Normalized defining polynomial
\( x^{18} - 9 x^{17} + 39 x^{16} - 18 x^{15} - 579 x^{14} + 2889 x^{13} - 7292 x^{12} - 207 x^{11} + \cdots + 385910388 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-3238832304182321509797314520003000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 17^{12}\)
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| Root discriminant: | \(229.98\) |
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| Galois root discriminant: | $2^{2/3}3^{13/6}5^{2/3}17^{2/3}\approx 331.6871648497297$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(17\)
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| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_3^2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{4}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{2}{9}a^{5}+\frac{1}{9}a^{4}-\frac{4}{9}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{18}a^{12}+\frac{1}{18}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{18}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{7}{18}a^{3}-\frac{1}{3}$, $\frac{1}{18}a^{13}+\frac{1}{18}a^{10}+\frac{1}{3}a^{8}-\frac{1}{18}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{7}{18}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{18}a^{14}-\frac{1}{18}a^{11}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{7}{18}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{7}{18}a^{5}-\frac{4}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{2626110}a^{15}+\frac{1199}{262611}a^{14}-\frac{12799}{1313055}a^{13}-\frac{32479}{1313055}a^{12}+\frac{10016}{262611}a^{11}-\frac{90478}{1313055}a^{10}-\frac{87806}{1313055}a^{9}+\frac{251621}{1313055}a^{8}+\frac{349108}{1313055}a^{7}+\frac{76829}{262611}a^{6}+\frac{456479}{1313055}a^{5}-\frac{76847}{1313055}a^{4}+\frac{179629}{875370}a^{3}-\frac{32404}{87537}a^{2}-\frac{122548}{437685}a-\frac{43163}{145895}$, $\frac{1}{97\cdots 90}a^{16}+\frac{34964761935076}{48\cdots 95}a^{15}-\frac{97\cdots 94}{48\cdots 95}a^{14}-\frac{13\cdots 82}{48\cdots 95}a^{13}+\frac{13\cdots 12}{48\cdots 95}a^{12}-\frac{59\cdots 16}{16\cdots 65}a^{11}+\frac{26\cdots 06}{16\cdots 65}a^{10}-\frac{34\cdots 61}{48\cdots 95}a^{9}+\frac{11\cdots 73}{97\cdots 79}a^{8}-\frac{14\cdots 37}{69\cdots 85}a^{7}+\frac{51\cdots 69}{48\cdots 95}a^{6}+\frac{61\cdots 02}{16\cdots 65}a^{5}+\frac{36\cdots 09}{97\cdots 90}a^{4}+\frac{10\cdots 17}{48\cdots 95}a^{3}-\frac{26\cdots 76}{53\cdots 55}a^{2}-\frac{11\cdots 19}{32\cdots 93}a+\frac{78\cdots 47}{16\cdots 65}$, $\frac{1}{31\cdots 30}a^{17}-\frac{18\cdots 79}{75\cdots 15}a^{16}+\frac{14\cdots 67}{10\cdots 10}a^{15}-\frac{38\cdots 12}{15\cdots 15}a^{14}-\frac{18\cdots 94}{15\cdots 15}a^{13}-\frac{28\cdots 43}{31\cdots 30}a^{12}+\frac{48\cdots 96}{52\cdots 05}a^{11}-\frac{16\cdots 68}{15\cdots 15}a^{10}-\frac{37\cdots 93}{31\cdots 30}a^{9}+\frac{13\cdots 52}{31\cdots 83}a^{8}+\frac{26\cdots 86}{15\cdots 15}a^{7}-\frac{90\cdots 03}{31\cdots 30}a^{6}+\frac{14\cdots 21}{31\cdots 30}a^{5}-\frac{31\cdots 21}{15\cdots 15}a^{4}-\frac{14\cdots 35}{31\cdots 83}a^{3}+\frac{40\cdots 47}{10\cdots 61}a^{2}-\frac{18\cdots 41}{10\cdots 61}a+\frac{14\cdots 04}{52\cdots 05}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $243$ (assuming GRH) |
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| Narrow class group: | $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $243$ (assuming GRH) |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -\frac{322613093892650873}{13540489471792812627627885} a^{17} + \frac{6416222392224513221}{27080978943585625255255770} a^{16} - \frac{3179461939910923999}{2708097894358562525525577} a^{15} + \frac{9296390486531632823}{5416195788717125051051154} a^{14} + \frac{34356121266500029119}{3008997660398402806139530} a^{13} - \frac{428983748103555977723}{5416195788717125051051154} a^{12} + \frac{776314451010393725379}{3008997660398402806139530} a^{11} - \frac{7687795363768885552441}{27080978943585625255255770} a^{10} - \frac{29635967101534573574633}{27080978943585625255255770} a^{9} + \frac{14186262304587396430969}{5416195788717125051051154} a^{8} + \frac{165733773107215295620363}{9026992981195208418418590} a^{7} - \frac{2773865953876668672112463}{27080978943585625255255770} a^{6} - \frac{1154013138366596768222449}{5416195788717125051051154} a^{5} + \frac{20921298042257590218660022}{13540489471792812627627885} a^{4} - \frac{35409505473404777045170151}{9026992981195208418418590} a^{3} + \frac{12092639429623220885692801}{4513496490597604209209295} a^{2} + \frac{3048315327674360689540792}{902699298119520841841859} a - \frac{11305076408578940112936246}{1504498830199201403069765} \)
(order $6$)
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| Fundamental units: |
$\frac{37\cdots 57}{31\cdots 30}a^{17}-\frac{55\cdots 91}{45\cdots 90}a^{16}+\frac{13\cdots 99}{21\cdots 22}a^{15}-\frac{22\cdots 13}{21\cdots 22}a^{14}-\frac{58\cdots 19}{10\cdots 10}a^{13}+\frac{87\cdots 41}{21\cdots 22}a^{12}-\frac{27\cdots 69}{19\cdots 10}a^{11}+\frac{58\cdots 67}{31\cdots 30}a^{10}+\frac{52\cdots 27}{10\cdots 10}a^{9}-\frac{10\cdots 59}{70\cdots 74}a^{8}-\frac{94\cdots 91}{10\cdots 10}a^{7}+\frac{57\cdots 87}{10\cdots 10}a^{6}+\frac{29\cdots 01}{31\cdots 83}a^{5}-\frac{13\cdots 49}{15\cdots 15}a^{4}+\frac{38\cdots 42}{17\cdots 35}a^{3}-\frac{10\cdots 42}{52\cdots 05}a^{2}-\frac{84\cdots 74}{10\cdots 61}a+\frac{55\cdots 87}{17\cdots 35}$, $\frac{77\cdots 33}{45\cdots 69}a^{17}-\frac{77\cdots 52}{45\cdots 69}a^{16}+\frac{37\cdots 86}{45\cdots 69}a^{15}-\frac{50\cdots 92}{45\cdots 69}a^{14}-\frac{44\cdots 00}{50\cdots 41}a^{13}+\frac{26\cdots 76}{45\cdots 69}a^{12}-\frac{91\cdots 24}{50\cdots 41}a^{11}+\frac{78\cdots 24}{45\cdots 69}a^{10}+\frac{57\cdots 71}{64\cdots 67}a^{9}-\frac{91\cdots 59}{45\cdots 69}a^{8}-\frac{20\cdots 02}{15\cdots 23}a^{7}+\frac{33\cdots 41}{45\cdots 69}a^{6}+\frac{74\cdots 19}{45\cdots 69}a^{5}-\frac{52\cdots 98}{45\cdots 69}a^{4}+\frac{58\cdots 31}{21\cdots 89}a^{3}-\frac{20\cdots 06}{15\cdots 23}a^{2}-\frac{55\cdots 44}{15\cdots 23}a+\frac{33\cdots 59}{50\cdots 41}$, $\frac{10\cdots 39}{31\cdots 83}a^{17}-\frac{27\cdots 07}{15\cdots 30}a^{16}+\frac{18\cdots 41}{52\cdots 05}a^{15}+\frac{85\cdots 61}{31\cdots 30}a^{14}-\frac{49\cdots 37}{31\cdots 30}a^{13}+\frac{12\cdots 17}{52\cdots 05}a^{12}+\frac{14\cdots 29}{10\cdots 10}a^{11}-\frac{13\cdots 47}{31\cdots 30}a^{10}+\frac{56\cdots 09}{52\cdots 05}a^{9}+\frac{31\cdots 07}{63\cdots 66}a^{8}-\frac{86\cdots 79}{31\cdots 30}a^{7}-\frac{28\cdots 76}{52\cdots 05}a^{6}+\frac{23\cdots 31}{31\cdots 30}a^{5}+\frac{37\cdots 66}{15\cdots 15}a^{4}-\frac{13\cdots 28}{52\cdots 05}a^{3}+\frac{14\cdots 51}{52\cdots 05}a^{2}+\frac{64\cdots 59}{10\cdots 61}a+\frac{67\cdots 37}{17\cdots 35}$, $\frac{34\cdots 48}{31\cdots 83}a^{17}-\frac{18\cdots 58}{22\cdots 45}a^{16}+\frac{89\cdots 22}{31\cdots 83}a^{15}+\frac{51\cdots 93}{15\cdots 15}a^{14}-\frac{93\cdots 92}{15\cdots 15}a^{13}+\frac{13\cdots 19}{63\cdots 66}a^{12}-\frac{70\cdots 76}{17\cdots 35}a^{11}-\frac{12\cdots 83}{15\cdots 15}a^{10}+\frac{17\cdots 21}{31\cdots 30}a^{9}+\frac{48\cdots 14}{15\cdots 15}a^{8}-\frac{31\cdots 30}{31\cdots 83}a^{7}+\frac{68\cdots 69}{31\cdots 30}a^{6}+\frac{30\cdots 34}{15\cdots 15}a^{5}-\frac{17\cdots 84}{52\cdots 05}a^{4}+\frac{53\cdots 65}{21\cdots 22}a^{3}+\frac{65\cdots 31}{52\cdots 05}a^{2}-\frac{22\cdots 19}{17\cdots 35}a-\frac{65\cdots 58}{35\cdots 87}$, $\frac{76\cdots 66}{15\cdots 15}a^{17}-\frac{10\cdots 23}{22\cdots 45}a^{16}+\frac{32\cdots 23}{31\cdots 30}a^{15}+\frac{12\cdots 97}{63\cdots 66}a^{14}-\frac{53\cdots 62}{17\cdots 35}a^{13}+\frac{15\cdots 13}{15\cdots 15}a^{12}-\frac{24\cdots 53}{35\cdots 70}a^{11}-\frac{17\cdots 08}{15\cdots 15}a^{10}+\frac{12\cdots 07}{31\cdots 83}a^{9}+\frac{79\cdots 91}{31\cdots 30}a^{8}-\frac{85\cdots 57}{10\cdots 61}a^{7}-\frac{10\cdots 22}{15\cdots 15}a^{6}+\frac{22\cdots 19}{31\cdots 30}a^{5}-\frac{51\cdots 84}{31\cdots 83}a^{4}-\frac{45\cdots 81}{10\cdots 10}a^{3}+\frac{22\cdots 23}{52\cdots 05}a^{2}-\frac{18\cdots 59}{52\cdots 05}a-\frac{10\cdots 27}{17\cdots 35}$, $\frac{29\cdots 09}{10\cdots 10}a^{17}-\frac{13\cdots 52}{75\cdots 15}a^{16}-\frac{10\cdots 88}{15\cdots 15}a^{15}+\frac{10\cdots 08}{15\cdots 15}a^{14}-\frac{34\cdots 14}{15\cdots 15}a^{13}-\frac{21\cdots 77}{31\cdots 30}a^{12}+\frac{15\cdots 57}{31\cdots 83}a^{11}+\frac{80\cdots 26}{15\cdots 15}a^{10}-\frac{19\cdots 41}{31\cdots 30}a^{9}+\frac{81\cdots 48}{15\cdots 15}a^{8}+\frac{10\cdots 78}{15\cdots 15}a^{7}-\frac{47\cdots 85}{63\cdots 66}a^{6}-\frac{14\cdots 97}{63\cdots 66}a^{5}+\frac{14\cdots 01}{15\cdots 15}a^{4}-\frac{62\cdots 99}{35\cdots 70}a^{3}-\frac{98\cdots 06}{52\cdots 05}a^{2}+\frac{10\cdots 76}{52\cdots 05}a+\frac{96\cdots 46}{35\cdots 87}$, $\frac{11\cdots 15}{10\cdots 61}a^{17}-\frac{59\cdots 47}{45\cdots 69}a^{16}+\frac{11\cdots 23}{21\cdots 22}a^{15}+\frac{21\cdots 14}{31\cdots 83}a^{14}-\frac{97\cdots 71}{63\cdots 66}a^{13}+\frac{45\cdots 59}{10\cdots 61}a^{12}+\frac{25\cdots 65}{31\cdots 83}a^{11}-\frac{36\cdots 49}{63\cdots 66}a^{10}-\frac{39\cdots 33}{10\cdots 61}a^{9}+\frac{23\cdots 49}{31\cdots 83}a^{8}-\frac{35\cdots 49}{63\cdots 66}a^{7}-\frac{20\cdots 14}{35\cdots 87}a^{6}+\frac{56\cdots 56}{31\cdots 83}a^{5}-\frac{86\cdots 81}{70\cdots 74}a^{4}-\frac{23\cdots 33}{70\cdots 74}a^{3}+\frac{63\cdots 07}{10\cdots 61}a^{2}+\frac{29\cdots 98}{35\cdots 87}a-\frac{21\cdots 08}{35\cdots 87}$, $\frac{97\cdots 29}{35\cdots 70}a^{17}-\frac{18\cdots 53}{45\cdots 90}a^{16}+\frac{47\cdots 78}{15\cdots 15}a^{15}-\frac{12\cdots 91}{10\cdots 61}a^{14}+\frac{24\cdots 71}{31\cdots 30}a^{13}+\frac{59\cdots 77}{31\cdots 30}a^{12}-\frac{66\cdots 41}{52\cdots 05}a^{11}+\frac{90\cdots 43}{21\cdots 22}a^{10}-\frac{16\cdots 19}{31\cdots 30}a^{9}-\frac{13\cdots 38}{52\cdots 05}a^{8}+\frac{49\cdots 27}{31\cdots 30}a^{7}-\frac{11\cdots 57}{31\cdots 30}a^{6}+\frac{29\cdots 71}{10\cdots 10}a^{5}+\frac{78\cdots 41}{15\cdots 15}a^{4}-\frac{28\cdots 53}{21\cdots 22}a^{3}+\frac{13\cdots 78}{17\cdots 35}a^{2}+\frac{52\cdots 42}{52\cdots 05}a-\frac{16\cdots 82}{17\cdots 35}$
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| Regulator: | \( 3978163078155.4883 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3978163078155.4883 \cdot 243}{6\cdot\sqrt{3238832304182321509797314520003000000000000}}\cr\approx \mathstrut & 1.36635071887234 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:C_6$ (as 18T23):
| A solvable group of order 54 |
| The 18 conjugacy class representatives for $C_3^2:C_6$ |
| Character table for $C_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.867.1 x3, 6.0.1771470000.6, 6.0.147954945870000.11, 6.0.147954945870000.12, 6.0.2255067.2 |
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 27 sibling: | 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 | R | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.3.4a1.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |
| 2.2.3.4a1.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
| 2.2.3.4a1.1 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
|
\(3\)
| 3.1.6.11a1.7 | $x^{6} + 9 x + 3$ | $6$ | $1$ | $11$ | $S_3\times C_3$ | $$[2, \frac{5}{2}]_{2}$$ |
| 3.1.6.11a1.7 | $x^{6} + 9 x + 3$ | $6$ | $1$ | $11$ | $S_3\times C_3$ | $$[2, \frac{5}{2}]_{2}$$ | |
| 3.1.6.11a1.7 | $x^{6} + 9 x + 3$ | $6$ | $1$ | $11$ | $S_3\times C_3$ | $$[2, \frac{5}{2}]_{2}$$ | |
|
\(5\)
| 5.2.3.4a1.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |
| 5.2.3.4a1.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
| 5.2.3.4a1.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
|
\(17\)
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |