Normalized defining polynomial
\( x^{17} - x^{16} - 112 x^{15} + 47 x^{14} + 3976 x^{13} - 4314 x^{12} - 64388 x^{11} + 136247 x^{10} + \cdots + 619 \)
Invariants
| Degree: | $17$ |
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| Signature: | $[17, 0]$ |
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| Discriminant: |
\(113335617496346216833223278514633468161\)
\(\medspace = 239^{16}\)
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| Root discriminant: | \(173.18\) |
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| Galois root discriminant: | $239^{16/17}\approx 173.17776964513632$ | ||
| Ramified primes: |
\(239\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{17}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(239\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{239}(128,·)$, $\chi_{239}(1,·)$, $\chi_{239}(67,·)$, $\chi_{239}(132,·)$, $\chi_{239}(6,·)$, $\chi_{239}(71,·)$, $\chi_{239}(75,·)$, $\chi_{239}(211,·)$, $\chi_{239}(22,·)$, $\chi_{239}(216,·)$, $\chi_{239}(163,·)$, $\chi_{239}(36,·)$, $\chi_{239}(101,·)$, $\chi_{239}(166,·)$, $\chi_{239}(40,·)$, $\chi_{239}(51,·)$, $\chi_{239}(187,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17044241}a^{15}+\frac{7923170}{17044241}a^{14}+\frac{689862}{17044241}a^{13}+\frac{6847608}{17044241}a^{12}-\frac{7864577}{17044241}a^{11}+\frac{293236}{17044241}a^{10}-\frac{5493659}{17044241}a^{9}+\frac{7262608}{17044241}a^{8}-\frac{1203353}{17044241}a^{7}+\frac{1583552}{17044241}a^{6}-\frac{1066023}{17044241}a^{5}-\frac{510254}{17044241}a^{4}+\frac{638656}{17044241}a^{3}+\frac{5875424}{17044241}a^{2}-\frac{1483120}{17044241}a+\frac{1127039}{17044241}$, $\frac{1}{31\cdots 09}a^{16}+\frac{32\cdots 12}{11\cdots 23}a^{15}+\frac{93\cdots 60}{31\cdots 09}a^{14}+\frac{55\cdots 64}{31\cdots 09}a^{13}+\frac{14\cdots 02}{31\cdots 09}a^{12}-\frac{97\cdots 08}{31\cdots 09}a^{11}+\frac{28\cdots 91}{31\cdots 09}a^{10}+\frac{69\cdots 00}{31\cdots 09}a^{9}-\frac{23\cdots 90}{31\cdots 09}a^{8}+\frac{13\cdots 75}{31\cdots 09}a^{7}-\frac{16\cdots 24}{31\cdots 09}a^{6}+\frac{14\cdots 41}{31\cdots 09}a^{5}+\frac{11\cdots 08}{31\cdots 09}a^{4}+\frac{64\cdots 08}{31\cdots 09}a^{3}+\frac{30\cdots 12}{31\cdots 09}a^{2}+\frac{65\cdots 77}{31\cdots 09}a-\frac{70\cdots 01}{50\cdots 11}$
| Monogenic: | No | |
| 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: | $16$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{36\cdots 80}{31\cdots 09}a^{16}-\frac{81\cdots 54}{31\cdots 09}a^{15}-\frac{42\cdots 58}{31\cdots 09}a^{14}+\frac{65\cdots 01}{31\cdots 09}a^{13}+\frac{16\cdots 93}{31\cdots 09}a^{12}-\frac{29\cdots 32}{31\cdots 09}a^{11}-\frac{29\cdots 49}{31\cdots 09}a^{10}+\frac{70\cdots 86}{31\cdots 09}a^{9}+\frac{21\cdots 90}{31\cdots 09}a^{8}-\frac{78\cdots 98}{31\cdots 09}a^{7}-\frac{14\cdots 43}{31\cdots 09}a^{6}+\frac{31\cdots 98}{31\cdots 09}a^{5}-\frac{42\cdots 92}{31\cdots 09}a^{4}+\frac{15\cdots 21}{31\cdots 09}a^{3}+\frac{41\cdots 38}{31\cdots 09}a^{2}-\frac{22\cdots 10}{31\cdots 09}a+\frac{98\cdots 95}{50\cdots 11}$, $\frac{18\cdots 51}{31\cdots 09}a^{16}+\frac{36\cdots 46}{31\cdots 09}a^{15}-\frac{20\cdots 11}{31\cdots 09}a^{14}-\frac{15\cdots 05}{31\cdots 09}a^{13}+\frac{69\cdots 97}{31\cdots 09}a^{12}+\frac{40\cdots 93}{31\cdots 09}a^{11}-\frac{11\cdots 46}{31\cdots 09}a^{10}+\frac{11\cdots 05}{31\cdots 09}a^{9}+\frac{86\cdots 26}{31\cdots 09}a^{8}-\frac{19\cdots 64}{31\cdots 09}a^{7}-\frac{44\cdots 98}{11\cdots 23}a^{6}+\frac{87\cdots 64}{31\cdots 09}a^{5}-\frac{11\cdots 65}{31\cdots 09}a^{4}+\frac{75\cdots 85}{31\cdots 09}a^{3}-\frac{22\cdots 13}{31\cdots 09}a^{2}+\frac{29\cdots 31}{31\cdots 09}a-\frac{16\cdots 04}{50\cdots 11}$, $\frac{22\cdots 92}{31\cdots 09}a^{16}+\frac{56\cdots 75}{31\cdots 09}a^{15}-\frac{24\cdots 43}{31\cdots 09}a^{14}-\frac{75\cdots 51}{31\cdots 09}a^{13}+\frac{75\cdots 97}{31\cdots 09}a^{12}+\frac{18\cdots 41}{31\cdots 09}a^{11}-\frac{12\cdots 65}{31\cdots 09}a^{10}-\frac{14\cdots 27}{31\cdots 09}a^{9}+\frac{11\cdots 78}{31\cdots 09}a^{8}-\frac{10\cdots 91}{31\cdots 09}a^{7}-\frac{47\cdots 63}{31\cdots 09}a^{6}+\frac{55\cdots 29}{31\cdots 09}a^{5}+\frac{35\cdots 09}{31\cdots 09}a^{4}-\frac{10\cdots 30}{31\cdots 09}a^{3}+\frac{69\cdots 91}{31\cdots 09}a^{2}-\frac{54\cdots 83}{11\cdots 43}a+\frac{32\cdots 67}{50\cdots 11}$, $\frac{10\cdots 01}{31\cdots 09}a^{16}+\frac{24\cdots 78}{13\cdots 21}a^{15}-\frac{10\cdots 70}{31\cdots 09}a^{14}-\frac{67\cdots 59}{31\cdots 09}a^{13}+\frac{24\cdots 47}{31\cdots 09}a^{12}+\frac{17\cdots 08}{31\cdots 09}a^{11}-\frac{31\cdots 29}{31\cdots 09}a^{10}-\frac{20\cdots 84}{31\cdots 09}a^{9}+\frac{30\cdots 82}{31\cdots 09}a^{8}+\frac{10\cdots 04}{31\cdots 09}a^{7}-\frac{20\cdots 16}{31\cdots 09}a^{6}-\frac{13\cdots 57}{31\cdots 09}a^{5}+\frac{54\cdots 82}{31\cdots 09}a^{4}-\frac{45\cdots 20}{31\cdots 09}a^{3}+\frac{14\cdots 47}{31\cdots 09}a^{2}-\frac{15\cdots 23}{31\cdots 09}a+\frac{69\cdots 10}{50\cdots 11}$, $\frac{27\cdots 24}{31\cdots 09}a^{16}+\frac{58\cdots 42}{31\cdots 09}a^{15}-\frac{30\cdots 37}{31\cdots 09}a^{14}-\frac{35\cdots 37}{13\cdots 21}a^{13}+\frac{94\cdots 10}{31\cdots 09}a^{12}+\frac{18\cdots 24}{31\cdots 09}a^{11}-\frac{15\cdots 02}{31\cdots 09}a^{10}-\frac{11\cdots 15}{31\cdots 09}a^{9}+\frac{12\cdots 13}{31\cdots 09}a^{8}-\frac{61\cdots 87}{31\cdots 09}a^{7}-\frac{45\cdots 79}{31\cdots 09}a^{6}+\frac{72\cdots 75}{31\cdots 09}a^{5}-\frac{14\cdots 61}{31\cdots 09}a^{4}-\frac{41\cdots 43}{31\cdots 09}a^{3}+\frac{30\cdots 00}{31\cdots 09}a^{2}-\frac{61\cdots 43}{31\cdots 09}a+\frac{15\cdots 30}{50\cdots 11}$, $\frac{12\cdots 83}{31\cdots 09}a^{16}+\frac{98\cdots 12}{31\cdots 09}a^{15}-\frac{14\cdots 08}{31\cdots 09}a^{14}-\frac{18\cdots 20}{31\cdots 09}a^{13}+\frac{46\cdots 30}{31\cdots 09}a^{12}+\frac{25\cdots 47}{31\cdots 09}a^{11}-\frac{75\cdots 27}{31\cdots 09}a^{10}+\frac{45\cdots 65}{31\cdots 09}a^{9}+\frac{57\cdots 09}{31\cdots 09}a^{8}-\frac{10\cdots 41}{31\cdots 09}a^{7}-\frac{96\cdots 67}{31\cdots 09}a^{6}+\frac{52\cdots 03}{31\cdots 09}a^{5}-\frac{70\cdots 08}{31\cdots 09}a^{4}+\frac{46\cdots 67}{31\cdots 09}a^{3}-\frac{15\cdots 02}{31\cdots 09}a^{2}+\frac{24\cdots 50}{31\cdots 09}a-\frac{17\cdots 94}{50\cdots 11}$, $\frac{24\cdots 49}{31\cdots 09}a^{16}+\frac{92\cdots 45}{31\cdots 09}a^{15}-\frac{27\cdots 90}{31\cdots 09}a^{14}-\frac{26\cdots 43}{31\cdots 09}a^{13}+\frac{94\cdots 90}{31\cdots 09}a^{12}+\frac{22\cdots 21}{31\cdots 09}a^{11}-\frac{15\cdots 85}{31\cdots 09}a^{10}+\frac{12\cdots 15}{31\cdots 09}a^{9}+\frac{12\cdots 85}{31\cdots 09}a^{8}-\frac{23\cdots 90}{31\cdots 09}a^{7}-\frac{21\cdots 81}{31\cdots 09}a^{6}+\frac{11\cdots 29}{31\cdots 09}a^{5}-\frac{14\cdots 25}{31\cdots 09}a^{4}+\frac{76\cdots 45}{31\cdots 09}a^{3}-\frac{17\cdots 48}{31\cdots 09}a^{2}+\frac{11\cdots 82}{31\cdots 09}a-\frac{21\cdots 25}{50\cdots 11}$, $\frac{27\cdots 57}{31\cdots 09}a^{16}+\frac{16\cdots 98}{31\cdots 09}a^{15}-\frac{27\cdots 56}{31\cdots 09}a^{14}-\frac{19\cdots 96}{31\cdots 09}a^{13}+\frac{56\cdots 39}{31\cdots 09}a^{12}+\frac{51\cdots 06}{31\cdots 09}a^{11}-\frac{59\cdots 19}{31\cdots 09}a^{10}-\frac{60\cdots 50}{31\cdots 09}a^{9}+\frac{58\cdots 82}{31\cdots 09}a^{8}+\frac{34\cdots 28}{31\cdots 09}a^{7}-\frac{46\cdots 95}{31\cdots 09}a^{6}-\frac{72\cdots 26}{31\cdots 09}a^{5}+\frac{16\cdots 57}{31\cdots 09}a^{4}-\frac{75\cdots 54}{31\cdots 09}a^{3}-\frac{83\cdots 74}{31\cdots 09}a^{2}+\frac{85\cdots 58}{31\cdots 09}a-\frac{84\cdots 13}{50\cdots 11}$, $\frac{64\cdots 09}{31\cdots 09}a^{16}-\frac{21\cdots 82}{31\cdots 09}a^{15}-\frac{71\cdots 78}{31\cdots 09}a^{14}-\frac{16\cdots 88}{31\cdots 09}a^{13}+\frac{24\cdots 05}{31\cdots 09}a^{12}-\frac{12\cdots 29}{31\cdots 09}a^{11}-\frac{40\cdots 77}{31\cdots 09}a^{10}+\frac{63\cdots 19}{31\cdots 09}a^{9}+\frac{28\cdots 92}{31\cdots 09}a^{8}-\frac{85\cdots 01}{31\cdots 09}a^{7}-\frac{45\cdots 06}{31\cdots 09}a^{6}+\frac{33\cdots 52}{31\cdots 09}a^{5}-\frac{60\cdots 19}{31\cdots 09}a^{4}+\frac{48\cdots 75}{31\cdots 09}a^{3}-\frac{18\cdots 76}{31\cdots 09}a^{2}+\frac{26\cdots 28}{31\cdots 09}a-\frac{62\cdots 90}{50\cdots 11}$, $\frac{43\cdots 16}{31\cdots 09}a^{16}-\frac{16\cdots 10}{31\cdots 09}a^{15}-\frac{48\cdots 34}{31\cdots 09}a^{14}-\frac{11\cdots 51}{31\cdots 09}a^{13}+\frac{17\cdots 79}{31\cdots 09}a^{12}-\frac{62\cdots 30}{31\cdots 09}a^{11}-\frac{29\cdots 83}{31\cdots 09}a^{10}+\frac{37\cdots 18}{31\cdots 09}a^{9}+\frac{22\cdots 63}{31\cdots 09}a^{8}-\frac{54\cdots 95}{31\cdots 09}a^{7}-\frac{29\cdots 10}{31\cdots 09}a^{6}+\frac{23\cdots 46}{31\cdots 09}a^{5}-\frac{32\cdots 91}{31\cdots 09}a^{4}+\frac{18\cdots 55}{31\cdots 09}a^{3}-\frac{43\cdots 74}{31\cdots 09}a^{2}+\frac{21\cdots 05}{31\cdots 09}a-\frac{30\cdots 92}{50\cdots 11}$, $\frac{35\cdots 04}{31\cdots 09}a^{16}-\frac{55\cdots 87}{31\cdots 09}a^{15}-\frac{40\cdots 79}{31\cdots 09}a^{14}+\frac{36\cdots 05}{31\cdots 09}a^{13}+\frac{15\cdots 09}{31\cdots 09}a^{12}-\frac{20\cdots 37}{31\cdots 09}a^{11}-\frac{26\cdots 05}{31\cdots 09}a^{10}+\frac{55\cdots 44}{31\cdots 09}a^{9}+\frac{18\cdots 33}{31\cdots 09}a^{8}-\frac{64\cdots 21}{31\cdots 09}a^{7}-\frac{61\cdots 85}{31\cdots 09}a^{6}+\frac{25\cdots 43}{31\cdots 09}a^{5}-\frac{39\cdots 14}{31\cdots 09}a^{4}+\frac{25\cdots 74}{31\cdots 09}a^{3}-\frac{65\cdots 28}{31\cdots 09}a^{2}+\frac{50\cdots 38}{31\cdots 09}a-\frac{77\cdots 31}{50\cdots 11}$, $\frac{39\cdots 47}{31\cdots 09}a^{16}-\frac{21\cdots 15}{31\cdots 09}a^{15}-\frac{44\cdots 09}{31\cdots 09}a^{14}-\frac{25\cdots 85}{31\cdots 09}a^{13}+\frac{15\cdots 49}{31\cdots 09}a^{12}-\frac{80\cdots 24}{31\cdots 09}a^{11}-\frac{26\cdots 84}{31\cdots 09}a^{10}+\frac{37\cdots 84}{31\cdots 09}a^{9}+\frac{20\cdots 92}{31\cdots 09}a^{8}-\frac{51\cdots 48}{31\cdots 09}a^{7}-\frac{26\cdots 40}{31\cdots 09}a^{6}+\frac{21\cdots 71}{31\cdots 09}a^{5}-\frac{29\cdots 64}{31\cdots 09}a^{4}+\frac{16\cdots 81}{31\cdots 09}a^{3}-\frac{37\cdots 62}{31\cdots 09}a^{2}+\frac{23\cdots 77}{31\cdots 09}a-\frac{50\cdots 27}{50\cdots 11}$, $\frac{17\cdots 04}{31\cdots 09}a^{16}+\frac{18\cdots 56}{31\cdots 09}a^{15}-\frac{19\cdots 03}{31\cdots 09}a^{14}-\frac{30\cdots 96}{31\cdots 09}a^{13}+\frac{63\cdots 00}{31\cdots 09}a^{12}+\frac{53\cdots 84}{31\cdots 09}a^{11}-\frac{10\cdots 11}{31\cdots 09}a^{10}+\frac{30\cdots 00}{31\cdots 09}a^{9}+\frac{80\cdots 04}{31\cdots 09}a^{8}-\frac{12\cdots 10}{31\cdots 09}a^{7}-\frac{17\cdots 26}{31\cdots 09}a^{6}+\frac{65\cdots 19}{31\cdots 09}a^{5}-\frac{72\cdots 42}{31\cdots 09}a^{4}+\frac{36\cdots 22}{31\cdots 09}a^{3}-\frac{82\cdots 61}{31\cdots 09}a^{2}+\frac{61\cdots 59}{31\cdots 09}a-\frac{22\cdots 47}{50\cdots 11}$, $\frac{75\cdots 70}{31\cdots 09}a^{16}-\frac{29\cdots 00}{31\cdots 09}a^{15}-\frac{84\cdots 82}{31\cdots 09}a^{14}-\frac{16\cdots 36}{31\cdots 09}a^{13}+\frac{30\cdots 76}{31\cdots 09}a^{12}-\frac{13\cdots 12}{31\cdots 09}a^{11}-\frac{19\cdots 92}{11\cdots 43}a^{10}+\frac{71\cdots 08}{31\cdots 09}a^{9}+\frac{37\cdots 35}{31\cdots 09}a^{8}-\frac{99\cdots 10}{31\cdots 09}a^{7}-\frac{40\cdots 13}{31\cdots 09}a^{6}+\frac{42\cdots 71}{31\cdots 09}a^{5}-\frac{60\cdots 80}{31\cdots 09}a^{4}+\frac{35\cdots 01}{31\cdots 09}a^{3}-\frac{80\cdots 71}{31\cdots 09}a^{2}+\frac{44\cdots 90}{31\cdots 09}a-\frac{95\cdots 43}{50\cdots 11}$, $\frac{47\cdots 63}{31\cdots 09}a^{16}+\frac{63\cdots 23}{31\cdots 09}a^{15}-\frac{51\cdots 66}{31\cdots 09}a^{14}-\frac{99\cdots 24}{31\cdots 09}a^{13}+\frac{16\cdots 66}{31\cdots 09}a^{12}+\frac{18\cdots 36}{31\cdots 09}a^{11}-\frac{26\cdots 41}{31\cdots 09}a^{10}+\frac{22\cdots 29}{31\cdots 09}a^{9}+\frac{20\cdots 90}{31\cdots 09}a^{8}-\frac{28\cdots 96}{31\cdots 09}a^{7}-\frac{49\cdots 74}{31\cdots 09}a^{6}+\frac{16\cdots 47}{31\cdots 09}a^{5}-\frac{17\cdots 05}{31\cdots 09}a^{4}+\frac{83\cdots 56}{31\cdots 09}a^{3}-\frac{17\cdots 56}{31\cdots 09}a^{2}+\frac{97\cdots 88}{31\cdots 09}a-\frac{16\cdots 33}{50\cdots 11}$, $\frac{60\cdots 44}{31\cdots 09}a^{16}+\frac{20\cdots 64}{31\cdots 09}a^{15}-\frac{66\cdots 15}{31\cdots 09}a^{14}-\frac{61\cdots 14}{31\cdots 09}a^{13}+\frac{22\cdots 92}{31\cdots 09}a^{12}+\frac{44\cdots 66}{31\cdots 09}a^{11}-\frac{37\cdots 88}{31\cdots 09}a^{10}+\frac{31\cdots 02}{31\cdots 09}a^{9}+\frac{28\cdots 05}{31\cdots 09}a^{8}-\frac{58\cdots 09}{31\cdots 09}a^{7}-\frac{45\cdots 56}{31\cdots 09}a^{6}+\frac{27\cdots 84}{31\cdots 09}a^{5}-\frac{35\cdots 21}{31\cdots 09}a^{4}+\frac{21\cdots 25}{31\cdots 09}a^{3}-\frac{62\cdots 23}{31\cdots 09}a^{2}+\frac{71\cdots 42}{31\cdots 09}a-\frac{25\cdots 42}{50\cdots 11}$
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| Regulator: | \( 24055588816300 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{0}\cdot 24055588816300 \cdot 1}{2\cdot\sqrt{113335617496346216833223278514633468161}}\cr\approx \mathstrut & 0.148085560941044 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 17 |
| The 17 conjugacy class representatives for $C_{17}$ |
| Character table for $C_{17}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
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\(239\)
| Deg $17$ | $17$ | $1$ | $16$ |