a^{9}-\frac{26\cdots 23}{26\cdots 07}a^{8}-\frac{32\cdots 68}{26\cdots 07}a^{7}+\frac{59\cdots 04}{26\cdots 07}a^{6}+\frac{48\cdots 79}{26\cdots 07}a^{5}-\frac{64\cdots 94}{26\cdots 07}a^{4}-\frac{28\cdots 31}{26\cdots 07}a^{3}+\frac{18\cdots 48}{17\cdots 57}a^{2}+\frac{22\cdots 84}{26\cdots 07}a-\frac{19\cdots 40}{26\cdots 07}$, $\frac{84\cdots 93}{26\cdots 07}a^{17}-\frac{55\cdots 74}{26\cdots 07}a^{16}-\frac{28\cdots 19}{26\cdots 07}a^{15}+\frac{23\cdots 59}{26\cdots 07}a^{14}+\frac{31\cdots 04}{26\cdots 07}a^{13}-\frac{38\cdots 41}{26\cdots 07}a^{12}-\frac{53\cdots 19}{26\cdots 07}a^{11}+\frac{31\cdots 58}{26\cdots 07}a^{10}-\frac{12\cdots 16}{26\cdots 07}a^{9}-\frac{12\cdots 02}{26\cdots 07}a^{8}+\frac{87\cdots 45}{26\cdots 07}a^{7}+\frac{23\cdots 82}{26\cdots 07}a^{6}-\frac{19\cdots 72}{26\cdots 07}a^{5}-\frac{16\cdots 28}{26\cdots 07}a^{4}+\frac{15\cdots 20}{26\cdots 07}a^{3}+\frac{11\cdots 17}{17\cdots 57}a^{2}-\frac{23\cdots 99}{26\cdots 07}a+\frac{39\cdots 77}{26\cdots 07}$, $\frac{20\cdots 31}{26\cdots 07}a^{17}-\frac{16\cdots 90}{26\cdots 07}a^{16}-\frac{45\cdots 67}{26\cdots 07}a^{15}+\frac{67\cdots 85}{26\cdots 07}a^{14}-\frac{23\cdots 52}{26\cdots 07}a^{13}-\frac{10\cdots 43}{26\cdots 07}a^{12}+\frac{14\cdots 28}{26\cdots 07}a^{11}+\frac{71\cdots 88}{26\cdots 07}a^{10}-\frac{15\cdots 00}{26\cdots 07}a^{9}-\frac{21\cdots 44}{26\cdots 07}a^{8}+\frac{68\cdots 95}{26\cdots 07}a^{7}+\frac{67\cdots 78}{26\cdots 07}a^{6}-\frac{12\cdots 91}{26\cdots 07}a^{5}+\frac{62\cdots 21}{26\cdots 07}a^{4}+\frac{61\cdots 13}{26\cdots 07}a^{3}-\frac{38\cdots 86}{17\cdots 57}a^{2}+\frac{12\cdots 75}{26\cdots 07}a-\frac{14\cdots 26}{26\cdots 07}$, $\frac{52\cdots 65}{26\cdots 07}a^{17}-\frac{23\cdots 16}{26\cdots 07}a^{16}-\frac{24\cdots 16}{26\cdots 07}a^{15}+\frac{10\cdots 35}{26\cdots 07}a^{14}+\frac{46\cdots 16}{26\cdots 07}a^{13}-\frac{17\cdots 65}{26\cdots 07}a^{12}-\frac{46\cdots 32}{26\cdots 07}a^{11}+\frac{14\cdots 34}{26\cdots 07}a^{10}+\frac{26\cdots 32}{26\cdots 07}a^{9}-\frac{64\cdots 29}{26\cdots 07}a^{8}-\frac{77\cdots 34}{26\cdots 07}a^{7}+\frac{14\cdots 69}{26\cdots 07}a^{6}+\frac{10\cdots 61}{26\cdots 07}a^{5}-\frac{16\cdots 60}{26\cdots 07}a^{4}-\frac{36\cdots 58}{26\cdots 07}a^{3}+\frac{45\cdots 97}{17\cdots 57}a^{2}-\frac{11\cdots 97}{26\cdots 07}a-\frac{10\cdots 10}{26\cdots 07}$, $\frac{19\cdots 90}{26\cdots 07}a^{17}-\frac{12\cdots 59}{26\cdots 07}a^{16}-\frac{65\cdots 57}{26\cdots 07}a^{15}+\frac{54\cdots 05}{26\cdots 07}a^{14}+\frac{68\cdots 09}{26\cdots 07}a^{13}-\frac{90\cdots 34}{26\cdots 07}a^{12}-\frac{55\cdots 91}{26\cdots 07}a^{11}+\frac{72\cdots 67}{26\cdots 07}a^{10}-\frac{34\cdots 21}{26\cdots 07}a^{9}-\frac{29\cdots 31}{26\cdots 07}a^{8}+\frac{22\cdots 10}{26\cdots 07}a^{7}+\frac{55\cdots 85}{26\cdots 07}a^{6}-\frac{50\cdots 43}{26\cdots 07}a^{5}-\frac{39\cdots 55}{26\cdots 07}a^{4}+\frac{39\cdots 68}{26\cdots 07}a^{3}+\frac{32\cdots 26}{17\cdots 57}a^{2}-\frac{70\cdots 85}{26\cdots 07}a-\frac{27\cdots 85}{26\cdots 07}$, $\frac{16\cdots 00}{26\cdots 07}a^{17}-\frac{11\cdots 13}{26\cdots 07}a^{16}-\frac{57\cdots 30}{26\cdots 07}a^{15}+\frac{48\cdots 29}{26\cdots 07}a^{14}+\frac{59\cdots 59}{26\cdots 07}a^{13}-\frac{79\cdots 71}{26\cdots 07}a^{12}-\frac{18\cdots 03}{26\cdots 07}a^{11}+\frac{64\cdots 64}{26\cdots 07}a^{10}-\frac{33\cdots 48}{26\cdots 07}a^{9}-\frac{26\cdots 23}{26\cdots 07}a^{8}+\frac{21\cdots 75}{26\cdots 07}a^{7}+\frac{49\cdots 64}{26\cdots 07}a^{6}-\frac{49\cdots 28}{26\cdots 07}a^{5}-\frac{34\cdots 97}{26\cdots 07}a^{4}+\frac{40\cdots 07}{26\cdots 07}a^{3}+\frac{22\cdots 05}{17\cdots 57}a^{2}-\frac{73\cdots 29}{26\cdots 07}a+\frac{10\cdots 39}{26\cdots 07}$, $\frac{38\cdots 40}{26\cdots 07}a^{17}-\frac{25\cdots 74}{26\cdots 07}a^{16}-\frac{13\cdots 88}{26\cdots 07}a^{15}+\frac{10\cdots 44}{26\cdots 07}a^{14}+\frac{15\cdots 64}{26\cdots 07}a^{13}-\frac{17\cdots 14}{26\cdots 07}a^{12}-\frac{36\cdots 88}{26\cdots 07}a^{11}+\frac{14\cdots 10}{26\cdots 07}a^{10}-\frac{48\cdots 04}{26\cdots 07}a^{9}-\frac{58\cdots 02}{26\cdots 07}a^{8}+\frac{35\cdots 92}{26\cdots 07}a^{7}+\frac{11\cdots 12}{26\cdots 07}a^{6}-\frac{84\cdots 50}{26\cdots 07}a^{5}-\frac{85\cdots 28}{26\cdots 07}a^{4}+\frac{65\cdots 90}{26\cdots 07}a^{3}+\frac{10\cdots 34}{17\cdots 57}a^{2}-\frac{93\cdots 68}{26\cdots 07}a+\frac{11\cdots 75}{26\cdots 07}$
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| Regulator: | \( 6552525312.42 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 6552525312.42 \cdot 1}{2\cdot\sqrt{34205654728777159191037355893457}}\cr\approx \mathstrut & 0.146848555802 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.3.361.1, 6.6.640267073.1, \(\Q(\zeta_{19})^+\) |
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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | R | R | $18$ | $18$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.9.2.9a1.2 | $x^{18} + 14 x^{11} + 16 x^{10} + 28 x^{9} + 49 x^{4} + 112 x^{3} + 260 x^{2} + 224 x + 213$ | $2$ | $9$ | $9$ | $C_{18}$ | $$[\ ]_{2}^{9}$$ |
|
\(19\)
| 19.1.9.8a1.1 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $$[\ ]_{9}$$ |
| 19.1.9.8a1.1 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $$[\ ]_{9}$$ |