Normalized defining polynomial
\( x^{18} - 18 x^{16} - x^{15} + 117 x^{14} - 27 x^{13} - 369 x^{12} + 366 x^{11} + 18 x^{10} - 768 x^{9} + \cdots - 256 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[6, 6]$ |
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| Discriminant: |
\(727702811914257020738325491712\)
\(\medspace = 2^{12}\cdot 3^{27}\cdot 13^{12}\)
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| Root discriminant: | \(45.60\) |
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| Galois root discriminant: | $2^{3/2}3^{355/162}13^{3/4}\approx 215.0532439469929$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{3}{8}a^{7}+\frac{1}{8}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{11}+\frac{1}{16}a^{10}+\frac{3}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{15}-\frac{1}{16}a^{13}-\frac{1}{32}a^{12}-\frac{3}{32}a^{11}-\frac{3}{32}a^{10}-\frac{1}{32}a^{9}+\frac{3}{16}a^{8}+\frac{1}{16}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{128}a^{16}-\frac{1}{64}a^{15}-\frac{1}{64}a^{14}-\frac{5}{128}a^{13}+\frac{7}{128}a^{12}-\frac{5}{128}a^{11}-\frac{3}{128}a^{10}-\frac{3}{16}a^{9}+\frac{7}{64}a^{8}-\frac{3}{32}a^{7}+\frac{5}{32}a^{6}-\frac{7}{16}a^{5}+\frac{5}{16}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{11\cdots 36}a^{17}+\frac{17\cdots 09}{55\cdots 68}a^{16}+\frac{35\cdots 07}{55\cdots 68}a^{15}-\frac{19\cdots 89}{11\cdots 36}a^{14}+\frac{21\cdots 75}{11\cdots 36}a^{13}+\frac{26\cdots 43}{11\cdots 36}a^{12}+\frac{12\cdots 93}{65\cdots 08}a^{11}+\frac{24\cdots 89}{27\cdots 84}a^{10}+\frac{69\cdots 19}{55\cdots 68}a^{9}+\frac{11\cdots 19}{27\cdots 84}a^{8}-\frac{10\cdots 43}{27\cdots 84}a^{7}-\frac{58\cdots 85}{13\cdots 92}a^{6}+\frac{17\cdots 53}{13\cdots 92}a^{5}-\frac{17\cdots 07}{34\cdots 48}a^{4}-\frac{31\cdots 81}{69\cdots 96}a^{3}-\frac{97\cdots 97}{34\cdots 48}a^{2}+\frac{42\cdots 17}{17\cdots 24}a+\frac{21\cdots 60}{43\cdots 81}$
| 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: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{11\cdots 43}{13\cdots 92}a^{17}+\frac{15\cdots 63}{27\cdots 84}a^{16}-\frac{47\cdots 41}{34\cdots 48}a^{15}-\frac{27\cdots 59}{34\cdots 48}a^{14}+\frac{22\cdots 39}{27\cdots 84}a^{13}-\frac{20\cdots 27}{27\cdots 84}a^{12}-\frac{39\cdots 79}{16\cdots 52}a^{11}+\frac{10\cdots 21}{27\cdots 84}a^{10}-\frac{91\cdots 69}{13\cdots 92}a^{9}-\frac{17\cdots 59}{13\cdots 92}a^{8}+\frac{18\cdots 59}{17\cdots 24}a^{7}+\frac{32\cdots 47}{69\cdots 96}a^{6}-\frac{18\cdots 43}{17\cdots 24}a^{5}+\frac{95\cdots 33}{34\cdots 48}a^{4}+\frac{73\cdots 85}{17\cdots 24}a^{3}+\frac{60\cdots 61}{17\cdots 24}a^{2}+\frac{27\cdots 77}{43\cdots 81}a+\frac{15\cdots 69}{43\cdots 81}$, $\frac{266828103905059}{43\cdots 81}a^{17}+\frac{31\cdots 13}{27\cdots 84}a^{16}-\frac{54\cdots 02}{43\cdots 81}a^{15}-\frac{76\cdots 67}{13\cdots 92}a^{14}+\frac{27\cdots 07}{27\cdots 84}a^{13}+\frac{14\cdots 41}{27\cdots 84}a^{12}-\frac{62\cdots 59}{16\cdots 52}a^{11}-\frac{36\cdots 53}{27\cdots 84}a^{10}+\frac{73\cdots 69}{13\cdots 92}a^{9}+\frac{30\cdots 99}{13\cdots 92}a^{8}+\frac{70\cdots 29}{34\cdots 48}a^{7}+\frac{19\cdots 49}{69\cdots 96}a^{6}-\frac{57\cdots 13}{17\cdots 24}a^{5}-\frac{71\cdots 01}{34\cdots 48}a^{4}-\frac{94\cdots 51}{17\cdots 24}a^{3}+\frac{16\cdots 57}{17\cdots 24}a^{2}-\frac{28\cdots 03}{43\cdots 81}a-\frac{69\cdots 49}{43\cdots 81}$, $\frac{4218397059}{1352705323744}a^{17}+\frac{1835443521}{2705410647488}a^{16}-\frac{9479030251}{169088165468}a^{15}-\frac{12082754517}{676352661872}a^{14}+\frac{987007156245}{2705410647488}a^{13}+\frac{111666948311}{2705410647488}a^{12}-\frac{3217085331609}{2705410647488}a^{11}+\frac{1572921395523}{2705410647488}a^{10}+\frac{686395515213}{1352705323744}a^{9}-\frac{1926538346361}{1352705323744}a^{8}+\frac{198663427467}{84544082734}a^{7}+\frac{848903710357}{676352661872}a^{6}-\frac{180959131983}{169088165468}a^{5}-\frac{385999061571}{338176330936}a^{4}+\frac{133474186171}{169088165468}a^{3}+\frac{16649627007}{169088165468}a^{2}-\frac{24193875537}{42272041367}a-\frac{4226039197}{42272041367}$, $\frac{15\cdots 67}{13\cdots 92}a^{17}+\frac{22\cdots 13}{55\cdots 68}a^{16}-\frac{58\cdots 47}{27\cdots 84}a^{15}-\frac{27\cdots 35}{27\cdots 84}a^{14}+\frac{82\cdots 99}{55\cdots 68}a^{13}+\frac{25\cdots 19}{55\cdots 68}a^{12}-\frac{17\cdots 89}{32\cdots 04}a^{11}+\frac{50\cdots 09}{55\cdots 68}a^{10}+\frac{81\cdots 07}{13\cdots 92}a^{9}-\frac{13\cdots 37}{27\cdots 84}a^{8}+\frac{89\cdots 19}{13\cdots 92}a^{7}+\frac{75\cdots 65}{13\cdots 92}a^{6}-\frac{60\cdots 21}{69\cdots 96}a^{5}-\frac{77\cdots 59}{69\cdots 96}a^{4}-\frac{24\cdots 87}{17\cdots 24}a^{3}-\frac{85\cdots 41}{34\cdots 48}a^{2}+\frac{18\cdots 27}{17\cdots 24}a+\frac{19\cdots 21}{87\cdots 62}$, $\frac{24\cdots 13}{11\cdots 36}a^{17}-\frac{17\cdots 45}{69\cdots 96}a^{16}-\frac{20\cdots 79}{55\cdots 68}a^{15}+\frac{39\cdots 31}{11\cdots 36}a^{14}+\frac{23\cdots 41}{11\cdots 36}a^{13}-\frac{25\cdots 35}{11\cdots 36}a^{12}-\frac{31\cdots 81}{65\cdots 08}a^{11}+\frac{49\cdots 41}{55\cdots 68}a^{10}-\frac{65\cdots 73}{55\cdots 68}a^{9}+\frac{96\cdots 09}{69\cdots 96}a^{8}+\frac{20\cdots 27}{27\cdots 84}a^{7}-\frac{17\cdots 57}{69\cdots 96}a^{6}+\frac{40\cdots 03}{13\cdots 92}a^{5}-\frac{13\cdots 75}{69\cdots 96}a^{4}-\frac{13\cdots 25}{69\cdots 96}a^{3}+\frac{10\cdots 55}{87\cdots 62}a^{2}-\frac{31\cdots 81}{43\cdots 81}a+\frac{96\cdots 45}{87\cdots 62}$, $\frac{20\cdots 93}{69\cdots 96}a^{17}-\frac{12\cdots 99}{27\cdots 84}a^{16}-\frac{37\cdots 03}{69\cdots 96}a^{15}+\frac{64\cdots 95}{13\cdots 92}a^{14}+\frac{10\cdots 51}{27\cdots 84}a^{13}-\frac{35\cdots 31}{27\cdots 84}a^{12}-\frac{20\cdots 35}{16\cdots 52}a^{11}+\frac{34\cdots 75}{27\cdots 84}a^{10}+\frac{83\cdots 35}{13\cdots 92}a^{9}-\frac{35\cdots 61}{13\cdots 92}a^{8}+\frac{94\cdots 87}{34\cdots 48}a^{7}+\frac{98\cdots 79}{69\cdots 96}a^{6}-\frac{57\cdots 17}{17\cdots 24}a^{5}+\frac{73\cdots 75}{34\cdots 48}a^{4}+\frac{44\cdots 59}{17\cdots 24}a^{3}-\frac{38\cdots 19}{17\cdots 24}a^{2}+\frac{61\cdots 80}{43\cdots 81}a-\frac{59\cdots 41}{43\cdots 81}$, $\frac{44\cdots 91}{11\cdots 36}a^{17}+\frac{21\cdots 45}{55\cdots 68}a^{16}-\frac{36\cdots 99}{55\cdots 68}a^{15}-\frac{81\cdots 63}{11\cdots 36}a^{14}-\frac{72\cdots 91}{11\cdots 36}a^{13}+\frac{59\cdots 05}{11\cdots 36}a^{12}+\frac{64\cdots 59}{65\cdots 08}a^{11}-\frac{14\cdots 11}{69\cdots 96}a^{10}+\frac{38\cdots 33}{55\cdots 68}a^{9}+\frac{67\cdots 07}{27\cdots 84}a^{8}-\frac{46\cdots 77}{27\cdots 84}a^{7}+\frac{15\cdots 75}{13\cdots 92}a^{6}+\frac{18\cdots 47}{13\cdots 92}a^{5}-\frac{16\cdots 89}{87\cdots 62}a^{4}-\frac{45\cdots 03}{69\cdots 96}a^{3}+\frac{16\cdots 19}{34\cdots 48}a^{2}+\frac{10\cdots 41}{17\cdots 24}a-\frac{47\cdots 07}{43\cdots 81}$, $\frac{87\cdots 63}{69\cdots 96}a^{17}-\frac{66\cdots 59}{27\cdots 84}a^{16}-\frac{18\cdots 01}{69\cdots 96}a^{15}+\frac{20\cdots 01}{13\cdots 92}a^{14}+\frac{58\cdots 15}{27\cdots 84}a^{13}-\frac{73\cdots 07}{27\cdots 84}a^{12}-\frac{14\cdots 67}{16\cdots 52}a^{11}+\frac{10\cdots 19}{27\cdots 84}a^{10}+\frac{19\cdots 29}{13\cdots 92}a^{9}-\frac{15\cdots 67}{13\cdots 92}a^{8}+\frac{19\cdots 83}{34\cdots 48}a^{7}+\frac{47\cdots 45}{69\cdots 96}a^{6}-\frac{41\cdots 29}{17\cdots 24}a^{5}+\frac{97\cdots 39}{34\cdots 48}a^{4}+\frac{49\cdots 13}{17\cdots 24}a^{3}-\frac{51\cdots 87}{17\cdots 24}a^{2}+\frac{24\cdots 26}{43\cdots 81}a+\frac{11\cdots 86}{43\cdots 81}$, $\frac{23\cdots 17}{55\cdots 68}a^{17}+\frac{10\cdots 55}{34\cdots 48}a^{16}-\frac{21\cdots 59}{27\cdots 84}a^{15}-\frac{31\cdots 93}{55\cdots 68}a^{14}+\frac{28\cdots 53}{55\cdots 68}a^{13}+\frac{13\cdots 57}{55\cdots 68}a^{12}-\frac{56\cdots 77}{32\cdots 04}a^{11}+\frac{10\cdots 29}{27\cdots 84}a^{10}+\frac{37\cdots 83}{27\cdots 84}a^{9}-\frac{10\cdots 09}{34\cdots 48}a^{8}+\frac{50\cdots 03}{13\cdots 92}a^{7}+\frac{13\cdots 97}{34\cdots 48}a^{6}-\frac{23\cdots 53}{69\cdots 96}a^{5}+\frac{63\cdots 61}{34\cdots 48}a^{4}-\frac{16\cdots 09}{34\cdots 48}a^{3}-\frac{18\cdots 29}{87\cdots 62}a^{2}-\frac{32\cdots 79}{43\cdots 81}a+\frac{52\cdots 68}{43\cdots 81}$, $\frac{39\cdots 55}{55\cdots 68}a^{17}+\frac{37\cdots 23}{27\cdots 84}a^{16}-\frac{32\cdots 47}{27\cdots 84}a^{15}-\frac{72\cdots 31}{55\cdots 68}a^{14}+\frac{36\cdots 97}{55\cdots 68}a^{13}-\frac{14\cdots 51}{55\cdots 68}a^{12}-\frac{69\cdots 01}{32\cdots 04}a^{11}+\frac{27\cdots 13}{13\cdots 92}a^{10}-\frac{17\cdots 11}{27\cdots 84}a^{9}-\frac{16\cdots 05}{13\cdots 92}a^{8}+\frac{57\cdots 83}{13\cdots 92}a^{7}+\frac{34\cdots 65}{69\cdots 96}a^{6}+\frac{69\cdots 87}{69\cdots 96}a^{5}-\frac{40\cdots 85}{17\cdots 24}a^{4}-\frac{87\cdots 39}{34\cdots 48}a^{3}+\frac{49\cdots 91}{17\cdots 24}a^{2}+\frac{20\cdots 35}{87\cdots 62}a-\frac{23\cdots 31}{43\cdots 81}$, $\frac{60\cdots 93}{11\cdots 36}a^{17}-\frac{42\cdots 53}{13\cdots 92}a^{16}-\frac{57\cdots 15}{55\cdots 68}a^{15}+\frac{54\cdots 79}{11\cdots 36}a^{14}+\frac{83\cdots 53}{11\cdots 36}a^{13}-\frac{53\cdots 11}{11\cdots 36}a^{12}-\frac{17\cdots 09}{65\cdots 08}a^{11}+\frac{17\cdots 69}{55\cdots 68}a^{10}+\frac{10\cdots 99}{55\cdots 68}a^{9}-\frac{20\cdots 89}{34\cdots 48}a^{8}+\frac{18\cdots 67}{27\cdots 84}a^{7}+\frac{10\cdots 41}{69\cdots 96}a^{6}-\frac{12\cdots 53}{13\cdots 92}a^{5}+\frac{17\cdots 01}{69\cdots 96}a^{4}+\frac{14\cdots 27}{69\cdots 96}a^{3}-\frac{73\cdots 23}{43\cdots 81}a^{2}+\frac{166834396244837}{87\cdots 62}a+\frac{11\cdots 77}{87\cdots 62}$
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| Regulator: | \( 289404566.926 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 289404566.926 \cdot 1}{2\cdot\sqrt{727702811914257020738325491712}}\cr\approx \mathstrut & 0.667970898720 \end{aligned}\] (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 | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.2.4a2.1 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ | |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 2.4.2.8a3.1 | $x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 3$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $$[2, 2]^{4}$$ | |
|
\(3\)
| 3.1.9.18c4.3 | $x^{9} + 6 x^{6} + 18 x + 21$ | $9$ | $1$ | $18$ | $C_3^2 : S_3 $ | $$[2, 2, \frac{7}{3}]^{2}$$ |
| 3.3.3.9a2.1 | $x^{9} + 6 x^{7} + 3 x^{6} + 12 x^{5} + 12 x^{4} + 14 x^{3} + 12 x^{2} + 12 x + 7$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $$[\frac{3}{2}]_{2}^{3}$$ | |
|
\(13\)
| 13.3.2.3a1.2 | $x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 13.3.4.9a1.3 | $x^{12} + 8 x^{10} + 44 x^{9} + 24 x^{8} + 264 x^{7} + 758 x^{6} + 528 x^{5} + 2920 x^{4} + 5676 x^{3} + 2904 x^{2} + 10648 x + 14654$ | $4$ | $3$ | $9$ | $C_{12}$ | $$[\ ]_{4}^{3}$$ |