Normalized defining polynomial
\( x^{20} - 5 x^{19} - 15 x^{18} + 90 x^{17} + 275 x^{16} - 1533 x^{15} + 635 x^{14} + 3760 x^{13} + \cdots + 73459824 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(2128822455717240071906250000000000000000\)
\(\medspace = 2^{16}\cdot 3^{15}\cdot 5^{21}\cdot 7^{15}\)
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| Root discriminant: | \(92.56\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{105}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{10}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+\frac{2}{5}$, $\frac{1}{10}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}+\frac{2}{5}a$, $\frac{1}{10}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}+\frac{2}{5}a^{2}$, $\frac{1}{10}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{10}a^{3}$, $\frac{1}{10}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{2}{5}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{100}a^{15}-\frac{1}{20}a^{14}-\frac{1}{20}a^{13}-\frac{1}{20}a^{11}+\frac{1}{100}a^{10}+\frac{3}{20}a^{9}-\frac{1}{2}a^{8}-\frac{7}{20}a^{7}-\frac{7}{20}a^{6}-\frac{11}{100}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{9}{25}$, $\frac{1}{100}a^{16}-\frac{1}{20}a^{13}-\frac{1}{20}a^{12}-\frac{1}{25}a^{11}-\frac{1}{4}a^{9}+\frac{3}{20}a^{8}+\frac{2}{5}a^{7}-\frac{9}{25}a^{6}+\frac{3}{20}a^{5}-\frac{1}{2}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{11}{25}a+\frac{2}{5}$, $\frac{1}{100}a^{17}-\frac{1}{20}a^{14}-\frac{1}{20}a^{13}-\frac{1}{25}a^{12}-\frac{1}{20}a^{10}+\frac{3}{20}a^{9}+\frac{2}{5}a^{8}-\frac{9}{25}a^{7}+\frac{3}{20}a^{6}-\frac{1}{2}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{11}{25}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{1000}a^{18}+\frac{3}{1000}a^{17}+\frac{3}{1000}a^{16}-\frac{1}{250}a^{15}+\frac{1}{200}a^{14}-\frac{49}{1000}a^{13}+\frac{13}{1000}a^{12}-\frac{21}{500}a^{11}+\frac{41}{1000}a^{10}-\frac{3}{40}a^{9}-\frac{221}{1000}a^{8}+\frac{221}{500}a^{7}+\frac{101}{500}a^{6}+\frac{1}{250}a^{5}-\frac{7}{25}a^{4}+\frac{38}{125}a^{3}+\frac{54}{125}a^{2}+\frac{34}{125}a-\frac{27}{125}$, $\frac{1}{12\cdots 00}a^{19}-\frac{22\cdots 67}{24\cdots 00}a^{18}-\frac{32\cdots 91}{12\cdots 00}a^{17}-\frac{11\cdots 09}{60\cdots 00}a^{16}-\frac{40\cdots 03}{12\cdots 00}a^{15}+\frac{51\cdots 61}{12\cdots 00}a^{14}-\frac{17\cdots 53}{48\cdots 60}a^{13}-\frac{90\cdots 79}{30\cdots 50}a^{12}-\frac{38\cdots 63}{12\cdots 00}a^{11}-\frac{19\cdots 93}{12\cdots 00}a^{10}+\frac{87\cdots 29}{12\cdots 00}a^{9}+\frac{33\cdots 36}{30\cdots 75}a^{8}+\frac{50\cdots 93}{60\cdots 00}a^{7}+\frac{13\cdots 57}{30\cdots 50}a^{6}-\frac{10\cdots 93}{30\cdots 50}a^{5}+\frac{75\cdots 93}{15\cdots 75}a^{4}+\frac{11\cdots 97}{30\cdots 75}a^{3}-\frac{62\cdots 58}{15\cdots 75}a^{2}-\frac{71\cdots 44}{15\cdots 75}a-\frac{26\cdots 79}{15\cdots 75}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{10}\times C_{410}$, which has order $16400$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{10}\times C_{410}$, which has order $16400$ (assuming GRH) |
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| Relative class number: | data not computed (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{20\cdots 84}{88\cdots 75}a^{19}+\frac{33\cdots 41}{35\cdots 90}a^{18}+\frac{18\cdots 41}{35\cdots 90}a^{17}-\frac{17\cdots 29}{88\cdots 75}a^{16}-\frac{17\cdots 89}{17\cdots 50}a^{15}+\frac{29\cdots 21}{88\cdots 75}a^{14}+\frac{34\cdots 71}{71\cdots 18}a^{13}-\frac{60\cdots 39}{35\cdots 90}a^{12}-\frac{58\cdots 63}{17\cdots 50}a^{11}+\frac{11\cdots 23}{88\cdots 75}a^{10}-\frac{25\cdots 81}{88\cdots 75}a^{9}+\frac{94\cdots 08}{17\cdots 95}a^{8}-\frac{75\cdots 97}{35\cdots 90}a^{7}+\frac{13\cdots 04}{88\cdots 75}a^{6}-\frac{98\cdots 18}{88\cdots 75}a^{5}+\frac{30\cdots 64}{88\cdots 75}a^{4}-\frac{13\cdots 23}{35\cdots 90}a^{3}+\frac{19\cdots 14}{17\cdots 95}a^{2}-\frac{40\cdots 31}{88\cdots 75}a-\frac{38\cdots 23}{88\cdots 75}$, $\frac{53\cdots 09}{12\cdots 00}a^{19}+\frac{65\cdots 56}{30\cdots 75}a^{18}+\frac{54\cdots 43}{60\cdots 50}a^{17}-\frac{13\cdots 92}{30\cdots 75}a^{16}-\frac{43\cdots 33}{24\cdots 80}a^{15}+\frac{49\cdots 73}{60\cdots 50}a^{14}+\frac{98\cdots 09}{12\cdots 00}a^{13}-\frac{27\cdots 17}{60\cdots 50}a^{12}-\frac{10\cdots 83}{12\cdots 00}a^{11}+\frac{44\cdots 37}{12\cdots 90}a^{10}-\frac{62\cdots 01}{12\cdots 00}a^{9}+\frac{33\cdots 49}{30\cdots 75}a^{8}-\frac{30\cdots 53}{60\cdots 50}a^{7}+\frac{24\cdots 57}{30\cdots 75}a^{6}-\frac{78\cdots 01}{24\cdots 80}a^{5}+\frac{13\cdots 61}{30\cdots 75}a^{4}-\frac{30\cdots 16}{30\cdots 75}a^{3}+\frac{33\cdots 16}{30\cdots 75}a^{2}-\frac{34\cdots 83}{30\cdots 75}a+\frac{24\cdots 03}{60\cdots 95}$, $\frac{23\cdots 27}{24\cdots 00}a^{19}+\frac{81\cdots 63}{24\cdots 00}a^{18}+\frac{56\cdots 21}{24\cdots 00}a^{17}-\frac{16\cdots 99}{24\cdots 80}a^{16}-\frac{21\cdots 63}{48\cdots 60}a^{15}+\frac{27\cdots 53}{24\cdots 00}a^{14}+\frac{62\cdots 93}{24\cdots 00}a^{13}-\frac{31\cdots 01}{60\cdots 50}a^{12}-\frac{15\cdots 67}{96\cdots 52}a^{11}+\frac{20\cdots 07}{48\cdots 60}a^{10}-\frac{24\cdots 93}{24\cdots 00}a^{9}+\frac{51\cdots 99}{30\cdots 75}a^{8}-\frac{94\cdots 93}{12\cdots 00}a^{7}+\frac{10\cdots 33}{48\cdots 76}a^{6}-\frac{11\cdots 37}{24\cdots 80}a^{5}-\frac{49\cdots 81}{30\cdots 75}a^{4}-\frac{52\cdots 86}{30\cdots 75}a^{3}-\frac{17\cdots 67}{30\cdots 75}a^{2}-\frac{12\cdots 71}{60\cdots 95}a-\frac{15\cdots 98}{60\cdots 95}$, $\frac{60\cdots 43}{35\cdots 00}a^{19}-\frac{16\cdots 17}{35\cdots 00}a^{18}-\frac{16\cdots 19}{35\cdots 00}a^{17}+\frac{31\cdots 69}{35\cdots 00}a^{16}+\frac{15\cdots 97}{17\cdots 50}a^{15}-\frac{12\cdots 98}{88\cdots 75}a^{14}-\frac{20\cdots 37}{35\cdots 00}a^{13}+\frac{21\cdots 21}{35\cdots 00}a^{12}+\frac{55\cdots 97}{17\cdots 50}a^{11}-\frac{47\cdots 34}{88\cdots 75}a^{10}+\frac{53\cdots 47}{35\cdots 00}a^{9}-\frac{63\cdots 43}{35\cdots 00}a^{8}+\frac{38\cdots 99}{35\cdots 00}a^{7}+\frac{22\cdots 61}{35\cdots 00}a^{6}+\frac{14\cdots 83}{17\cdots 50}a^{5}+\frac{86\cdots 33}{88\cdots 75}a^{4}+\frac{28\cdots 13}{88\cdots 75}a^{3}+\frac{29\cdots 96}{88\cdots 75}a^{2}+\frac{29\cdots 84}{88\cdots 75}a+\frac{66\cdots 49}{88\cdots 75}$, $\frac{13\cdots 07}{24\cdots 00}a^{19}-\frac{96\cdots 97}{60\cdots 50}a^{18}-\frac{18\cdots 41}{12\cdots 00}a^{17}+\frac{77\cdots 71}{24\cdots 00}a^{16}+\frac{70\cdots 99}{24\cdots 00}a^{15}-\frac{15\cdots 01}{30\cdots 75}a^{14}-\frac{23\cdots 69}{12\cdots 00}a^{13}+\frac{67\cdots 63}{24\cdots 00}a^{12}+\frac{25\cdots 01}{24\cdots 00}a^{11}-\frac{27\cdots 13}{12\cdots 00}a^{10}+\frac{58\cdots 79}{12\cdots 00}a^{9}-\frac{74\cdots 67}{24\cdots 00}a^{8}+\frac{17\cdots 83}{60\cdots 50}a^{7}+\frac{34\cdots 77}{12\cdots 00}a^{6}+\frac{29\cdots 93}{12\cdots 00}a^{5}+\frac{20\cdots 87}{60\cdots 50}a^{4}+\frac{25\cdots 51}{30\cdots 75}a^{3}+\frac{37\cdots 14}{30\cdots 75}a^{2}+\frac{22\cdots 78}{30\cdots 75}a+\frac{79\cdots 17}{30\cdots 75}$, $\frac{44\cdots 21}{24\cdots 00}a^{19}+\frac{22\cdots 92}{30\cdots 75}a^{18}+\frac{48\cdots 51}{12\cdots 00}a^{17}-\frac{37\cdots 71}{24\cdots 00}a^{16}-\frac{18\cdots 01}{24\cdots 00}a^{15}+\frac{16\cdots 71}{60\cdots 50}a^{14}+\frac{11\cdots 37}{30\cdots 75}a^{13}-\frac{32\cdots 43}{24\cdots 00}a^{12}-\frac{65\cdots 71}{24\cdots 00}a^{11}+\frac{12\cdots 47}{12\cdots 00}a^{10}-\frac{13\cdots 41}{60\cdots 50}a^{9}+\frac{10\cdots 79}{24\cdots 00}a^{8}-\frac{52\cdots 64}{30\cdots 75}a^{7}+\frac{17\cdots 53}{12\cdots 00}a^{6}-\frac{28\cdots 63}{30\cdots 75}a^{5}+\frac{12\cdots 82}{30\cdots 75}a^{4}-\frac{94\cdots 32}{30\cdots 75}a^{3}+\frac{37\cdots 76}{30\cdots 75}a^{2}-\frac{11\cdots 43}{30\cdots 75}a-\frac{31\cdots 03}{30\cdots 75}$, $\frac{79\cdots 89}{24\cdots 00}a^{19}-\frac{11\cdots 03}{24\cdots 00}a^{18}-\frac{24\cdots 69}{24\cdots 00}a^{17}+\frac{82\cdots 37}{12\cdots 00}a^{16}+\frac{45\cdots 69}{24\cdots 00}a^{15}-\frac{17\cdots 41}{24\cdots 00}a^{14}-\frac{35\cdots 83}{24\cdots 00}a^{13}-\frac{36\cdots 81}{60\cdots 50}a^{12}+\frac{17\cdots 69}{24\cdots 00}a^{11}-\frac{60\cdots 71}{24\cdots 00}a^{10}+\frac{43\cdots 51}{24\cdots 00}a^{9}+\frac{14\cdots 96}{30\cdots 75}a^{8}+\frac{13\cdots 17}{12\cdots 00}a^{7}+\frac{55\cdots 23}{12\cdots 00}a^{6}+\frac{45\cdots 02}{30\cdots 75}a^{5}+\frac{25\cdots 89}{60\cdots 50}a^{4}+\frac{40\cdots 87}{60\cdots 50}a^{3}+\frac{42\cdots 33}{30\cdots 75}a^{2}+\frac{19\cdots 72}{30\cdots 75}a+\frac{61\cdots 07}{30\cdots 75}$, $\frac{21\cdots 97}{12\cdots 00}a^{19}-\frac{42\cdots 29}{24\cdots 00}a^{18}-\frac{14\cdots 39}{24\cdots 00}a^{17}+\frac{87\cdots 07}{48\cdots 60}a^{16}+\frac{66\cdots 51}{60\cdots 50}a^{15}-\frac{73\cdots 31}{24\cdots 00}a^{14}-\frac{21\cdots 09}{24\cdots 00}a^{13}-\frac{45\cdots 59}{24\cdots 00}a^{12}+\frac{98\cdots 79}{24\cdots 80}a^{11}-\frac{55\cdots 51}{24\cdots 00}a^{10}+\frac{19\cdots 81}{24\cdots 00}a^{9}+\frac{39\cdots 19}{24\cdots 00}a^{8}+\frac{15\cdots 51}{60\cdots 50}a^{7}+\frac{42\cdots 03}{12\cdots 19}a^{6}+\frac{87\cdots 53}{12\cdots 00}a^{5}+\frac{86\cdots 07}{30\cdots 75}a^{4}+\frac{19\cdots 31}{60\cdots 50}a^{3}+\frac{29\cdots 28}{30\cdots 75}a^{2}+\frac{15\cdots 33}{60\cdots 95}a+\frac{48\cdots 17}{30\cdots 75}$, $\frac{64\cdots 67}{60\cdots 50}a^{19}+\frac{73\cdots 01}{24\cdots 00}a^{18}+\frac{67\cdots 79}{24\cdots 00}a^{17}-\frac{14\cdots 41}{24\cdots 00}a^{16}-\frac{64\cdots 11}{12\cdots 00}a^{15}+\frac{22\cdots 87}{24\cdots 00}a^{14}+\frac{82\cdots 01}{24\cdots 00}a^{13}-\frac{97\cdots 11}{24\cdots 00}a^{12}-\frac{11\cdots 29}{60\cdots 50}a^{11}+\frac{85\cdots 63}{24\cdots 00}a^{10}-\frac{23\cdots 57}{24\cdots 00}a^{9}+\frac{29\cdots 39}{24\cdots 00}a^{8}-\frac{42\cdots 81}{60\cdots 50}a^{7}-\frac{92\cdots 08}{30\cdots 75}a^{6}-\frac{61\cdots 39}{12\cdots 00}a^{5}-\frac{16\cdots 39}{30\cdots 75}a^{4}-\frac{59\cdots 42}{30\cdots 75}a^{3}-\frac{55\cdots 98}{30\cdots 75}a^{2}-\frac{63\cdots 58}{30\cdots 75}a-\frac{13\cdots 86}{30\cdots 75}$
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| Regulator: | \( 63728758.83290176 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 63728758.83290176 \cdot 16400}{2\cdot\sqrt{2128822455717240071906250000000000000000}}\cr\approx \mathstrut & 1.08612116045538 \end{aligned}\] (assuming GRH)
Galois group
$C_{20}:C_4$ (as 20T18):
| A solvable group of order 80 |
| The 14 conjugacy class representatives for $C_{20}:C_4$ |
| Character table for $C_{20}:C_4$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-42 +2 \sqrt{21}})\), 5.5.2450000.1, 10.10.10210252500000000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 sibling: | 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 | R | R | R | $20$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(2\)
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
|
\(3\)
| 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
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\(5\)
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
| 5.10.11.1 | $x^{10} + 20 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
|
\(7\)
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 7.16.12.1 | $x^{16} + 20 x^{14} + 16 x^{13} + 162 x^{12} + 240 x^{11} + 776 x^{10} + 1344 x^{9} + 2539 x^{8} + 3696 x^{7} + 5016 x^{6} + 5312 x^{5} + 4594 x^{4} + 2928 x^{3} + 1404 x^{2} + 432 x + 88$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ |