Normalized defining polynomial
\( x^{18} - 9 x^{17} + 45 x^{16} - 126 x^{15} + 189 x^{14} - 27 x^{13} - 216 x^{12} - 729 x^{11} + \cdots + 1156 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-450283905890997363000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{37}\cdot 5^{12}\)
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| Root discriminant: | \(44.40\) |
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| Galois root discriminant: | $2^{2/3}3^{37/18}5^{2/3}\approx 44.40336798307826$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_3:S_3$ |
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| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{7}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{45}a^{9}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{7}{45}$, $\frac{1}{45}a^{10}+\frac{1}{5}a^{5}+\frac{4}{9}a-\frac{1}{5}$, $\frac{1}{45}a^{11}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{11}{45}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{450}a^{12}+\frac{2}{225}a^{11}-\frac{2}{225}a^{10}-\frac{1}{90}a^{9}-\frac{2}{25}a^{7}-\frac{1}{50}a^{6}-\frac{7}{25}a^{5}-\frac{1}{5}a^{4}-\frac{1}{18}a^{3}-\frac{77}{225}a^{2}-\frac{13}{225}a-\frac{77}{225}$, $\frac{1}{1350}a^{13}-\frac{1}{1350}a^{12}+\frac{1}{225}a^{11}+\frac{1}{270}a^{10}-\frac{1}{270}a^{9}+\frac{1}{25}a^{8}-\frac{1}{150}a^{7}+\frac{11}{150}a^{6}+\frac{1}{5}a^{5}-\frac{77}{270}a^{4}-\frac{659}{1350}a^{3}+\frac{74}{225}a^{2}+\frac{203}{675}a-\frac{64}{135}$, $\frac{1}{1350}a^{14}-\frac{1}{1350}a^{12}-\frac{13}{1350}a^{11}-\frac{1}{225}a^{10}-\frac{11}{1350}a^{9}+\frac{1}{30}a^{8}+\frac{2}{75}a^{7}-\frac{13}{150}a^{6}+\frac{641}{1350}a^{5}+\frac{17}{75}a^{4}+\frac{41}{270}a^{3}+\frac{77}{675}a^{2}-\frac{68}{225}a-\frac{83}{675}$, $\frac{1}{1350}a^{15}+\frac{1}{1350}a^{12}-\frac{1}{225}a^{10}-\frac{1}{270}a^{9}+\frac{1}{15}a^{8}-\frac{7}{75}a^{7}+\frac{13}{270}a^{6}+\frac{32}{75}a^{5}-\frac{2}{15}a^{4}-\frac{61}{135}a^{3}+\frac{2}{75}a^{2}+\frac{8}{45}a+\frac{62}{135}$, $\frac{1}{83700}a^{16}+\frac{1}{20925}a^{15}+\frac{23}{83700}a^{14}+\frac{1}{83700}a^{13}+\frac{1}{1674}a^{12}-\frac{629}{83700}a^{11}-\frac{623}{83700}a^{10}-\frac{8}{2325}a^{9}-\frac{203}{3100}a^{8}-\frac{1337}{16740}a^{7}+\frac{191}{20925}a^{6}+\frac{34363}{83700}a^{5}-\frac{6268}{20925}a^{4}+\frac{6233}{41850}a^{3}-\frac{763}{8370}a^{2}+\frac{4328}{20925}a+\frac{11}{2325}$, $\frac{1}{21\cdots 00}a^{17}-\frac{59536140920819}{10\cdots 50}a^{16}-\frac{356602764606103}{24\cdots 00}a^{15}-\frac{21177735745259}{24\cdots 00}a^{14}+\frac{27\cdots 11}{10\cdots 50}a^{13}+\frac{21\cdots 99}{21\cdots 00}a^{12}+\frac{24\cdots 69}{21\cdots 00}a^{11}-\frac{15\cdots 63}{18\cdots 75}a^{10}+\frac{57\cdots 91}{21\cdots 00}a^{9}+\frac{84\cdots 43}{21\cdots 00}a^{8}+\frac{44\cdots 64}{54\cdots 25}a^{7}+\frac{77\cdots 33}{80\cdots 00}a^{6}-\frac{13\cdots 13}{20\cdots 75}a^{5}+\frac{20\cdots 36}{54\cdots 25}a^{4}-\frac{13\cdots 82}{54\cdots 25}a^{3}-\frac{637242093008258}{70\cdots 39}a^{2}+\frac{15\cdots 73}{18\cdots 75}a-\frac{66\cdots 31}{32\cdots 25}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{3}\times C_{6}\times C_{6}$, which has order $108$ (assuming GRH) |
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| Narrow class group: | $C_{3}\times C_{6}\times C_{6}$, which has order $108$ (assuming GRH) |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -\frac{16580722975945}{218223796955517909} a^{17} + \frac{616288653199925}{872895187822071636} a^{16} - \frac{1549449380180957}{436447593911035818} a^{15} + \frac{8822454428403755}{872895187822071636} a^{14} - \frac{13264872400894195}{872895187822071636} a^{13} + \frac{543259332915545}{218223796955517909} a^{12} + \frac{16120294174521395}{872895187822071636} a^{11} + \frac{49458692589293621}{872895187822071636} a^{10} - \frac{132822660357257615}{436447593911035818} a^{9} + \frac{379599875740379095}{872895187822071636} a^{8} + \frac{24401006559029995}{872895187822071636} a^{7} - \frac{617704859030346235}{436447593911035818} a^{6} + \frac{2629026047845200643}{872895187822071636} a^{5} - \frac{218289606409466330}{218223796955517909} a^{4} - \frac{3459436774271394085}{436447593911035818} a^{3} + \frac{6591770227628359775}{436447593911035818} a^{2} - \frac{2089576757630355020}{218223796955517909} a + \frac{54616452757346}{414086901243867} \)
(order $6$)
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| Fundamental units: |
$\frac{20932347410989}{72\cdots 30}a^{17}-\frac{30\cdots 89}{10\cdots 50}a^{16}+\frac{78\cdots 72}{54\cdots 25}a^{15}-\frac{46\cdots 81}{10\cdots 45}a^{14}+\frac{15\cdots 61}{21\cdots 90}a^{13}-\frac{27\cdots 77}{10\cdots 45}a^{12}-\frac{43\cdots 51}{54\cdots 25}a^{11}-\frac{64\cdots 93}{36\cdots 50}a^{10}+\frac{14\cdots 06}{12\cdots 05}a^{9}-\frac{73\cdots 41}{36\cdots 15}a^{8}+\frac{67\cdots 23}{21\cdots 90}a^{7}+\frac{32\cdots 21}{54\cdots 25}a^{6}-\frac{14\cdots 71}{10\cdots 50}a^{5}+\frac{16\cdots 70}{21\cdots 09}a^{4}+\frac{32\cdots 48}{10\cdots 45}a^{3}-\frac{15\cdots 81}{21\cdots 09}a^{2}+\frac{99\cdots 19}{18\cdots 75}a+\frac{388896603587753}{71\cdots 65}$, $\frac{18812513506075}{43\cdots 18}a^{17}-\frac{21\cdots 42}{54\cdots 25}a^{16}+\frac{851170432952231}{43\cdots 18}a^{15}-\frac{60\cdots 69}{10\cdots 45}a^{14}+\frac{65\cdots 63}{72\cdots 30}a^{13}-\frac{127672504144048}{40\cdots 35}a^{12}-\frac{13\cdots 86}{18\cdots 75}a^{11}-\frac{31\cdots 51}{10\cdots 50}a^{10}+\frac{17\cdots 38}{10\cdots 45}a^{9}-\frac{28\cdots 08}{10\cdots 45}a^{8}+\frac{94\cdots 61}{21\cdots 90}a^{7}+\frac{41\cdots 18}{54\cdots 25}a^{6}-\frac{17\cdots 59}{10\cdots 50}a^{5}+\frac{36\cdots 89}{46\cdots 26}a^{4}+\frac{30\cdots 73}{80\cdots 70}a^{3}-\frac{32\cdots 21}{36\cdots 15}a^{2}+\frac{75\cdots 19}{10\cdots 45}a-\frac{16\cdots 24}{32\cdots 25}$, $\frac{13014558088057}{60\cdots 25}a^{17}-\frac{42\cdots 69}{21\cdots 00}a^{16}+\frac{651564821100217}{72\cdots 30}a^{15}-\frac{19\cdots 07}{87\cdots 36}a^{14}+\frac{14\cdots 27}{72\cdots 00}a^{13}+\frac{19\cdots 79}{72\cdots 30}a^{12}-\frac{20\cdots 63}{43\cdots 80}a^{11}-\frac{50\cdots 03}{21\cdots 00}a^{10}+\frac{24\cdots 39}{36\cdots 15}a^{9}-\frac{62\cdots 79}{24\cdots 00}a^{8}-\frac{30\cdots 81}{43\cdots 80}a^{7}+\frac{86\cdots 91}{36\cdots 15}a^{6}-\frac{17\cdots 07}{21\cdots 00}a^{5}+\frac{18\cdots 63}{72\cdots 30}a^{4}+\frac{51\cdots 63}{18\cdots 75}a^{3}-\frac{34\cdots 39}{10\cdots 50}a^{2}+\frac{14\cdots 59}{54\cdots 25}a+\frac{36\cdots 83}{10\cdots 75}$, $\frac{70102800686131}{36\cdots 50}a^{17}-\frac{32\cdots 57}{21\cdots 00}a^{16}+\frac{36\cdots 98}{54\cdots 25}a^{15}-\frac{23254193961797}{15\cdots 20}a^{14}+\frac{21\cdots 43}{21\cdots 00}a^{13}+\frac{15\cdots 07}{36\cdots 50}a^{12}-\frac{14\cdots 21}{14\cdots 60}a^{11}-\frac{24\cdots 47}{72\cdots 00}a^{10}+\frac{38\cdots 71}{10\cdots 45}a^{9}-\frac{10\cdots 81}{72\cdots 00}a^{8}-\frac{25\cdots 27}{21\cdots 00}a^{7}+\frac{47\cdots 37}{10\cdots 45}a^{6}-\frac{53\cdots 07}{80\cdots 00}a^{5}+\frac{40\cdots 36}{10\cdots 45}a^{4}+\frac{77\cdots 83}{36\cdots 50}a^{3}-\frac{12\cdots 67}{36\cdots 50}a^{2}+\frac{42\cdots 54}{18\cdots 75}a+\frac{48\cdots 02}{32\cdots 25}$, $\frac{113699694883274}{54\cdots 25}a^{17}-\frac{52\cdots 03}{21\cdots 00}a^{16}+\frac{13\cdots 83}{10\cdots 50}a^{15}-\frac{28\cdots 21}{72\cdots 00}a^{14}+\frac{13\cdots 67}{21\cdots 00}a^{13}-\frac{65\cdots 43}{21\cdots 90}a^{12}-\frac{11\cdots 71}{21\cdots 00}a^{11}-\frac{65\cdots 29}{24\cdots 00}a^{10}+\frac{62\cdots 17}{54\cdots 25}a^{9}-\frac{33\cdots 07}{21\cdots 00}a^{8}-\frac{49\cdots 91}{43\cdots 80}a^{7}+\frac{22\cdots 44}{54\cdots 25}a^{6}-\frac{86\cdots 53}{72\cdots 00}a^{5}+\frac{64\cdots 71}{10\cdots 50}a^{4}+\frac{94\cdots 88}{54\cdots 25}a^{3}-\frac{41\cdots 67}{10\cdots 50}a^{2}+\frac{22\cdots 26}{60\cdots 25}a-\frac{15\cdots 17}{64\cdots 85}$, $\frac{618685898753113}{10\cdots 50}a^{17}-\frac{12\cdots 31}{24\cdots 00}a^{16}+\frac{30\cdots 37}{12\cdots 50}a^{15}-\frac{15\cdots 13}{21\cdots 00}a^{14}+\frac{22\cdots 77}{21\cdots 00}a^{13}-\frac{78\cdots 89}{10\cdots 50}a^{12}-\frac{27\cdots 97}{21\cdots 00}a^{11}-\frac{94\cdots 43}{21\cdots 00}a^{10}+\frac{11\cdots 67}{54\cdots 25}a^{9}-\frac{62\cdots 37}{21\cdots 00}a^{8}-\frac{15\cdots 33}{24\cdots 00}a^{7}+\frac{20\cdots 49}{20\cdots 75}a^{6}-\frac{43\cdots 47}{21\cdots 00}a^{5}+\frac{33\cdots 63}{54\cdots 25}a^{4}+\frac{30\cdots 48}{54\cdots 25}a^{3}-\frac{11\cdots 11}{10\cdots 50}a^{2}+\frac{33\cdots 51}{54\cdots 25}a+\frac{12\cdots 56}{32\cdots 25}$, $\frac{322603739731933}{43\cdots 80}a^{17}-\frac{73\cdots 37}{21\cdots 00}a^{16}+\frac{23\cdots 27}{21\cdots 00}a^{15}+\frac{26\cdots 11}{10\cdots 50}a^{14}-\frac{10\cdots 07}{21\cdots 00}a^{13}+\frac{24\cdots 03}{21\cdots 00}a^{12}+\frac{18\cdots 09}{10\cdots 50}a^{11}-\frac{12\cdots 41}{21\cdots 00}a^{10}-\frac{75\cdots 87}{21\cdots 00}a^{9}+\frac{11\cdots 48}{54\cdots 25}a^{8}-\frac{44\cdots 03}{21\cdots 00}a^{7}+\frac{93\cdots 77}{21\cdots 00}a^{6}+\frac{23\cdots 41}{21\cdots 00}a^{5}-\frac{23\cdots 09}{10\cdots 50}a^{4}+\frac{15\cdots 89}{10\cdots 50}a^{3}+\frac{22\cdots 19}{35\cdots 50}a^{2}-\frac{12\cdots 01}{54\cdots 25}a-\frac{26\cdots 83}{32\cdots 25}$, $\frac{78109384333}{10\cdots 30}a^{17}-\frac{242492714687}{407323932721452}a^{16}+\frac{1362086689583}{509154915901815}a^{15}-\frac{12550185402211}{20\cdots 60}a^{14}+\frac{12646209136217}{20\cdots 60}a^{13}+\frac{6003521529667}{10\cdots 30}a^{12}-\frac{16357800586819}{20\cdots 60}a^{11}-\frac{28399617019967}{407323932721452}a^{10}+\frac{98943780907409}{509154915901815}a^{9}-\frac{292007102936771}{20\cdots 60}a^{8}-\frac{447210112519109}{20\cdots 60}a^{7}+\frac{528496642922611}{509154915901815}a^{6}-\frac{30\cdots 13}{20\cdots 60}a^{5}-\frac{610173757355222}{509154915901815}a^{4}+\frac{61\cdots 23}{10\cdots 30}a^{3}-\frac{13\cdots 05}{203661966360726}a^{2}+\frac{157729312514}{101830983180363}a-\frac{348638392172}{5990057834139}$
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| Regulator: | \( 2396291.4248 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 2396291.4248 \cdot 108}{6\cdot\sqrt{450283905890997363000000000000}}\cr\approx \mathstrut & 0.98104231852 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 18 |
| The 6 conjugacy class representatives for $C_3^2 : C_2$ |
| Character table for $C_3^2 : C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.24300.1 x3, 3.1.24300.2 x3, 3.1.108.1 x3, 3.1.6075.2 x3, 6.0.1771470000.3, 6.0.1771470000.4, 6.0.34992.1, 6.0.110716875.2, 9.1.387420489000000.21 x9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
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 | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.1.0.1}{1} }^{18}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 2.2.3.4a1.2 | $x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(3\)
| 3.1.18.37c2.16 | $x^{18} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ | $18$ | $1$ | $37$ | not computed | not computed |
|
\(5\)
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |