Normalized defining polynomial
\( x^{20} - 40 x^{17} + 500 x^{16} + 4208 x^{15} + 22860 x^{14} + 89480 x^{13} + 347650 x^{12} + \cdots + 3773156 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(388231026043026145280000000000000000000000\)
\(\medspace = 2^{59}\cdot 5^{22}\cdot 7^{10}\)
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| Root discriminant: | \(120.08\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{20}a^{10}-\frac{1}{10}a^{8}+\frac{2}{5}a^{7}+\frac{3}{10}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}-\frac{9}{20}a^{2}-\frac{2}{5}a+\frac{1}{10}$, $\frac{1}{20}a^{11}-\frac{1}{10}a^{9}-\frac{1}{10}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{4}a^{3}-\frac{1}{5}a^{2}-\frac{1}{2}a-\frac{2}{5}$, $\frac{1}{20}a^{12}-\frac{1}{10}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{4}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{20}a^{13}-\frac{1}{10}a^{9}+\frac{1}{5}a^{7}-\frac{1}{20}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{20}a^{14}-\frac{1}{5}a^{7}-\frac{9}{20}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{10}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{100}a^{15}+\frac{1}{100}a^{10}-\frac{1}{10}a^{8}+\frac{3}{20}a^{7}+\frac{3}{10}a^{6}+\frac{3}{25}a^{5}+\frac{3}{10}a^{3}-\frac{9}{20}a^{2}-\frac{2}{5}a-\frac{11}{50}$, $\frac{1}{100}a^{16}+\frac{1}{100}a^{11}-\frac{1}{10}a^{9}-\frac{1}{20}a^{8}+\frac{1}{10}a^{7}-\frac{7}{25}a^{6}+\frac{2}{5}a^{5}+\frac{3}{10}a^{4}+\frac{3}{20}a^{3}+\frac{1}{5}a^{2}-\frac{1}{50}a-\frac{1}{5}$, $\frac{1}{100}a^{17}+\frac{1}{100}a^{12}-\frac{1}{20}a^{9}-\frac{1}{10}a^{8}-\frac{12}{25}a^{7}+\frac{1}{10}a^{5}+\frac{3}{20}a^{4}-\frac{1}{5}a^{3}+\frac{2}{25}a^{2}+\frac{1}{5}$, $\frac{1}{100}a^{18}+\frac{1}{100}a^{13}-\frac{1}{10}a^{9}+\frac{1}{50}a^{8}-\frac{2}{5}a^{6}+\frac{7}{20}a^{5}-\frac{1}{5}a^{4}+\frac{2}{25}a^{3}-\frac{1}{4}a^{2}+\frac{1}{5}a-\frac{3}{10}$, $\frac{1}{52\cdots 00}a^{19}+\frac{15\cdots 73}{66\cdots 75}a^{18}+\frac{27\cdots 83}{13\cdots 50}a^{17}-\frac{13\cdots 94}{66\cdots 75}a^{16}+\frac{43\cdots 66}{66\cdots 75}a^{15}-\frac{55\cdots 47}{26\cdots 00}a^{14}-\frac{88\cdots 72}{66\cdots 75}a^{13}-\frac{13\cdots 09}{26\cdots 00}a^{12}-\frac{14\cdots 13}{13\cdots 50}a^{11}+\frac{76\cdots 67}{13\cdots 50}a^{10}-\frac{87\cdots 29}{26\cdots 00}a^{9}-\frac{99\cdots 63}{13\cdots 50}a^{8}+\frac{11\cdots 71}{13\cdots 50}a^{7}-\frac{76\cdots 17}{26\cdots 00}a^{6}-\frac{61\cdots 81}{13\cdots 50}a^{5}-\frac{55\cdots 91}{26\cdots 00}a^{4}-\frac{56\cdots 53}{52\cdots 00}a^{3}-\frac{24\cdots 88}{66\cdots 75}a^{2}-\frac{69\cdots 13}{26\cdots 00}a-\frac{22\cdots 37}{66\cdots 75}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
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| Narrow class group: | $C_{4}$, which has order $4$ (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: |
$\frac{12\cdots 11}{13\cdots 50}a^{19}-\frac{61\cdots 29}{10\cdots 24}a^{18}+\frac{10\cdots 67}{26\cdots 10}a^{17}-\frac{48\cdots 69}{13\cdots 50}a^{16}+\frac{31\cdots 08}{66\cdots 75}a^{15}+\frac{93\cdots 27}{26\cdots 00}a^{14}+\frac{49\cdots 87}{26\cdots 10}a^{13}+\frac{37\cdots 29}{52\cdots 20}a^{12}+\frac{36\cdots 21}{13\cdots 50}a^{11}+\frac{28\cdots 67}{26\cdots 00}a^{10}+\frac{52\cdots 87}{13\cdots 50}a^{9}+\frac{15\cdots 01}{13\cdots 55}a^{8}+\frac{32\cdots 83}{13\cdots 55}a^{7}+\frac{98\cdots 59}{26\cdots 00}a^{6}+\frac{54\cdots 07}{13\cdots 50}a^{5}+\frac{10\cdots 91}{26\cdots 00}a^{4}+\frac{54\cdots 08}{13\cdots 55}a^{3}+\frac{83\cdots 54}{26\cdots 31}a^{2}+\frac{12\cdots 74}{66\cdots 75}a+\frac{36\cdots 54}{66\cdots 75}$, $\frac{17\cdots 61}{26\cdots 10}a^{19}-\frac{10\cdots 76}{26\cdots 31}a^{18}+\frac{19\cdots 21}{66\cdots 75}a^{17}-\frac{13\cdots 71}{52\cdots 20}a^{16}+\frac{44\cdots 27}{13\cdots 50}a^{15}+\frac{33\cdots 88}{13\cdots 55}a^{14}+\frac{17\cdots 67}{13\cdots 55}a^{13}+\frac{13\cdots 19}{26\cdots 00}a^{12}+\frac{51\cdots 41}{26\cdots 10}a^{11}+\frac{51\cdots 31}{66\cdots 75}a^{10}+\frac{37\cdots 93}{13\cdots 55}a^{9}+\frac{42\cdots 41}{52\cdots 20}a^{8}+\frac{22\cdots 39}{13\cdots 50}a^{7}+\frac{34\cdots 94}{13\cdots 55}a^{6}+\frac{19\cdots 47}{66\cdots 75}a^{5}+\frac{15\cdots 27}{52\cdots 20}a^{4}+\frac{77\cdots 72}{26\cdots 31}a^{3}+\frac{14\cdots 63}{66\cdots 75}a^{2}+\frac{17\cdots 44}{13\cdots 55}a+\frac{26\cdots 78}{66\cdots 75}$, $\frac{39\cdots 09}{26\cdots 00}a^{19}+\frac{14\cdots 49}{13\cdots 50}a^{18}-\frac{31\cdots 03}{26\cdots 00}a^{17}-\frac{15\cdots 61}{26\cdots 00}a^{16}+\frac{39\cdots 61}{52\cdots 20}a^{15}+\frac{16\cdots 99}{26\cdots 00}a^{14}+\frac{22\cdots 47}{66\cdots 75}a^{13}+\frac{85\cdots 28}{66\cdots 75}a^{12}+\frac{13\cdots 99}{26\cdots 00}a^{11}+\frac{52\cdots 29}{26\cdots 10}a^{10}+\frac{19\cdots 83}{26\cdots 00}a^{9}+\frac{57\cdots 61}{26\cdots 00}a^{8}+\frac{12\cdots 89}{26\cdots 00}a^{7}+\frac{19\cdots 93}{26\cdots 00}a^{6}+\frac{22\cdots 17}{26\cdots 10}a^{5}+\frac{10\cdots 21}{13\cdots 50}a^{4}+\frac{11\cdots 27}{13\cdots 50}a^{3}+\frac{86\cdots 03}{13\cdots 50}a^{2}+\frac{24\cdots 63}{66\cdots 75}a+\frac{15\cdots 16}{13\cdots 55}$, $\frac{31\cdots 97}{26\cdots 10}a^{19}-\frac{18\cdots 11}{52\cdots 20}a^{18}+\frac{36\cdots 57}{52\cdots 20}a^{17}-\frac{75\cdots 39}{13\cdots 50}a^{16}+\frac{19\cdots 69}{26\cdots 00}a^{15}+\frac{15\cdots 19}{52\cdots 20}a^{14}+\frac{86\cdots 69}{52\cdots 20}a^{13}+\frac{13\cdots 81}{26\cdots 31}a^{12}+\frac{58\cdots 47}{26\cdots 00}a^{11}+\frac{23\cdots 49}{26\cdots 00}a^{10}+\frac{15\cdots 39}{52\cdots 20}a^{9}+\frac{19\cdots 43}{26\cdots 10}a^{8}+\frac{13\cdots 73}{10\cdots 24}a^{7}+\frac{37\cdots 59}{26\cdots 00}a^{6}+\frac{39\cdots 73}{26\cdots 00}a^{5}+\frac{20\cdots 56}{13\cdots 55}a^{4}+\frac{67\cdots 93}{52\cdots 20}a^{3}+\frac{10\cdots 34}{13\cdots 55}a^{2}+\frac{39\cdots 03}{13\cdots 50}a+\frac{30\cdots 63}{66\cdots 75}$, $\frac{88\cdots 99}{52\cdots 20}a^{19}-\frac{35\cdots 31}{26\cdots 00}a^{18}+\frac{69\cdots 69}{66\cdots 75}a^{17}-\frac{44\cdots 64}{66\cdots 75}a^{16}+\frac{58\cdots 54}{66\cdots 75}a^{15}+\frac{66\cdots 71}{10\cdots 24}a^{14}+\frac{87\cdots 69}{26\cdots 00}a^{13}+\frac{81\cdots 19}{66\cdots 75}a^{12}+\frac{63\cdots 27}{13\cdots 50}a^{11}+\frac{50\cdots 01}{26\cdots 00}a^{10}+\frac{92\cdots 01}{13\cdots 55}a^{9}+\frac{13\cdots 62}{66\cdots 75}a^{8}+\frac{27\cdots 38}{66\cdots 75}a^{7}+\frac{16\cdots 23}{26\cdots 00}a^{6}+\frac{17\cdots 07}{26\cdots 00}a^{5}+\frac{17\cdots 12}{26\cdots 31}a^{4}+\frac{17\cdots 47}{26\cdots 00}a^{3}+\frac{32\cdots 52}{66\cdots 75}a^{2}+\frac{36\cdots 91}{13\cdots 50}a+\frac{48\cdots 97}{66\cdots 75}$, $\frac{38\cdots 51}{52\cdots 20}a^{19}-\frac{92\cdots 91}{52\cdots 20}a^{18}+\frac{20\cdots 19}{66\cdots 75}a^{17}-\frac{84\cdots 01}{26\cdots 00}a^{16}+\frac{11\cdots 09}{26\cdots 00}a^{15}+\frac{55\cdots 03}{26\cdots 10}a^{14}+\frac{57\cdots 05}{52\cdots 62}a^{13}+\frac{97\cdots 31}{26\cdots 00}a^{12}+\frac{39\cdots 69}{26\cdots 00}a^{11}+\frac{16\cdots 99}{26\cdots 00}a^{10}+\frac{27\cdots 46}{13\cdots 55}a^{9}+\frac{56\cdots 95}{10\cdots 24}a^{8}+\frac{26\cdots 37}{26\cdots 00}a^{7}+\frac{81\cdots 82}{66\cdots 75}a^{6}+\frac{16\cdots 89}{13\cdots 50}a^{5}+\frac{69\cdots 87}{52\cdots 20}a^{4}+\frac{31\cdots 11}{26\cdots 10}a^{3}+\frac{52\cdots 02}{66\cdots 75}a^{2}+\frac{23\cdots 18}{66\cdots 75}a+\frac{48\cdots 58}{66\cdots 75}$, $\frac{21\cdots 87}{13\cdots 50}a^{19}+\frac{42\cdots 33}{66\cdots 75}a^{18}-\frac{43\cdots 62}{66\cdots 75}a^{17}-\frac{18\cdots 73}{26\cdots 00}a^{16}+\frac{21\cdots 63}{26\cdots 00}a^{15}+\frac{19\cdots 69}{26\cdots 00}a^{14}+\frac{10\cdots 87}{26\cdots 00}a^{13}+\frac{42\cdots 07}{26\cdots 00}a^{12}+\frac{16\cdots 77}{26\cdots 00}a^{11}+\frac{63\cdots 93}{26\cdots 00}a^{10}+\frac{59\cdots 02}{66\cdots 75}a^{9}+\frac{71\cdots 29}{26\cdots 00}a^{8}+\frac{16\cdots 39}{26\cdots 00}a^{7}+\frac{25\cdots 09}{26\cdots 00}a^{6}+\frac{25\cdots 01}{26\cdots 00}a^{5}+\frac{13\cdots 07}{26\cdots 00}a^{4}-\frac{42\cdots 09}{26\cdots 00}a^{3}-\frac{60\cdots 59}{26\cdots 00}a^{2}-\frac{21\cdots 87}{13\cdots 50}a-\frac{54\cdots 83}{13\cdots 50}$, $\frac{38\cdots 57}{26\cdots 00}a^{19}-\frac{25\cdots 79}{13\cdots 55}a^{18}+\frac{30\cdots 89}{26\cdots 00}a^{17}-\frac{73\cdots 59}{13\cdots 55}a^{16}+\frac{21\cdots 17}{26\cdots 00}a^{15}+\frac{13\cdots 37}{26\cdots 00}a^{14}+\frac{33\cdots 42}{13\cdots 55}a^{13}+\frac{24\cdots 69}{26\cdots 00}a^{12}+\frac{96\cdots 79}{26\cdots 10}a^{11}+\frac{19\cdots 01}{13\cdots 50}a^{10}+\frac{13\cdots 09}{26\cdots 00}a^{9}+\frac{37\cdots 93}{26\cdots 10}a^{8}+\frac{74\cdots 63}{26\cdots 00}a^{7}+\frac{20\cdots 87}{52\cdots 20}a^{6}+\frac{51\cdots 37}{13\cdots 50}a^{5}+\frac{10\cdots 51}{26\cdots 00}a^{4}+\frac{20\cdots 27}{52\cdots 20}a^{3}+\frac{17\cdots 03}{66\cdots 75}a^{2}+\frac{34\cdots 27}{26\cdots 10}a+\frac{19\cdots 44}{66\cdots 75}$, $\frac{70\cdots 67}{10\cdots 24}a^{19}-\frac{98\cdots 33}{13\cdots 50}a^{18}-\frac{15\cdots 57}{26\cdots 00}a^{17}-\frac{13\cdots 07}{66\cdots 75}a^{16}+\frac{89\cdots 39}{26\cdots 00}a^{15}+\frac{33\cdots 28}{13\cdots 55}a^{14}+\frac{14\cdots 87}{13\cdots 50}a^{13}+\frac{58\cdots 89}{13\cdots 50}a^{12}+\frac{21\cdots 91}{13\cdots 50}a^{11}+\frac{45\cdots 06}{66\cdots 75}a^{10}+\frac{12\cdots 59}{52\cdots 20}a^{9}+\frac{43\cdots 67}{66\cdots 75}a^{8}+\frac{33\cdots 81}{26\cdots 00}a^{7}+\frac{11\cdots 51}{66\cdots 75}a^{6}+\frac{11\cdots 57}{66\cdots 75}a^{5}+\frac{97\cdots 79}{52\cdots 62}a^{4}+\frac{45\cdots 37}{26\cdots 00}a^{3}+\frac{16\cdots 47}{13\cdots 50}a^{2}+\frac{78\cdots 03}{13\cdots 50}a+\frac{89\cdots 18}{66\cdots 75}$
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| Regulator: | \( 3399389331400 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 3399389331400 \cdot 4}{2\cdot\sqrt{388231026043026145280000000000000000000000}}\cr\approx \mathstrut & 1.04636804067995 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.(F_5\times S_4)$ (as 20T811):
| A solvable group of order 122880 |
| The 80 conjugacy class representatives for $C_2^8.(F_5\times S_4)$ |
| Character table for $C_2^8.(F_5\times S_4)$ |
Intermediate fields
| 5.1.200000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 siblings: | data not computed |
| Degree 40 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | $15{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.10 | $x^{4} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ |
| 2.2.8.48b6.49 | $x^{16} + 8 x^{15} + 40 x^{14} + 144 x^{13} + 406 x^{12} + 920 x^{11} + 1716 x^{10} + 2664 x^{9} + 3475 x^{8} + 3816 x^{7} + 3524 x^{6} + 2712 x^{5} + 1714 x^{4} + 872 x^{3} + 348 x^{2} + 112 x + 23$ | $8$ | $2$ | $48$ | 16T863 | $$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}]^{4}$$ | |
|
\(5\)
| 5.1.5.5a1.4 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.1.15.17a1.4 | $x^{15} + 20 x^{3} + 5$ | $15$ | $1$ | $17$ | $F_5 \times S_3$ | $$[\frac{5}{4}]_{12}^{2}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 7.1.3.2a1.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.4.1.0a1.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 7.4.3.8a1.2 | $x^{12} + 15 x^{10} + 12 x^{9} + 84 x^{8} + 120 x^{7} + 263 x^{6} + 372 x^{5} + 492 x^{4} + 424 x^{3} + 279 x^{2} + 115 x + 27$ | $3$ | $4$ | $8$ | $C_{12}$ | $$[\ ]_{3}^{4}$$ |