Normalized defining polynomial
\( x^{18} - 9 x^{17} + 39 x^{16} - 18 x^{15} - 489 x^{14} + 2259 x^{13} - 46082 x^{12} + 294723 x^{11} + \cdots + 57871188 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-1955476758561888850288369070881021250289363000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 163^{12}\)
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| Root discriminant: | \(1037.96\) |
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| Galois root discriminant: | $2^{2/3}3^{13/6}5^{2/3}163^{2/3}\approx 1496.9976600338653$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(163\)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{5}$, $\frac{1}{6}a^{12}-\frac{1}{6}a^{9}-\frac{1}{6}a^{6}+\frac{1}{6}a^{3}$, $\frac{1}{342}a^{13}-\frac{17}{342}a^{12}+\frac{23}{171}a^{11}-\frac{17}{114}a^{10}-\frac{7}{114}a^{9}-\frac{7}{57}a^{8}+\frac{11}{342}a^{7}-\frac{73}{342}a^{6}+\frac{79}{171}a^{5}+\frac{17}{114}a^{4}+\frac{53}{114}a^{3}-\frac{16}{57}a^{2}-\frac{7}{19}a$, $\frac{1}{342}a^{14}-\frac{5}{114}a^{12}+\frac{47}{342}a^{11}+\frac{4}{57}a^{10}-\frac{1}{6}a^{9}-\frac{1}{18}a^{8}+\frac{1}{3}a^{7}+\frac{1}{6}a^{6}+\frac{1}{342}a^{5}+\frac{1}{3}a^{4}-\frac{43}{114}a^{3}-\frac{8}{57}a^{2}-\frac{5}{19}a$, $\frac{1}{1930530527910}a^{15}-\frac{142077403}{193053052791}a^{14}-\frac{141424027}{101606869890}a^{13}-\frac{54470793683}{1930530527910}a^{12}+\frac{7447568584}{64351017597}a^{11}-\frac{463145459}{8815207890}a^{10}+\frac{161453387783}{1930530527910}a^{9}+\frac{153857835911}{965265263955}a^{8}+\frac{405892017301}{1930530527910}a^{7}+\frac{73685575699}{386106105582}a^{6}-\frac{3986263939}{107251695995}a^{5}+\frac{187705159217}{643510175970}a^{4}+\frac{146435901382}{321755087985}a^{3}+\frac{8758557529}{21450339199}a^{2}+\frac{28837731329}{107251695995}a+\frac{28407726}{77326385}$, $\frac{1}{18\cdots 90}a^{16}-\frac{1005698551}{18\cdots 90}a^{15}+\frac{17\cdots 37}{18\cdots 90}a^{14}-\frac{50\cdots 69}{41\cdots 22}a^{13}-\frac{73\cdots 16}{92\cdots 95}a^{12}+\frac{63\cdots 69}{18\cdots 90}a^{11}-\frac{10\cdots 91}{18\cdots 90}a^{10}-\frac{93\cdots 41}{56\cdots 65}a^{9}-\frac{89\cdots 91}{18\cdots 90}a^{8}-\frac{23\cdots 97}{61\cdots 30}a^{7}+\frac{32\cdots 09}{92\cdots 95}a^{6}-\frac{48\cdots 53}{97\cdots 10}a^{5}-\frac{43\cdots 78}{10\cdots 55}a^{4}-\frac{30\cdots 59}{61\cdots 30}a^{3}+\frac{84\cdots 97}{30\cdots 65}a^{2}+\frac{21\cdots 17}{54\cdots 45}a-\frac{86\cdots 64}{39\cdots 35}$, $\frac{1}{60\cdots 30}a^{17}+\frac{5329092690962}{30\cdots 15}a^{16}+\frac{73\cdots 73}{30\cdots 15}a^{15}+\frac{79\cdots 54}{67\cdots 47}a^{14}-\frac{42\cdots 61}{31\cdots 70}a^{13}-\frac{13\cdots 93}{60\cdots 30}a^{12}+\frac{37\cdots 37}{30\cdots 15}a^{11}-\frac{94\cdots 23}{12\cdots 46}a^{10}+\frac{63\cdots 11}{60\cdots 30}a^{9}+\frac{29\cdots 52}{10\cdots 05}a^{8}-\frac{19\cdots 43}{60\cdots 30}a^{7}-\frac{14\cdots 87}{60\cdots 30}a^{6}-\frac{10\cdots 57}{60\cdots 30}a^{5}+\frac{39\cdots 39}{20\cdots 10}a^{4}+\frac{18\cdots 03}{40\cdots 82}a^{3}+\frac{33\cdots 89}{10\cdots 05}a^{2}+\frac{47\cdots 01}{17\cdots 65}a-\frac{34\cdots 24}{12\cdots 95}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{18}\times C_{18}$, which has order $26244$ (assuming GRH) |
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| Narrow class group: | $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{18}\times C_{18}$, which has order $26244$ (assuming GRH) |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( \frac{24921398623776617}{1292465022871357038411315} a^{17} - \frac{23484685646650511}{136048949775932319832770} a^{16} + \frac{106061711367566811}{143607224763484115379035} a^{15} - \frac{139639680644256479}{516986009148542815364526} a^{14} - \frac{24825151132399014223}{2584930045742714076822630} a^{13} + \frac{37076948998947826237}{861643348580904692274210} a^{12} - \frac{2285057096889175725019}{2584930045742714076822630} a^{11} + \frac{14556420704941145007221}{2584930045742714076822630} a^{10} - \frac{14003991285178525320553}{861643348580904692274210} a^{9} - \frac{46872085491917501703749}{2584930045742714076822630} a^{8} + \frac{461745366397073566222087}{2584930045742714076822630} a^{7} - \frac{335686957168797603244451}{861643348580904692274210} a^{6} + \frac{552216233951117157459817}{45349649925310773277590} a^{5} - \frac{15054430257045689532699793}{430821674290452346137105} a^{4} + \frac{36387771993734541661154891}{861643348580904692274210} a^{3} - \frac{4338365191898794239331168}{143607224763484115379035} a^{2} + \frac{2085078308652011258361722}{143607224763484115379035} a - \frac{49920774496919951370}{20707602705621357661} \)
(order $6$)
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| Fundamental units: |
$\frac{12\cdots 89}{60\cdots 23}a^{17}-\frac{47\cdots 08}{30\cdots 15}a^{16}+\frac{17\cdots 71}{30\cdots 15}a^{15}+\frac{23\cdots 73}{60\cdots 30}a^{14}-\frac{29\cdots 57}{31\cdots 70}a^{13}+\frac{20\cdots 19}{60\cdots 30}a^{12}-\frac{53\cdots 39}{60\cdots 30}a^{11}+\frac{58\cdots 91}{12\cdots 46}a^{10}-\frac{68\cdots 31}{60\cdots 30}a^{9}-\frac{19\cdots 67}{60\cdots 30}a^{8}+\frac{88\cdots 47}{60\cdots 30}a^{7}-\frac{14\cdots 13}{60\cdots 30}a^{6}+\frac{16\cdots 65}{13\cdots 94}a^{5}-\frac{86\cdots 55}{40\cdots 82}a^{4}+\frac{40\cdots 83}{20\cdots 10}a^{3}-\frac{44\cdots 99}{33\cdots 35}a^{2}+\frac{83\cdots 44}{17\cdots 65}a-\frac{10\cdots 32}{12\cdots 95}$, $\frac{50\cdots 62}{30\cdots 15}a^{17}-\frac{28\cdots 69}{20\cdots 10}a^{16}+\frac{17\cdots 64}{30\cdots 15}a^{15}+\frac{14\cdots 72}{30\cdots 15}a^{14}-\frac{44\cdots 06}{53\cdots 95}a^{13}+\frac{10\cdots 08}{30\cdots 15}a^{12}-\frac{22\cdots 48}{30\cdots 15}a^{11}+\frac{45\cdots 67}{10\cdots 05}a^{10}-\frac{71\cdots 62}{60\cdots 23}a^{9}-\frac{66\cdots 91}{30\cdots 15}a^{8}+\frac{15\cdots 82}{10\cdots 05}a^{7}-\frac{81\cdots 26}{30\cdots 15}a^{6}+\frac{10\cdots 16}{10\cdots 05}a^{5}-\frac{51\cdots 83}{20\cdots 10}a^{4}+\frac{76\cdots 86}{33\cdots 35}a^{3}-\frac{44\cdots 45}{67\cdots 47}a^{2}+\frac{28\cdots 09}{17\cdots 65}a+\frac{71\cdots 65}{25\cdots 99}$, $\frac{96\cdots 69}{35\cdots 90}a^{17}-\frac{40\cdots 82}{17\cdots 95}a^{16}+\frac{22\cdots 59}{24\cdots 70}a^{15}+\frac{26\cdots 52}{35\cdots 19}a^{14}-\frac{12\cdots 61}{94\cdots 05}a^{13}+\frac{31\cdots 86}{59\cdots 65}a^{12}-\frac{21\cdots 22}{17\cdots 95}a^{11}+\frac{12\cdots 83}{17\cdots 95}a^{10}-\frac{14\cdots 86}{75\cdots 35}a^{9}-\frac{61\cdots 72}{17\cdots 95}a^{8}+\frac{40\cdots 26}{17\cdots 95}a^{7}-\frac{25\cdots 73}{59\cdots 65}a^{6}+\frac{20\cdots 03}{11\cdots 30}a^{5}-\frac{23\cdots 38}{59\cdots 65}a^{4}+\frac{49\cdots 21}{11\cdots 30}a^{3}-\frac{55\cdots 38}{19\cdots 55}a^{2}+\frac{13\cdots 08}{10\cdots 45}a-\frac{23\cdots 59}{15\cdots 47}$, $\frac{16\cdots 31}{35\cdots 90}a^{17}-\frac{51\cdots 51}{35\cdots 90}a^{16}-\frac{73\cdots 37}{39\cdots 10}a^{15}+\frac{41\cdots 57}{71\cdots 38}a^{14}-\frac{22\cdots 43}{18\cdots 10}a^{13}-\frac{32\cdots 07}{11\cdots 30}a^{12}-\frac{60\cdots 11}{35\cdots 90}a^{11}+\frac{76\cdots 79}{35\cdots 90}a^{10}+\frac{23\cdots 03}{11\cdots 30}a^{9}-\frac{53\cdots 61}{35\cdots 90}a^{8}-\frac{41\cdots 37}{35\cdots 90}a^{7}+\frac{11\cdots 11}{11\cdots 30}a^{6}+\frac{16\cdots 61}{59\cdots 65}a^{5}+\frac{44\cdots 33}{59\cdots 65}a^{4}-\frac{69\cdots 68}{59\cdots 65}a^{3}-\frac{28\cdots 62}{19\cdots 55}a^{2}+\frac{11\cdots 62}{10\cdots 45}a-\frac{43\cdots 15}{15\cdots 47}$, $\frac{55\cdots 31}{60\cdots 30}a^{17}-\frac{51\cdots 41}{60\cdots 30}a^{16}+\frac{22\cdots 10}{60\cdots 23}a^{15}-\frac{25\cdots 69}{12\cdots 46}a^{14}-\frac{72\cdots 12}{15\cdots 85}a^{13}+\frac{13\cdots 02}{60\cdots 23}a^{12}-\frac{25\cdots 91}{60\cdots 30}a^{11}+\frac{84\cdots 98}{30\cdots 15}a^{10}-\frac{25\cdots 31}{30\cdots 15}a^{9}-\frac{11\cdots 67}{16\cdots 02}a^{8}+\frac{26\cdots 83}{30\cdots 15}a^{7}-\frac{62\cdots 76}{30\cdots 15}a^{6}+\frac{39\cdots 78}{67\cdots 47}a^{5}-\frac{36\cdots 73}{20\cdots 10}a^{4}+\frac{81\cdots 66}{33\cdots 35}a^{3}-\frac{63\cdots 27}{33\cdots 35}a^{2}+\frac{35\cdots 57}{35\cdots 13}a-\frac{36\cdots 23}{12\cdots 95}$, $\frac{74\cdots 51}{67\cdots 70}a^{17}-\frac{87\cdots 24}{10\cdots 05}a^{16}-\frac{10\cdots 69}{20\cdots 10}a^{15}+\frac{33\cdots 88}{67\cdots 47}a^{14}-\frac{31\cdots 99}{10\cdots 90}a^{13}+\frac{24\cdots 02}{33\cdots 35}a^{12}+\frac{39\cdots 31}{10\cdots 05}a^{11}-\frac{13\cdots 21}{67\cdots 70}a^{10}+\frac{99\cdots 46}{10\cdots 05}a^{9}-\frac{22\cdots 23}{33\cdots 35}a^{8}+\frac{30\cdots 93}{27\cdots 70}a^{7}+\frac{65\cdots 16}{42\cdots 65}a^{6}-\frac{97\cdots 47}{20\cdots 10}a^{5}+\frac{13\cdots 79}{20\cdots 10}a^{4}-\frac{10\cdots 09}{20\cdots 10}a^{3}+\frac{10\cdots 07}{33\cdots 35}a^{2}-\frac{17\cdots 77}{17\cdots 65}a+\frac{38\cdots 09}{25\cdots 99}$, $\frac{27\cdots 34}{60\cdots 23}a^{17}-\frac{11\cdots 87}{30\cdots 15}a^{16}+\frac{28\cdots 59}{20\cdots 10}a^{15}+\frac{44\cdots 47}{60\cdots 30}a^{14}-\frac{34\cdots 23}{15\cdots 85}a^{13}+\frac{80\cdots 94}{10\cdots 05}a^{12}-\frac{12\cdots 31}{60\cdots 30}a^{11}+\frac{34\cdots 73}{30\cdots 15}a^{10}-\frac{27\cdots 77}{10\cdots 05}a^{9}-\frac{83\cdots 01}{12\cdots 46}a^{8}+\frac{10\cdots 96}{30\cdots 15}a^{7}-\frac{58\cdots 52}{10\cdots 05}a^{6}+\frac{57\cdots 29}{20\cdots 10}a^{5}-\frac{18\cdots 37}{33\cdots 35}a^{4}+\frac{10\cdots 93}{20\cdots 10}a^{3}-\frac{11\cdots 71}{33\cdots 35}a^{2}+\frac{44\cdots 77}{35\cdots 13}a-\frac{28\cdots 72}{12\cdots 95}$, $\frac{14\cdots 35}{40\cdots 82}a^{17}-\frac{18\cdots 79}{60\cdots 30}a^{16}+\frac{78\cdots 73}{60\cdots 30}a^{15}-\frac{20\cdots 03}{10\cdots 05}a^{14}-\frac{28\cdots 89}{15\cdots 85}a^{13}+\frac{22\cdots 93}{30\cdots 15}a^{12}-\frac{53\cdots 72}{33\cdots 35}a^{11}+\frac{60\cdots 13}{60\cdots 23}a^{10}-\frac{83\cdots 47}{30\cdots 15}a^{9}-\frac{13\cdots 71}{33\cdots 35}a^{8}+\frac{99\cdots 29}{30\cdots 15}a^{7}-\frac{19\cdots 36}{30\cdots 15}a^{6}+\frac{89\cdots 47}{40\cdots 82}a^{5}-\frac{24\cdots 57}{40\cdots 82}a^{4}+\frac{12\cdots 17}{20\cdots 10}a^{3}-\frac{90\cdots 01}{33\cdots 35}a^{2}+\frac{82\cdots 36}{17\cdots 65}a+\frac{18\cdots 77}{12\cdots 95}$
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| Regulator: | \( 78531702270513380 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 78531702270513380 \cdot 26244}{6\cdot\sqrt{1955476758561888850288369070881021250289363000000000000}}\cr\approx \mathstrut & 3.74900721342608 \end{aligned}\] (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
| \(\Q(\sqrt{-3}) \), 3.1.79707.1 x3, 6.0.1250501507258670000.1, 6.0.1250501507258670000.2, 6.0.1771470000.6, 6.0.19059617547.1 |
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 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 | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.1.0.1}{1} }^{18}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 53 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
|
\(163\)
| 163.1.3.2a1.1 | $x^{3} + 163$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 163.1.3.2a1.1 | $x^{3} + 163$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 163.1.3.2a1.1 | $x^{3} + 163$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 163.1.3.2a1.1 | $x^{3} + 163$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 163.1.3.2a1.1 | $x^{3} + 163$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 163.1.3.2a1.1 | $x^{3} + 163$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |