Normalized defining polynomial
\( x^{17} - x^{16} - 448 x^{15} + 1309 x^{14} + 75494 x^{13} - 374314 x^{12} - 5597667 x^{11} + \cdots + 117028501127 \)
Invariants
| Degree: | $17$ |
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| Signature: | $[17, 0]$ |
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| Discriminant: |
\(462899178676311160288470191817767102492054133121\)
\(\medspace = 953^{16}\)
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| Root discriminant: | \(636.58\) |
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| Galois root discriminant: | $953^{16/17}\approx 636.5787568645812$ | ||
| Ramified primes: |
\(953\)
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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: | \(953\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{953}(256,·)$, $\chi_{953}(1,·)$, $\chi_{953}(770,·)$, $\chi_{953}(134,·)$, $\chi_{953}(16,·)$, $\chi_{953}(276,·)$, $\chi_{953}(732,·)$, $\chi_{953}(604,·)$, $\chi_{953}(417,·)$, $\chi_{953}(802,·)$, $\chi_{953}(284,·)$, $\chi_{953}(238,·)$, $\chi_{953}(882,·)$, $\chi_{953}(884,·)$, $\chi_{953}(949,·)$, $\chi_{953}(889,·)$, $\chi_{953}(443,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{109}a^{15}-\frac{29}{109}a^{14}-\frac{2}{109}a^{13}-\frac{11}{109}a^{12}+\frac{16}{109}a^{11}-\frac{27}{109}a^{10}+\frac{51}{109}a^{9}+\frac{8}{109}a^{8}-\frac{50}{109}a^{7}-\frac{37}{109}a^{6}-\frac{28}{109}a^{5}+\frac{28}{109}a^{4}-\frac{2}{109}a^{3}+\frac{41}{109}a^{2}+\frac{52}{109}a-\frac{1}{109}$, $\frac{1}{14\cdots 97}a^{16}-\frac{37\cdots 38}{14\cdots 97}a^{15}+\frac{43\cdots 68}{14\cdots 97}a^{14}+\frac{11\cdots 46}{14\cdots 97}a^{13}-\frac{56\cdots 91}{14\cdots 97}a^{12}-\frac{52\cdots 74}{14\cdots 97}a^{11}-\frac{86\cdots 81}{14\cdots 97}a^{10}+\frac{45\cdots 12}{14\cdots 97}a^{9}-\frac{52\cdots 76}{14\cdots 97}a^{8}-\frac{53\cdots 71}{14\cdots 97}a^{7}-\frac{32\cdots 68}{14\cdots 97}a^{6}+\frac{38\cdots 13}{14\cdots 97}a^{5}-\frac{19\cdots 38}{14\cdots 97}a^{4}+\frac{58\cdots 20}{14\cdots 97}a^{3}-\frac{65\cdots 17}{14\cdots 97}a^{2}-\frac{33\cdots 80}{14\cdots 97}a-\frac{32\cdots 35}{14\cdots 97}$
| 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: |
$\frac{86\cdots 32}{14\cdots 97}a^{16}+\frac{26\cdots 87}{14\cdots 97}a^{15}-\frac{37\cdots 29}{14\cdots 97}a^{14}-\frac{39\cdots 68}{14\cdots 97}a^{13}+\frac{63\cdots 87}{14\cdots 97}a^{12}-\frac{64\cdots 56}{14\cdots 97}a^{11}-\frac{50\cdots 54}{14\cdots 97}a^{10}+\frac{15\cdots 20}{14\cdots 97}a^{9}+\frac{18\cdots 53}{14\cdots 97}a^{8}-\frac{92\cdots 68}{14\cdots 97}a^{7}-\frac{15\cdots 64}{14\cdots 97}a^{6}+\frac{17\cdots 67}{14\cdots 97}a^{5}-\frac{21\cdots 55}{14\cdots 97}a^{4}-\frac{59\cdots 03}{14\cdots 97}a^{3}+\frac{15\cdots 28}{14\cdots 97}a^{2}-\frac{72\cdots 70}{14\cdots 97}a-\frac{24\cdots 01}{14\cdots 97}$, $\frac{43\cdots 68}{14\cdots 97}a^{16}+\frac{13\cdots 15}{14\cdots 97}a^{15}-\frac{18\cdots 29}{14\cdots 97}a^{14}-\frac{20\cdots 05}{14\cdots 97}a^{13}+\frac{31\cdots 61}{14\cdots 97}a^{12}-\frac{31\cdots 09}{14\cdots 97}a^{11}-\frac{25\cdots 84}{14\cdots 97}a^{10}+\frac{76\cdots 19}{14\cdots 97}a^{9}+\frac{91\cdots 54}{14\cdots 97}a^{8}-\frac{45\cdots 03}{14\cdots 97}a^{7}-\frac{81\cdots 99}{14\cdots 97}a^{6}+\frac{85\cdots 49}{14\cdots 97}a^{5}-\frac{10\cdots 65}{14\cdots 97}a^{4}-\frac{29\cdots 38}{14\cdots 97}a^{3}+\frac{75\cdots 74}{14\cdots 97}a^{2}-\frac{35\cdots 97}{14\cdots 97}a-\frac{11\cdots 35}{14\cdots 97}$, $\frac{74\cdots 11}{14\cdots 97}a^{16}+\frac{20\cdots 16}{14\cdots 97}a^{15}-\frac{32\cdots 10}{14\cdots 97}a^{14}-\frac{23\cdots 08}{14\cdots 97}a^{13}+\frac{55\cdots 99}{14\cdots 97}a^{12}-\frac{74\cdots 31}{14\cdots 97}a^{11}-\frac{44\cdots 47}{14\cdots 97}a^{10}+\frac{14\cdots 69}{14\cdots 97}a^{9}+\frac{15\cdots 46}{14\cdots 97}a^{8}-\frac{85\cdots 61}{14\cdots 97}a^{7}-\frac{12\cdots 41}{14\cdots 97}a^{6}+\frac{15\cdots 00}{14\cdots 97}a^{5}-\frac{21\cdots 83}{14\cdots 97}a^{4}-\frac{52\cdots 83}{14\cdots 97}a^{3}+\frac{14\cdots 60}{14\cdots 97}a^{2}-\frac{71\cdots 02}{14\cdots 97}a-\frac{23\cdots 47}{14\cdots 97}$, $\frac{52\cdots 31}{14\cdots 97}a^{16}+\frac{13\cdots 40}{14\cdots 97}a^{15}-\frac{22\cdots 89}{14\cdots 97}a^{14}-\frac{15\cdots 90}{14\cdots 97}a^{13}+\frac{38\cdots 97}{14\cdots 97}a^{12}-\frac{53\cdots 00}{14\cdots 97}a^{11}-\frac{31\cdots 29}{14\cdots 97}a^{10}+\frac{10\cdots 96}{14\cdots 97}a^{9}+\frac{10\cdots 16}{14\cdots 97}a^{8}-\frac{60\cdots 09}{14\cdots 97}a^{7}-\frac{87\cdots 21}{14\cdots 97}a^{6}+\frac{10\cdots 00}{14\cdots 97}a^{5}-\frac{15\cdots 90}{14\cdots 97}a^{4}-\frac{37\cdots 01}{14\cdots 97}a^{3}+\frac{10\cdots 92}{14\cdots 97}a^{2}-\frac{50\cdots 64}{14\cdots 97}a-\frac{16\cdots 01}{14\cdots 97}$, $\frac{11\cdots 55}{14\cdots 97}a^{16}+\frac{38\cdots 04}{14\cdots 97}a^{15}-\frac{50\cdots 74}{14\cdots 97}a^{14}-\frac{64\cdots 10}{14\cdots 97}a^{13}+\frac{84\cdots 93}{14\cdots 97}a^{12}-\frac{70\cdots 52}{14\cdots 97}a^{11}-\frac{68\cdots 71}{14\cdots 97}a^{10}+\frac{19\cdots 18}{14\cdots 97}a^{9}+\frac{24\cdots 98}{14\cdots 97}a^{8}-\frac{11\cdots 05}{14\cdots 97}a^{7}-\frac{21\cdots 53}{14\cdots 97}a^{6}+\frac{22\cdots 11}{14\cdots 97}a^{5}-\frac{27\cdots 13}{14\cdots 97}a^{4}-\frac{76\cdots 37}{14\cdots 97}a^{3}+\frac{20\cdots 69}{14\cdots 97}a^{2}-\frac{99\cdots 87}{14\cdots 97}a-\frac{32\cdots 45}{14\cdots 97}$, $\frac{13\cdots 61}{14\cdots 97}a^{16}-\frac{16\cdots 75}{14\cdots 97}a^{15}-\frac{59\cdots 20}{14\cdots 97}a^{14}+\frac{18\cdots 46}{14\cdots 97}a^{13}+\frac{10\cdots 84}{14\cdots 97}a^{12}-\frac{50\cdots 77}{14\cdots 97}a^{11}-\frac{79\cdots 71}{14\cdots 97}a^{10}+\frac{56\cdots 90}{14\cdots 97}a^{9}+\frac{21\cdots 61}{14\cdots 97}a^{8}-\frac{26\cdots 21}{14\cdots 97}a^{7}+\frac{20\cdots 37}{14\cdots 97}a^{6}+\frac{42\cdots 85}{14\cdots 97}a^{5}-\frac{12\cdots 47}{14\cdots 97}a^{4}-\frac{65\cdots 71}{14\cdots 97}a^{3}+\frac{65\cdots 05}{14\cdots 97}a^{2}-\frac{59\cdots 92}{14\cdots 97}a-\frac{15\cdots 98}{14\cdots 97}$, $\frac{49\cdots 14}{14\cdots 97}a^{16}+\frac{15\cdots 57}{14\cdots 97}a^{15}-\frac{21\cdots 15}{14\cdots 97}a^{14}-\frac{21\cdots 39}{13\cdots 33}a^{13}+\frac{36\cdots 82}{14\cdots 97}a^{12}-\frac{36\cdots 69}{14\cdots 97}a^{11}-\frac{29\cdots 58}{14\cdots 97}a^{10}+\frac{81\cdots 25}{13\cdots 33}a^{9}+\frac{10\cdots 86}{14\cdots 97}a^{8}-\frac{53\cdots 20}{14\cdots 97}a^{7}-\frac{89\cdots 23}{14\cdots 97}a^{6}+\frac{98\cdots 15}{14\cdots 97}a^{5}-\frac{12\cdots 92}{14\cdots 97}a^{4}-\frac{33\cdots 04}{14\cdots 97}a^{3}+\frac{89\cdots 01}{14\cdots 97}a^{2}-\frac{43\cdots 95}{14\cdots 97}a-\frac{14\cdots 07}{14\cdots 97}$, $\frac{52\cdots 19}{14\cdots 97}a^{16}+\frac{43\cdots 34}{14\cdots 97}a^{15}-\frac{23\cdots 86}{14\cdots 97}a^{14}+\frac{23\cdots 44}{14\cdots 97}a^{13}+\frac{41\cdots 46}{14\cdots 97}a^{12}-\frac{11\cdots 02}{14\cdots 97}a^{11}-\frac{33\cdots 61}{14\cdots 97}a^{10}+\frac{15\cdots 95}{14\cdots 97}a^{9}+\frac{11\cdots 18}{14\cdots 97}a^{8}-\frac{79\cdots 61}{14\cdots 97}a^{7}-\frac{64\cdots 14}{14\cdots 97}a^{6}+\frac{13\cdots 51}{14\cdots 97}a^{5}-\frac{22\cdots 13}{14\cdots 97}a^{4}-\frac{44\cdots 61}{14\cdots 97}a^{3}+\frac{13\cdots 67}{14\cdots 97}a^{2}-\frac{72\cdots 36}{14\cdots 97}a-\frac{23\cdots 86}{14\cdots 97}$, $\frac{23\cdots 82}{14\cdots 97}a^{16}+\frac{87\cdots 55}{14\cdots 97}a^{15}-\frac{10\cdots 95}{14\cdots 97}a^{14}-\frac{16\cdots 55}{14\cdots 97}a^{13}+\frac{17\cdots 32}{14\cdots 97}a^{12}-\frac{93\cdots 67}{14\cdots 97}a^{11}-\frac{13\cdots 73}{14\cdots 97}a^{10}+\frac{35\cdots 34}{14\cdots 97}a^{9}+\frac{49\cdots 35}{14\cdots 97}a^{8}-\frac{23\cdots 80}{14\cdots 97}a^{7}-\frac{45\cdots 23}{14\cdots 97}a^{6}+\frac{43\cdots 93}{14\cdots 97}a^{5}-\frac{50\cdots 51}{14\cdots 97}a^{4}-\frac{15\cdots 22}{14\cdots 97}a^{3}+\frac{38\cdots 16}{14\cdots 97}a^{2}-\frac{18\cdots 87}{14\cdots 97}a-\frac{59\cdots 94}{14\cdots 97}$, $\frac{54\cdots 74}{14\cdots 97}a^{16}+\frac{12\cdots 85}{14\cdots 97}a^{15}-\frac{23\cdots 59}{14\cdots 97}a^{14}-\frac{63\cdots 75}{14\cdots 97}a^{13}+\frac{38\cdots 83}{14\cdots 97}a^{12}-\frac{75\cdots 10}{14\cdots 97}a^{11}-\frac{28\cdots 57}{14\cdots 97}a^{10}+\frac{12\cdots 95}{14\cdots 97}a^{9}+\frac{83\cdots 60}{14\cdots 97}a^{8}-\frac{65\cdots 20}{14\cdots 97}a^{7}+\frac{27\cdots 10}{14\cdots 97}a^{6}+\frac{96\cdots 64}{14\cdots 97}a^{5}-\frac{32\cdots 70}{14\cdots 97}a^{4}+\frac{15\cdots 01}{14\cdots 97}a^{3}+\frac{13\cdots 68}{14\cdots 97}a^{2}-\frac{30\cdots 45}{14\cdots 97}a+\frac{18\cdots 78}{14\cdots 97}$, $\frac{22\cdots 97}{14\cdots 97}a^{16}+\frac{69\cdots 32}{14\cdots 97}a^{15}-\frac{99\cdots 97}{14\cdots 97}a^{14}-\frac{10\cdots 10}{14\cdots 97}a^{13}+\frac{16\cdots 43}{14\cdots 97}a^{12}-\frac{17\cdots 86}{14\cdots 97}a^{11}-\frac{13\cdots 36}{14\cdots 97}a^{10}+\frac{40\cdots 03}{14\cdots 97}a^{9}+\frac{47\cdots 22}{14\cdots 97}a^{8}-\frac{24\cdots 33}{14\cdots 97}a^{7}-\frac{40\cdots 76}{14\cdots 97}a^{6}+\frac{45\cdots 66}{14\cdots 97}a^{5}-\frac{57\cdots 29}{14\cdots 97}a^{4}-\frac{15\cdots 76}{14\cdots 97}a^{3}+\frac{40\cdots 22}{14\cdots 97}a^{2}-\frac{19\cdots 46}{14\cdots 97}a-\frac{65\cdots 91}{14\cdots 97}$, $\frac{62\cdots 28}{14\cdots 97}a^{16}+\frac{16\cdots 39}{14\cdots 97}a^{15}-\frac{27\cdots 23}{14\cdots 97}a^{14}-\frac{19\cdots 31}{14\cdots 97}a^{13}+\frac{46\cdots 53}{14\cdots 97}a^{12}-\frac{62\cdots 81}{14\cdots 97}a^{11}-\frac{36\cdots 25}{14\cdots 97}a^{10}+\frac{12\cdots 92}{14\cdots 97}a^{9}+\frac{12\cdots 82}{14\cdots 97}a^{8}-\frac{71\cdots 56}{14\cdots 97}a^{7}-\frac{10\cdots 56}{14\cdots 97}a^{6}+\frac{12\cdots 86}{14\cdots 97}a^{5}-\frac{17\cdots 38}{14\cdots 97}a^{4}-\frac{43\cdots 18}{14\cdots 97}a^{3}+\frac{12\cdots 73}{14\cdots 97}a^{2}-\frac{59\cdots 23}{14\cdots 97}a-\frac{19\cdots 84}{14\cdots 97}$, $\frac{82\cdots 10}{14\cdots 97}a^{16}+\frac{25\cdots 49}{14\cdots 97}a^{15}-\frac{35\cdots 49}{14\cdots 97}a^{14}-\frac{37\cdots 70}{14\cdots 97}a^{13}+\frac{60\cdots 96}{14\cdots 97}a^{12}-\frac{62\cdots 91}{14\cdots 97}a^{11}-\frac{48\cdots 75}{14\cdots 97}a^{10}+\frac{14\cdots 76}{14\cdots 97}a^{9}+\frac{17\cdots 34}{14\cdots 97}a^{8}-\frac{87\cdots 09}{14\cdots 97}a^{7}-\frac{14\cdots 47}{14\cdots 97}a^{6}+\frac{16\cdots 58}{14\cdots 97}a^{5}-\frac{20\cdots 39}{14\cdots 97}a^{4}-\frac{56\cdots 83}{14\cdots 97}a^{3}+\frac{14\cdots 03}{14\cdots 97}a^{2}-\frac{70\cdots 02}{14\cdots 97}a-\frac{23\cdots 91}{14\cdots 97}$, $\frac{11\cdots 69}{14\cdots 97}a^{16}+\frac{57\cdots 46}{14\cdots 97}a^{15}-\frac{52\cdots 33}{14\cdots 97}a^{14}+\frac{75\cdots 36}{14\cdots 97}a^{13}+\frac{89\cdots 65}{14\cdots 97}a^{12}-\frac{30\cdots 02}{14\cdots 97}a^{11}-\frac{70\cdots 73}{14\cdots 97}a^{10}+\frac{38\cdots 87}{14\cdots 97}a^{9}+\frac{21\cdots 04}{14\cdots 97}a^{8}-\frac{19\cdots 63}{14\cdots 97}a^{7}+\frac{25\cdots 76}{14\cdots 97}a^{6}+\frac{31\cdots 05}{14\cdots 97}a^{5}-\frac{78\cdots 07}{14\cdots 97}a^{4}-\frac{66\cdots 27}{14\cdots 97}a^{3}+\frac{43\cdots 79}{14\cdots 97}a^{2}-\frac{35\cdots 21}{14\cdots 97}a-\frac{10\cdots 17}{14\cdots 97}$, $\frac{23\cdots 69}{14\cdots 97}a^{16}+\frac{61\cdots 50}{14\cdots 97}a^{15}-\frac{10\cdots 83}{14\cdots 97}a^{14}-\frac{65\cdots 58}{14\cdots 97}a^{13}+\frac{17\cdots 30}{14\cdots 97}a^{12}-\frac{24\cdots 20}{14\cdots 97}a^{11}-\frac{14\cdots 04}{14\cdots 97}a^{10}+\frac{47\cdots 06}{14\cdots 97}a^{9}+\frac{49\cdots 30}{14\cdots 97}a^{8}-\frac{27\cdots 34}{14\cdots 97}a^{7}-\frac{38\cdots 73}{14\cdots 97}a^{6}+\frac{49\cdots 59}{14\cdots 97}a^{5}-\frac{68\cdots 41}{14\cdots 97}a^{4}-\frac{16\cdots 81}{14\cdots 97}a^{3}+\frac{46\cdots 44}{14\cdots 97}a^{2}-\frac{23\cdots 65}{14\cdots 97}a-\frac{76\cdots 02}{14\cdots 97}$, $\frac{96\cdots 83}{14\cdots 97}a^{16}+\frac{30\cdots 44}{14\cdots 97}a^{15}-\frac{41\cdots 18}{14\cdots 97}a^{14}-\frac{49\cdots 30}{14\cdots 97}a^{13}+\frac{70\cdots 66}{14\cdots 97}a^{12}-\frac{64\cdots 21}{14\cdots 97}a^{11}-\frac{56\cdots 55}{14\cdots 97}a^{10}+\frac{16\cdots 19}{14\cdots 97}a^{9}+\frac{18\cdots 40}{13\cdots 33}a^{8}-\frac{10\cdots 84}{14\cdots 97}a^{7}-\frac{17\cdots 64}{14\cdots 97}a^{6}+\frac{18\cdots 45}{14\cdots 97}a^{5}-\frac{23\cdots 10}{14\cdots 97}a^{4}-\frac{64\cdots 49}{14\cdots 97}a^{3}+\frac{16\cdots 44}{14\cdots 97}a^{2}-\frac{80\cdots 83}{14\cdots 97}a-\frac{26\cdots 81}{14\cdots 97}$
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| Regulator: | \( 1792851846429171700 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{0}\cdot 1792851846429171700 \cdot 1}{2\cdot\sqrt{462899178676311160288470191817767102492054133121}}\cr\approx \mathstrut & 0.172695542633050 \end{aligned}\] (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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\(953\)
| Deg $17$ | $17$ | $1$ | $16$ |