Normalized defining polynomial
\( x^{18} - x^{17} + 3 x^{16} + 10 x^{15} + 2 x^{14} + 4 x^{13} + 24 x^{12} + 142 x^{11} - 225 x^{10} + \cdots - 257 \)
Invariants
| Degree: | $18$ |
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| Signature: | $(2, 8)$ |
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| Discriminant: |
\(11389168905865078884007936\)
\(\medspace = 2^{20}\cdot 11^{6}\cdot 19^{10}\)
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| Root discriminant: | \(24.66\) |
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| Galois root discriminant: | $2^{19/12}11^{1/2}19^{3/4}\approx 90.4466258093427$ | ||
| Ramified primes: |
\(2\), \(11\), \(19\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{64\cdots 36}a^{17}+\frac{17\cdots 19}{64\cdots 36}a^{16}+\frac{20\cdots 11}{32\cdots 18}a^{15}-\frac{17\cdots 79}{32\cdots 18}a^{14}+\frac{39\cdots 66}{16\cdots 09}a^{13}-\frac{75\cdots 45}{32\cdots 18}a^{12}-\frac{55\cdots 51}{16\cdots 09}a^{11}-\frac{36\cdots 13}{16\cdots 09}a^{10}+\frac{25\cdots 03}{64\cdots 36}a^{9}-\frac{68\cdots 01}{64\cdots 36}a^{8}+\frac{42\cdots 80}{16\cdots 09}a^{7}-\frac{19\cdots 08}{16\cdots 09}a^{6}-\frac{75\cdots 82}{16\cdots 09}a^{5}+\frac{15\cdots 31}{32\cdots 18}a^{4}+\frac{53\cdots 66}{16\cdots 09}a^{3}-\frac{45\cdots 05}{16\cdots 09}a^{2}-\frac{86\cdots 71}{64\cdots 36}a+\frac{22\cdots 19}{64\cdots 36}$
| 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: | $C_{2}$, which has order $2$ (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: |
$\frac{39\cdots 80}{16\cdots 09}a^{17}-\frac{11\cdots 85}{32\cdots 18}a^{16}+\frac{14\cdots 64}{16\cdots 09}a^{15}+\frac{33\cdots 53}{16\cdots 09}a^{14}-\frac{11\cdots 19}{32\cdots 18}a^{13}+\frac{36\cdots 87}{32\cdots 18}a^{12}+\frac{17\cdots 09}{32\cdots 18}a^{11}+\frac{10\cdots 79}{32\cdots 18}a^{10}-\frac{11\cdots 31}{16\cdots 09}a^{9}+\frac{32\cdots 91}{32\cdots 18}a^{8}+\frac{16\cdots 65}{32\cdots 18}a^{7}-\frac{71\cdots 49}{32\cdots 18}a^{6}+\frac{74\cdots 38}{16\cdots 09}a^{5}-\frac{21\cdots 47}{16\cdots 09}a^{4}+\frac{28\cdots 79}{32\cdots 18}a^{3}+\frac{16\cdots 89}{32\cdots 18}a^{2}-\frac{13\cdots 67}{32\cdots 18}a-\frac{25\cdots 79}{16\cdots 09}$, $\frac{38\cdots 59}{16\cdots 09}a^{17}-\frac{23\cdots 39}{64\cdots 36}a^{16}+\frac{28\cdots 79}{32\cdots 18}a^{15}+\frac{31\cdots 03}{16\cdots 09}a^{14}-\frac{17\cdots 17}{32\cdots 18}a^{13}+\frac{18\cdots 86}{16\cdots 09}a^{12}+\frac{82\cdots 41}{16\cdots 09}a^{11}+\frac{10\cdots 23}{32\cdots 18}a^{10}-\frac{11\cdots 80}{16\cdots 09}a^{9}+\frac{68\cdots 67}{64\cdots 36}a^{8}+\frac{58\cdots 13}{16\cdots 09}a^{7}-\frac{70\cdots 79}{32\cdots 18}a^{6}+\frac{72\cdots 62}{16\cdots 09}a^{5}-\frac{21\cdots 02}{16\cdots 09}a^{4}+\frac{33\cdots 75}{32\cdots 18}a^{3}+\frac{56\cdots 18}{16\cdots 09}a^{2}-\frac{13\cdots 71}{32\cdots 18}a-\frac{78\cdots 75}{64\cdots 36}$, $\frac{10\cdots 89}{32\cdots 18}a^{17}+\frac{14\cdots 05}{64\cdots 36}a^{16}-\frac{25\cdots 65}{32\cdots 18}a^{15}-\frac{61\cdots 22}{16\cdots 09}a^{14}-\frac{41\cdots 47}{32\cdots 18}a^{13}-\frac{12\cdots 06}{16\cdots 09}a^{12}-\frac{14\cdots 42}{16\cdots 09}a^{11}-\frac{15\cdots 13}{32\cdots 18}a^{10}+\frac{20\cdots 71}{32\cdots 18}a^{9}-\frac{39\cdots 49}{64\cdots 36}a^{8}-\frac{29\cdots 10}{16\cdots 09}a^{7}+\frac{73\cdots 75}{32\cdots 18}a^{6}-\frac{96\cdots 04}{16\cdots 09}a^{5}+\frac{22\cdots 04}{16\cdots 09}a^{4}+\frac{88\cdots 83}{32\cdots 18}a^{3}-\frac{31\cdots 08}{16\cdots 09}a^{2}+\frac{10\cdots 74}{16\cdots 09}a+\frac{36\cdots 97}{64\cdots 36}$, $\frac{20\cdots 89}{16\cdots 09}a^{17}-\frac{11\cdots 43}{64\cdots 36}a^{16}+\frac{14\cdots 81}{32\cdots 18}a^{15}+\frac{17\cdots 27}{16\cdots 09}a^{14}-\frac{63\cdots 95}{32\cdots 18}a^{13}+\frac{19\cdots 73}{32\cdots 18}a^{12}+\frac{45\cdots 78}{16\cdots 09}a^{11}+\frac{27\cdots 62}{16\cdots 09}a^{10}-\frac{58\cdots 40}{16\cdots 09}a^{9}+\frac{34\cdots 99}{64\cdots 36}a^{8}+\frac{41\cdots 86}{16\cdots 09}a^{7}-\frac{18\cdots 50}{16\cdots 09}a^{6}+\frac{38\cdots 87}{16\cdots 09}a^{5}-\frac{11\cdots 14}{16\cdots 09}a^{4}+\frac{15\cdots 57}{32\cdots 18}a^{3}+\frac{83\cdots 39}{32\cdots 18}a^{2}-\frac{72\cdots 71}{32\cdots 18}a-\frac{51\cdots 65}{64\cdots 36}$, $\frac{33\cdots 09}{32\cdots 18}a^{17}+\frac{96\cdots 51}{64\cdots 36}a^{16}-\frac{12\cdots 13}{32\cdots 18}a^{15}-\frac{27\cdots 47}{32\cdots 18}a^{14}+\frac{59\cdots 99}{32\cdots 18}a^{13}-\frac{15\cdots 07}{32\cdots 18}a^{12}-\frac{36\cdots 23}{16\cdots 09}a^{11}-\frac{44\cdots 29}{32\cdots 18}a^{10}+\frac{94\cdots 91}{32\cdots 18}a^{9}-\frac{28\cdots 09}{64\cdots 36}a^{8}-\frac{31\cdots 77}{16\cdots 09}a^{7}+\frac{30\cdots 17}{32\cdots 18}a^{6}-\frac{31\cdots 86}{16\cdots 09}a^{5}+\frac{18\cdots 05}{32\cdots 18}a^{4}-\frac{13\cdots 81}{32\cdots 18}a^{3}-\frac{61\cdots 01}{32\cdots 18}a^{2}+\frac{29\cdots 28}{16\cdots 09}a+\frac{38\cdots 63}{64\cdots 36}$, $\frac{37\cdots 88}{16\cdots 09}a^{17}+\frac{95\cdots 01}{32\cdots 18}a^{16}-\frac{25\cdots 55}{32\cdots 18}a^{15}-\frac{34\cdots 58}{16\cdots 09}a^{14}+\frac{27\cdots 91}{32\cdots 18}a^{13}-\frac{18\cdots 54}{16\cdots 09}a^{12}-\frac{90\cdots 78}{16\cdots 09}a^{11}-\frac{51\cdots 50}{16\cdots 09}a^{10}+\frac{19\cdots 27}{32\cdots 18}a^{9}-\frac{28\cdots 59}{32\cdots 18}a^{8}-\frac{10\cdots 74}{16\cdots 09}a^{7}+\frac{29\cdots 63}{16\cdots 09}a^{6}-\frac{14\cdots 15}{32\cdots 18}a^{5}+\frac{19\cdots 69}{16\cdots 09}a^{4}-\frac{22\cdots 85}{32\cdots 18}a^{3}-\frac{99\cdots 75}{16\cdots 09}a^{2}+\frac{71\cdots 84}{16\cdots 09}a+\frac{65\cdots 31}{32\cdots 18}$, $\frac{49\cdots 65}{64\cdots 36}a^{17}+\frac{17\cdots 17}{16\cdots 09}a^{16}-\frac{88\cdots 19}{32\cdots 18}a^{15}-\frac{10\cdots 42}{16\cdots 09}a^{14}+\frac{35\cdots 23}{32\cdots 18}a^{13}-\frac{56\cdots 85}{16\cdots 09}a^{12}-\frac{54\cdots 83}{32\cdots 18}a^{11}-\frac{32\cdots 55}{32\cdots 18}a^{10}+\frac{13\cdots 31}{64\cdots 36}a^{9}-\frac{50\cdots 42}{16\cdots 09}a^{8}-\frac{51\cdots 29}{32\cdots 18}a^{7}+\frac{22\cdots 67}{32\cdots 18}a^{6}-\frac{46\cdots 53}{32\cdots 18}a^{5}+\frac{67\cdots 33}{16\cdots 09}a^{4}-\frac{89\cdots 53}{32\cdots 18}a^{3}-\frac{51\cdots 97}{32\cdots 18}a^{2}+\frac{87\cdots 47}{64\cdots 36}a+\frac{78\cdots 88}{16\cdots 09}$, $\frac{22\cdots 35}{64\cdots 36}a^{17}-\frac{15\cdots 73}{32\cdots 18}a^{16}+\frac{40\cdots 57}{32\cdots 18}a^{15}+\frac{94\cdots 25}{32\cdots 18}a^{14}-\frac{17\cdots 75}{32\cdots 18}a^{13}+\frac{25\cdots 62}{16\cdots 09}a^{12}+\frac{24\cdots 89}{32\cdots 18}a^{11}+\frac{74\cdots 63}{16\cdots 09}a^{10}-\frac{62\cdots 25}{64\cdots 36}a^{9}+\frac{23\cdots 56}{16\cdots 09}a^{8}+\frac{22\cdots 73}{32\cdots 18}a^{7}-\frac{50\cdots 88}{16\cdots 09}a^{6}+\frac{20\cdots 75}{32\cdots 18}a^{5}-\frac{61\cdots 85}{32\cdots 18}a^{4}+\frac{41\cdots 99}{32\cdots 18}a^{3}+\frac{22\cdots 31}{32\cdots 18}a^{2}-\frac{39\cdots 89}{64\cdots 36}a-\frac{34\cdots 71}{16\cdots 09}$, $\frac{50\cdots 73}{64\cdots 36}a^{17}+\frac{17\cdots 80}{16\cdots 09}a^{16}-\frac{89\cdots 39}{32\cdots 18}a^{15}-\frac{21\cdots 99}{32\cdots 18}a^{14}+\frac{16\cdots 02}{16\cdots 09}a^{13}-\frac{11\cdots 29}{32\cdots 18}a^{12}-\frac{28\cdots 49}{16\cdots 09}a^{11}-\frac{33\cdots 15}{32\cdots 18}a^{10}+\frac{13\cdots 27}{64\cdots 36}a^{9}-\frac{10\cdots 29}{32\cdots 18}a^{8}-\frac{27\cdots 93}{16\cdots 09}a^{7}+\frac{22\cdots 33}{32\cdots 18}a^{6}-\frac{47\cdots 65}{32\cdots 18}a^{5}+\frac{13\cdots 59}{32\cdots 18}a^{4}-\frac{44\cdots 39}{16\cdots 09}a^{3}-\frac{27\cdots 10}{16\cdots 09}a^{2}+\frac{89\cdots 25}{64\cdots 36}a+\frac{82\cdots 69}{16\cdots 09}$
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| Regulator: | \( 223875.4921708372 \) (assuming GRH) |
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| Unit signature rank: | \( 1 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 223875.4921708372 \cdot 1}{2\cdot\sqrt{11389168905865078884007936}}\cr\approx \mathstrut & 0.322277123940902 \end{aligned}\] (assuming GRH)
Galois group
$C_2^4.S_4^2$ (as 18T545):
| A solvable group of order 9216 |
| The 60 conjugacy class representatives for $C_2^4.S_4^2$ |
| Character table for $C_2^4.S_4^2$ |
Intermediate fields
| 3.1.44.1, 3.1.76.1, 9.1.584277056.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | 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 | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 2.2.6.16a1.8 | $x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 233 x^{6} + 234 x^{5} + 192 x^{4} + 124 x^{3} + 63 x^{2} + 26 x + 7$ | $6$ | $2$ | $16$ | 12T30 | $$[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$$ | |
|
\(11\)
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 11.3.2.3a1.2 | $x^{6} + 4 x^{4} + 18 x^{3} + 4 x^{2} + 36 x + 92$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 11.3.2.3a1.2 | $x^{6} + 4 x^{4} + 18 x^{3} + 4 x^{2} + 36 x + 92$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(19\)
| 19.1.2.1a1.2 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 19.1.4.3a1.1 | $x^{4} + 19$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
| 19.4.2.4a1.2 | $x^{8} + 4 x^{6} + 22 x^{5} + 8 x^{4} + 44 x^{3} + 129 x^{2} + 44 x + 23$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |