Normalized defining polynomial
\( x^{18} - 57 x^{16} - 56 x^{15} + 1017 x^{14} + 2532 x^{13} - 1491 x^{12} - 24102 x^{11} - 32877 x^{10} + \cdots + 4034368 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $[2, 8]$ |
| |
| Discriminant: |
\(1165498283698428929755771705757331456\)
\(\medspace = 2^{12}\cdot 3^{33}\cdot 13^{15}\)
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| |
| Root discriminant: | \(100.85\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{39}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{52}a^{12}+\frac{2}{13}a^{10}-\frac{1}{13}a^{9}-\frac{23}{52}a^{8}-\frac{4}{13}a^{7}-\frac{11}{26}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{104}a^{13}+\frac{1}{13}a^{11}-\frac{1}{26}a^{10}-\frac{23}{104}a^{9}-\frac{2}{13}a^{8}+\frac{15}{52}a^{7}-\frac{1}{4}a^{6}-\frac{3}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{312}a^{14}+\frac{1}{312}a^{13}-\frac{1}{156}a^{12}-\frac{2}{13}a^{11}-\frac{1}{104}a^{10}+\frac{35}{104}a^{9}+\frac{35}{78}a^{8}+\frac{1}{39}a^{7}+\frac{155}{312}a^{6}+\frac{11}{24}a^{5}-\frac{1}{24}a^{4}-\frac{1}{24}a^{3}-\frac{1}{12}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{624}a^{15}-\frac{1}{624}a^{14}+\frac{1}{312}a^{13}+\frac{1}{156}a^{12}+\frac{47}{208}a^{11}-\frac{51}{208}a^{10}-\frac{11}{78}a^{9}-\frac{14}{39}a^{8}-\frac{137}{624}a^{7}+\frac{181}{624}a^{6}+\frac{7}{48}a^{5}+\frac{13}{48}a^{4}+\frac{3}{8}a^{3}+\frac{5}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{624}a^{16}-\frac{1}{624}a^{14}-\frac{1}{312}a^{13}+\frac{5}{624}a^{12}+\frac{3}{52}a^{11}-\frac{115}{624}a^{10}+\frac{4}{13}a^{9}+\frac{271}{624}a^{8}+\frac{35}{78}a^{7}+\frac{83}{312}a^{6}+\frac{1}{3}a^{5}+\frac{7}{16}a^{4}-\frac{1}{24}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{54\cdots 84}a^{17}-\frac{20\cdots 21}{20\cdots 84}a^{16}-\frac{16\cdots 41}{54\cdots 84}a^{15}-\frac{13\cdots 03}{27\cdots 92}a^{14}-\frac{89\cdots 07}{54\cdots 84}a^{13}-\frac{13\cdots 07}{27\cdots 92}a^{12}-\frac{85\cdots 95}{54\cdots 84}a^{11}-\frac{27\cdots 50}{65\cdots 87}a^{10}+\frac{56\cdots 19}{41\cdots 68}a^{9}-\frac{56\cdots 53}{20\cdots 84}a^{8}+\frac{17\cdots 35}{20\cdots 84}a^{7}-\frac{78\cdots 37}{20\cdots 84}a^{6}-\frac{49\cdots 77}{13\cdots 56}a^{5}-\frac{14\cdots 97}{53\cdots 56}a^{4}-\frac{65\cdots 99}{67\cdots 32}a^{3}-\frac{13\cdots 51}{40\cdots 92}a^{2}-\frac{42\cdots 23}{10\cdots 98}a-\frac{17\cdots 65}{10\cdots 98}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
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| Narrow class group: | $C_{3}$, which has order $3$ (assuming GRH) |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{12\cdots 47}{13\cdots 56}a^{17}-\frac{20\cdots 05}{69\cdots 28}a^{16}-\frac{72\cdots 87}{13\cdots 56}a^{15}+\frac{81\cdots 85}{69\cdots 28}a^{14}+\frac{15\cdots 07}{13\cdots 56}a^{13}-\frac{49\cdots 79}{69\cdots 28}a^{12}-\frac{12\cdots 45}{13\cdots 56}a^{11}-\frac{63\cdots 65}{34\cdots 64}a^{10}+\frac{51\cdots 61}{13\cdots 56}a^{9}+\frac{55\cdots 31}{69\cdots 28}a^{8}+\frac{61\cdots 57}{69\cdots 28}a^{7}-\frac{10\cdots 47}{69\cdots 28}a^{6}+\frac{53\cdots 71}{10\cdots 12}a^{5}+\frac{13\cdots 61}{53\cdots 56}a^{4}+\frac{81\cdots 29}{13\cdots 64}a^{3}-\frac{11\cdots 33}{13\cdots 64}a^{2}-\frac{52\cdots 11}{33\cdots 66}a-\frac{19\cdots 75}{33\cdots 66}$, $\frac{32\cdots 89}{13\cdots 56}a^{17}-\frac{97\cdots 69}{69\cdots 28}a^{16}-\frac{56\cdots 99}{41\cdots 68}a^{15}-\frac{21\cdots 83}{53\cdots 56}a^{14}+\frac{36\cdots 05}{13\cdots 56}a^{13}+\frac{86\cdots 27}{20\cdots 84}a^{12}-\frac{13\cdots 63}{13\cdots 56}a^{11}-\frac{94\cdots 93}{17\cdots 32}a^{10}-\frac{14\cdots 11}{41\cdots 68}a^{9}-\frac{70\cdots 57}{69\cdots 28}a^{8}-\frac{40\cdots 05}{69\cdots 28}a^{7}-\frac{39\cdots 37}{69\cdots 28}a^{6}-\frac{23\cdots 39}{10\cdots 12}a^{5}+\frac{23\cdots 89}{53\cdots 56}a^{4}+\frac{39\cdots 83}{40\cdots 92}a^{3}+\frac{33\cdots 43}{13\cdots 64}a^{2}-\frac{40\cdots 59}{33\cdots 66}a+\frac{50\cdots 79}{10\cdots 98}$, $\frac{29\cdots 93}{16\cdots 52}a^{17}+\frac{41\cdots 39}{10\cdots 88}a^{16}-\frac{26\cdots 01}{20\cdots 76}a^{15}-\frac{32\cdots 43}{10\cdots 88}a^{14}+\frac{64\cdots 53}{20\cdots 76}a^{13}+\frac{88\cdots 25}{10\cdots 88}a^{12}-\frac{48\cdots 67}{20\cdots 76}a^{11}-\frac{18\cdots 51}{26\cdots 72}a^{10}+\frac{68\cdots 15}{20\cdots 76}a^{9}+\frac{96\cdots 39}{80\cdots 76}a^{8}-\frac{44\cdots 17}{80\cdots 76}a^{7}-\frac{66\cdots 45}{80\cdots 76}a^{6}+\frac{25\cdots 25}{16\cdots 52}a^{5}+\frac{37\cdots 81}{80\cdots 76}a^{4}+\frac{59\cdots 69}{20\cdots 44}a^{3}-\frac{13\cdots 11}{20\cdots 44}a^{2}+\frac{33\cdots 77}{50\cdots 86}a+\frac{16\cdots 95}{50\cdots 86}$, $\frac{90\cdots 65}{17\cdots 62}a^{17}-\frac{71\cdots 89}{10\cdots 92}a^{16}-\frac{14\cdots 77}{45\cdots 32}a^{15}+\frac{16\cdots 77}{11\cdots 08}a^{14}+\frac{46\cdots 41}{68\cdots 48}a^{13}+\frac{57\cdots 51}{13\cdots 96}a^{12}-\frac{63\cdots 23}{13\cdots 96}a^{11}-\frac{48\cdots 41}{52\cdots 96}a^{10}+\frac{22\cdots 69}{13\cdots 64}a^{9}+\frac{22\cdots 85}{10\cdots 92}a^{8}-\frac{10\cdots 13}{10\cdots 92}a^{7}-\frac{87\cdots 61}{10\cdots 92}a^{6}+\frac{48\cdots 33}{10\cdots 92}a^{5}+\frac{30\cdots 83}{67\cdots 32}a^{4}+\frac{25\cdots 71}{40\cdots 92}a^{3}-\frac{54\cdots 88}{16\cdots 33}a^{2}+\frac{27\cdots 55}{50\cdots 99}a-\frac{50\cdots 40}{50\cdots 99}$, $\frac{18\cdots 33}{54\cdots 84}a^{17}+\frac{18\cdots 53}{20\cdots 84}a^{16}-\frac{38\cdots 79}{18\cdots 28}a^{15}-\frac{64\cdots 47}{27\cdots 92}a^{14}+\frac{24\cdots 25}{54\cdots 84}a^{13}+\frac{33\cdots 75}{27\cdots 92}a^{12}-\frac{10\cdots 19}{54\cdots 84}a^{11}-\frac{45\cdots 21}{26\cdots 48}a^{10}-\frac{33\cdots 11}{13\cdots 56}a^{9}+\frac{75\cdots 49}{20\cdots 84}a^{8}+\frac{46\cdots 21}{20\cdots 84}a^{7}+\frac{57\cdots 93}{20\cdots 84}a^{6}-\frac{20\cdots 05}{13\cdots 56}a^{5}-\frac{56\cdots 57}{53\cdots 56}a^{4}-\frac{17\cdots 15}{20\cdots 96}a^{3}-\frac{69\cdots 13}{40\cdots 92}a^{2}+\frac{12\cdots 63}{50\cdots 99}a-\frac{84\cdots 11}{10\cdots 98}$, $\frac{29\cdots 41}{90\cdots 64}a^{17}-\frac{61\cdots 73}{65\cdots 87}a^{16}-\frac{15\cdots 93}{90\cdots 64}a^{15}+\frac{76\cdots 41}{22\cdots 16}a^{14}+\frac{78\cdots 23}{27\cdots 92}a^{13}-\frac{26\cdots 03}{22\cdots 16}a^{12}-\frac{98\cdots 91}{90\cdots 64}a^{11}-\frac{36\cdots 59}{80\cdots 84}a^{10}+\frac{32\cdots 07}{69\cdots 28}a^{9}-\frac{24\cdots 69}{17\cdots 32}a^{8}+\frac{49\cdots 69}{34\cdots 64}a^{7}-\frac{30\cdots 93}{34\cdots 64}a^{6}+\frac{11\cdots 25}{69\cdots 28}a^{5}+\frac{66\cdots 59}{20\cdots 96}a^{4}+\frac{78\cdots 65}{16\cdots 33}a^{3}-\frac{10\cdots 93}{67\cdots 32}a^{2}+\frac{59\cdots 05}{50\cdots 99}a+\frac{34\cdots 73}{16\cdots 33}$, $\frac{30\cdots 75}{27\cdots 92}a^{17}+\frac{59\cdots 93}{26\cdots 48}a^{16}-\frac{54\cdots 79}{90\cdots 64}a^{15}-\frac{83\cdots 15}{45\cdots 32}a^{14}+\frac{21\cdots 75}{27\cdots 92}a^{13}+\frac{30\cdots 29}{68\cdots 48}a^{12}+\frac{20\cdots 41}{27\cdots 92}a^{11}-\frac{64\cdots 59}{52\cdots 96}a^{10}-\frac{33\cdots 61}{53\cdots 56}a^{9}-\frac{94\cdots 29}{52\cdots 96}a^{8}-\frac{21\cdots 45}{52\cdots 96}a^{7}-\frac{33\cdots 29}{26\cdots 48}a^{6}-\frac{58\cdots 67}{20\cdots 84}a^{5}-\frac{11\cdots 95}{26\cdots 28}a^{4}-\frac{12\cdots 91}{40\cdots 92}a^{3}-\frac{78\cdots 25}{67\cdots 32}a^{2}-\frac{17\cdots 13}{50\cdots 99}a-\frac{11\cdots 03}{50\cdots 99}$, $\frac{23\cdots 39}{90\cdots 64}a^{17}-\frac{11\cdots 91}{10\cdots 92}a^{16}-\frac{14\cdots 91}{90\cdots 64}a^{15}-\frac{35\cdots 01}{13\cdots 96}a^{14}+\frac{91\cdots 85}{27\cdots 92}a^{13}+\frac{40\cdots 51}{13\cdots 96}a^{12}-\frac{15\cdots 37}{90\cdots 64}a^{11}-\frac{18\cdots 43}{52\cdots 96}a^{10}-\frac{30\cdots 67}{69\cdots 28}a^{9}+\frac{90\cdots 85}{10\cdots 92}a^{8}-\frac{24\cdots 93}{10\cdots 92}a^{7}-\frac{57\cdots 89}{10\cdots 92}a^{6}-\frac{61\cdots 51}{20\cdots 84}a^{5}+\frac{13\cdots 85}{26\cdots 28}a^{4}+\frac{10\cdots 71}{20\cdots 96}a^{3}+\frac{46\cdots 35}{20\cdots 96}a^{2}-\frac{13\cdots 13}{50\cdots 99}a+\frac{24\cdots 34}{50\cdots 99}$, $\frac{73\cdots 91}{18\cdots 28}a^{17}-\frac{76\cdots 49}{20\cdots 84}a^{16}-\frac{43\cdots 83}{18\cdots 28}a^{15}-\frac{21\cdots 87}{90\cdots 64}a^{14}+\frac{25\cdots 53}{54\cdots 84}a^{13}+\frac{65\cdots 11}{90\cdots 64}a^{12}-\frac{42\cdots 21}{18\cdots 28}a^{11}-\frac{12\cdots 35}{10\cdots 92}a^{10}-\frac{46\cdots 19}{13\cdots 56}a^{9}+\frac{11\cdots 13}{69\cdots 28}a^{8}+\frac{16\cdots 83}{53\cdots 56}a^{7}-\frac{71\cdots 97}{69\cdots 28}a^{6}+\frac{15\cdots 47}{13\cdots 56}a^{5}+\frac{13\cdots 31}{16\cdots 68}a^{4}+\frac{25\cdots 09}{13\cdots 64}a^{3}-\frac{17\cdots 59}{13\cdots 64}a^{2}-\frac{25\cdots 18}{50\cdots 99}a-\frac{19\cdots 23}{33\cdots 66}$
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| Regulator: | \( 51867970070.3 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 51867970070.3 \cdot 3}{2\cdot\sqrt{1165498283698428929755771705757331456}}\cr\approx \mathstrut & 0.700218578805 \end{aligned}\] (assuming GRH)
Galois group
$C_3^4:(C_6^2:C_4)$ (as 18T576):
| A solvable group of order 11664 |
| The 49 conjugacy class representatives for $C_3^4:(C_6^2:C_4)$ |
| Character table for $C_3^4:(C_6^2:C_4)$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 6.2.2313441.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.2.4a2.2 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 7$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ | |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 2.4.2.8a3.1 | $x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 3$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $$[2, 2]^{4}$$ | |
|
\(3\)
| 3.1.9.21a2.14 | $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{4} + 21$ | $9$ | $1$ | $21$ | $C_3^2 : S_3 $ | $$[2, \frac{5}{2}, \frac{17}{6}]_{2}$$ |
| 3.3.3.12a1.1 | $x^{9} + 6 x^{7} + 6 x^{6} + 12 x^{5} + 24 x^{4} + 17 x^{3} + 24 x^{2} + 18 x + 7$ | $3$ | $3$ | $12$ | $S_3\times C_3$ | $$[2]^{6}$$ | |
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.4.3a1.1 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 13.1.12.11a1.1 | $x^{12} + 13$ | $12$ | $1$ | $11$ | $C_{12}$ | $$[\ ]_{12}$$ |