Normalized defining polynomial
\( x^{20} - 10 x^{19} + 60 x^{18} - 255 x^{17} + 1100 x^{16} - 4272 x^{15} + 12470 x^{14} - 28900 x^{13} + \cdots + 14016234 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(10644112278586200359531250000000000000000\)
\(\medspace = 2^{16}\cdot 3^{15}\cdot 5^{22}\cdot 7^{15}\)
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| Root discriminant: | \(100.31\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{5}-\frac{2}{5}$, $\frac{1}{5}a^{11}-\frac{1}{5}a^{6}-\frac{2}{5}a$, $\frac{1}{35}a^{12}-\frac{1}{35}a^{11}-\frac{2}{35}a^{10}+\frac{2}{7}a^{9}+\frac{3}{7}a^{8}+\frac{2}{5}a^{7}+\frac{1}{35}a^{6}-\frac{3}{35}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{17}{35}a^{2}+\frac{12}{35}a+\frac{4}{35}$, $\frac{1}{35}a^{13}-\frac{3}{35}a^{11}+\frac{1}{35}a^{10}-\frac{2}{7}a^{9}-\frac{6}{35}a^{8}+\frac{3}{7}a^{7}-\frac{2}{35}a^{6}-\frac{6}{35}a^{5}-\frac{1}{7}a^{4}-\frac{12}{35}a^{3}-\frac{1}{7}a^{2}+\frac{16}{35}a-\frac{17}{35}$, $\frac{1}{35}a^{14}-\frac{2}{35}a^{11}-\frac{2}{35}a^{10}-\frac{11}{35}a^{9}-\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{3}{35}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{7}a^{3}-\frac{16}{35}a-\frac{16}{35}$, $\frac{1}{35}a^{15}+\frac{3}{35}a^{11}-\frac{1}{35}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{7}+\frac{2}{35}a^{6}+\frac{8}{35}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{3}{7}a^{2}-\frac{6}{35}a+\frac{3}{7}$, $\frac{1}{175}a^{16}+\frac{2}{175}a^{15}+\frac{2}{175}a^{14}+\frac{1}{175}a^{13}+\frac{1}{175}a^{12}+\frac{1}{25}a^{11}-\frac{1}{35}a^{10}+\frac{3}{175}a^{9}+\frac{4}{175}a^{8}-\frac{3}{25}a^{7}+\frac{6}{35}a^{6}-\frac{71}{175}a^{5}-\frac{44}{175}a^{4}-\frac{67}{175}a^{3}+\frac{9}{25}a^{2}-\frac{51}{175}a+\frac{1}{175}$, $\frac{1}{175}a^{17}-\frac{2}{175}a^{15}+\frac{2}{175}a^{14}-\frac{1}{175}a^{13}+\frac{11}{175}a^{11}+\frac{13}{175}a^{10}+\frac{68}{175}a^{9}+\frac{3}{25}a^{8}+\frac{27}{175}a^{7}-\frac{11}{175}a^{6}-\frac{27}{175}a^{5}+\frac{36}{175}a^{4}+\frac{47}{175}a^{3}+\frac{83}{175}a^{2}+\frac{68}{175}a+\frac{73}{175}$, $\frac{1}{18\cdots 25}a^{18}+\frac{28\cdots 41}{18\cdots 25}a^{17}+\frac{97\cdots 24}{18\cdots 25}a^{16}-\frac{23\cdots 88}{18\cdots 25}a^{15}-\frac{90\cdots 07}{18\cdots 25}a^{14}+\frac{54\cdots 02}{54\cdots 35}a^{13}+\frac{31\cdots 76}{27\cdots 75}a^{12}+\frac{50\cdots 86}{18\cdots 25}a^{11}+\frac{70\cdots 86}{18\cdots 25}a^{10}-\frac{55\cdots 98}{18\cdots 25}a^{9}+\frac{47\cdots 57}{18\cdots 25}a^{8}+\frac{39\cdots 46}{37\cdots 45}a^{7}+\frac{94\cdots 66}{27\cdots 75}a^{6}-\frac{21\cdots 36}{27\cdots 75}a^{5}-\frac{53\cdots 16}{18\cdots 25}a^{4}+\frac{72\cdots 98}{18\cdots 25}a^{3}+\frac{86\cdots 19}{18\cdots 25}a^{2}-\frac{18\cdots 89}{37\cdots 45}a+\frac{10\cdots 59}{18\cdots 25}$, $\frac{1}{29\cdots 25}a^{19}+\frac{63\cdots 06}{29\cdots 25}a^{18}+\frac{29\cdots 39}{29\cdots 25}a^{17}-\frac{76\cdots 66}{29\cdots 25}a^{16}+\frac{38\cdots 97}{29\cdots 25}a^{15}-\frac{57\cdots 48}{42\cdots 75}a^{14}+\frac{58\cdots 62}{42\cdots 75}a^{13}-\frac{25\cdots 67}{29\cdots 25}a^{12}-\frac{43\cdots 37}{59\cdots 05}a^{11}+\frac{10\cdots 87}{29\cdots 25}a^{10}+\frac{45\cdots 73}{29\cdots 25}a^{9}+\frac{82\cdots 68}{29\cdots 25}a^{8}-\frac{28\cdots 77}{85\cdots 15}a^{7}-\frac{96\cdots 56}{42\cdots 75}a^{6}-\frac{10\cdots 78}{29\cdots 25}a^{5}-\frac{35\cdots 74}{59\cdots 05}a^{4}+\frac{22\cdots 26}{59\cdots 05}a^{3}-\frac{14\cdots 14}{29\cdots 25}a^{2}-\frac{71\cdots 63}{29\cdots 25}a-\frac{24\cdots 37}{60\cdots 25}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{10}\times C_{820}$, which has order $32800$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{10}\times C_{820}$, which has order $32800$ (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: |
$\frac{10\cdots 52}{85\cdots 15}a^{19}+\frac{10\cdots 94}{85\cdots 15}a^{18}-\frac{63\cdots 04}{85\cdots 15}a^{17}+\frac{27\cdots 07}{85\cdots 15}a^{16}-\frac{11\cdots 88}{85\cdots 15}a^{15}+\frac{46\cdots 48}{85\cdots 15}a^{14}-\frac{19\cdots 84}{12\cdots 45}a^{13}+\frac{32\cdots 74}{85\cdots 15}a^{12}-\frac{69\cdots 68}{12\cdots 45}a^{11}-\frac{72\cdots 32}{85\cdots 15}a^{10}+\frac{94\cdots 48}{85\cdots 15}a^{9}-\frac{27\cdots 24}{85\cdots 15}a^{8}+\frac{37\cdots 28}{85\cdots 15}a^{7}-\frac{14\cdots 50}{24\cdots 09}a^{6}-\frac{39\cdots 52}{12\cdots 45}a^{5}+\frac{11\cdots 68}{85\cdots 15}a^{4}-\frac{28\cdots 56}{12\cdots 45}a^{3}-\frac{31\cdots 92}{24\cdots 09}a^{2}-\frac{31\cdots 17}{85\cdots 15}a+\frac{33\cdots 55}{17\cdots 63}$, $\frac{52\cdots 28}{59\cdots 05}a^{19}-\frac{28\cdots 77}{29\cdots 25}a^{18}+\frac{69\cdots 75}{11\cdots 41}a^{17}-\frac{76\cdots 78}{29\cdots 25}a^{16}+\frac{32\cdots 82}{29\cdots 25}a^{15}-\frac{18\cdots 61}{42\cdots 75}a^{14}+\frac{79\cdots 87}{60\cdots 25}a^{13}-\frac{93\cdots 74}{29\cdots 25}a^{12}+\frac{55\cdots 01}{11\cdots 41}a^{11}+\frac{19\cdots 29}{29\cdots 25}a^{10}-\frac{26\cdots 68}{29\cdots 25}a^{9}+\frac{82\cdots 48}{29\cdots 25}a^{8}-\frac{23\cdots 39}{60\cdots 25}a^{7}+\frac{19\cdots 17}{42\cdots 75}a^{6}+\frac{14\cdots 54}{59\cdots 05}a^{5}-\frac{35\cdots 11}{29\cdots 25}a^{4}+\frac{16\cdots 38}{29\cdots 25}a^{3}+\frac{45\cdots 63}{29\cdots 25}a^{2}+\frac{12\cdots 86}{29\cdots 25}a-\frac{16\cdots 31}{42\cdots 75}$, $\frac{55\cdots 22}{59\cdots 05}a^{19}-\frac{54\cdots 96}{59\cdots 05}a^{18}+\frac{16\cdots 94}{29\cdots 25}a^{17}-\frac{72\cdots 84}{29\cdots 25}a^{16}+\frac{31\cdots 64}{29\cdots 25}a^{15}-\frac{24\cdots 58}{59\cdots 05}a^{14}+\frac{53\cdots 26}{42\cdots 75}a^{13}-\frac{88\cdots 89}{29\cdots 25}a^{12}+\frac{13\cdots 36}{29\cdots 25}a^{11}+\frac{19\cdots 37}{29\cdots 25}a^{10}-\frac{50\cdots 56}{59\cdots 05}a^{9}+\frac{73\cdots 18}{29\cdots 25}a^{8}-\frac{10\cdots 68}{29\cdots 25}a^{7}+\frac{18\cdots 08}{42\cdots 75}a^{6}+\frac{72\cdots 21}{29\cdots 25}a^{5}-\frac{60\cdots 19}{59\cdots 05}a^{4}+\frac{12\cdots 11}{29\cdots 25}a^{3}+\frac{74\cdots 49}{59\cdots 05}a^{2}+\frac{10\cdots 96}{29\cdots 25}a-\frac{85\cdots 77}{29\cdots 25}$, $\frac{31\cdots 04}{85\cdots 15}a^{19}-\frac{33\cdots 76}{85\cdots 15}a^{18}+\frac{41\cdots 08}{17\cdots 63}a^{17}-\frac{91\cdots 66}{85\cdots 15}a^{16}+\frac{78\cdots 32}{17\cdots 63}a^{15}-\frac{15\cdots 56}{85\cdots 15}a^{14}+\frac{67\cdots 24}{12\cdots 45}a^{13}-\frac{11\cdots 82}{85\cdots 15}a^{12}+\frac{18\cdots 36}{85\cdots 15}a^{11}+\frac{25\cdots 92}{12\cdots 45}a^{10}-\frac{29\cdots 46}{85\cdots 15}a^{9}+\frac{19\cdots 79}{17\cdots 63}a^{8}-\frac{13\cdots 48}{85\cdots 15}a^{7}+\frac{27\cdots 86}{12\cdots 45}a^{6}+\frac{10\cdots 98}{12\cdots 45}a^{5}-\frac{39\cdots 36}{85\cdots 15}a^{4}+\frac{15\cdots 86}{85\cdots 15}a^{3}+\frac{47\cdots 59}{85\cdots 15}a^{2}+\frac{27\cdots 86}{17\cdots 63}a+\frac{11\cdots 57}{85\cdots 15}$, $\frac{29\cdots 72}{59\cdots 05}a^{19}-\frac{14\cdots 73}{29\cdots 25}a^{18}+\frac{18\cdots 67}{59\cdots 05}a^{17}-\frac{39\cdots 67}{29\cdots 25}a^{16}+\frac{17\cdots 58}{29\cdots 25}a^{15}-\frac{96\cdots 89}{42\cdots 75}a^{14}+\frac{41\cdots 13}{60\cdots 25}a^{13}-\frac{48\cdots 41}{29\cdots 25}a^{12}+\frac{14\cdots 07}{59\cdots 05}a^{11}+\frac{10\cdots 81}{29\cdots 25}a^{10}-\frac{13\cdots 57}{29\cdots 25}a^{9}+\frac{41\cdots 02}{29\cdots 25}a^{8}-\frac{83\cdots 47}{42\cdots 75}a^{7}+\frac{10\cdots 58}{42\cdots 75}a^{6}+\frac{15\cdots 20}{11\cdots 41}a^{5}-\frac{17\cdots 14}{29\cdots 25}a^{4}+\frac{62\cdots 87}{29\cdots 25}a^{3}+\frac{20\cdots 97}{29\cdots 25}a^{2}+\frac{59\cdots 34}{29\cdots 25}a+\frac{50\cdots 31}{42\cdots 75}$, $\frac{15\cdots 04}{85\cdots 15}a^{19}+\frac{16\cdots 34}{85\cdots 15}a^{18}-\frac{10\cdots 78}{85\cdots 15}a^{17}+\frac{92\cdots 17}{17\cdots 63}a^{16}-\frac{19\cdots 28}{85\cdots 15}a^{15}+\frac{77\cdots 26}{85\cdots 15}a^{14}-\frac{67\cdots 24}{24\cdots 09}a^{13}+\frac{11\cdots 73}{17\cdots 63}a^{12}-\frac{86\cdots 36}{85\cdots 15}a^{11}-\frac{21\cdots 46}{17\cdots 63}a^{10}+\frac{15\cdots 11}{85\cdots 15}a^{9}-\frac{70\cdots 56}{12\cdots 45}a^{8}+\frac{99\cdots 72}{12\cdots 45}a^{7}-\frac{12\cdots 17}{12\cdots 45}a^{6}-\frac{59\cdots 38}{12\cdots 45}a^{5}+\frac{20\cdots 11}{85\cdots 15}a^{4}-\frac{90\cdots 43}{85\cdots 15}a^{3}-\frac{26\cdots 34}{85\cdots 15}a^{2}-\frac{75\cdots 38}{85\cdots 15}a+\frac{55\cdots 61}{85\cdots 15}$, $\frac{14\cdots 28}{59\cdots 05}a^{19}-\frac{31\cdots 17}{11\cdots 41}a^{18}+\frac{69\cdots 93}{42\cdots 75}a^{17}-\frac{21\cdots 11}{29\cdots 25}a^{16}+\frac{92\cdots 76}{29\cdots 25}a^{15}-\frac{72\cdots 62}{59\cdots 05}a^{14}+\frac{15\cdots 84}{42\cdots 75}a^{13}-\frac{26\cdots 56}{29\cdots 25}a^{12}+\frac{39\cdots 84}{29\cdots 25}a^{11}+\frac{10\cdots 87}{60\cdots 25}a^{10}-\frac{14\cdots 99}{59\cdots 05}a^{9}+\frac{22\cdots 02}{29\cdots 25}a^{8}-\frac{32\cdots 22}{29\cdots 25}a^{7}+\frac{57\cdots 87}{42\cdots 75}a^{6}+\frac{19\cdots 09}{29\cdots 25}a^{5}-\frac{19\cdots 51}{59\cdots 05}a^{4}+\frac{82\cdots 41}{60\cdots 25}a^{3}+\frac{23\cdots 96}{59\cdots 05}a^{2}+\frac{34\cdots 79}{29\cdots 25}a-\frac{12\cdots 53}{29\cdots 25}$, $\frac{39\cdots 64}{17\cdots 63}a^{19}-\frac{69\cdots 83}{29\cdots 25}a^{18}+\frac{43\cdots 39}{29\cdots 25}a^{17}-\frac{18\cdots 31}{29\cdots 25}a^{16}+\frac{81\cdots 62}{29\cdots 25}a^{15}-\frac{31\cdots 83}{29\cdots 25}a^{14}+\frac{13\cdots 87}{42\cdots 75}a^{13}-\frac{64\cdots 98}{85\cdots 15}a^{12}+\frac{33\cdots 21}{29\cdots 25}a^{11}+\frac{49\cdots 28}{29\cdots 25}a^{10}-\frac{64\cdots 07}{29\cdots 25}a^{9}+\frac{38\cdots 34}{59\cdots 05}a^{8}-\frac{27\cdots 27}{29\cdots 25}a^{7}+\frac{47\cdots 11}{42\cdots 75}a^{6}+\frac{26\cdots 38}{42\cdots 75}a^{5}-\frac{79\cdots 54}{29\cdots 25}a^{4}+\frac{32\cdots 03}{29\cdots 25}a^{3}+\frac{98\cdots 97}{29\cdots 25}a^{2}+\frac{56\cdots 83}{59\cdots 05}a-\frac{14\cdots 07}{59\cdots 05}$, $\frac{14\cdots 24}{59\cdots 05}a^{19}-\frac{77\cdots 31}{29\cdots 25}a^{18}+\frac{47\cdots 27}{29\cdots 25}a^{17}-\frac{20\cdots 06}{29\cdots 25}a^{16}+\frac{90\cdots 28}{29\cdots 25}a^{15}-\frac{50\cdots 78}{42\cdots 75}a^{14}+\frac{61\cdots 26}{17\cdots 63}a^{13}-\frac{25\cdots 04}{29\cdots 25}a^{12}+\frac{39\cdots 68}{29\cdots 25}a^{11}+\frac{49\cdots 28}{29\cdots 25}a^{10}-\frac{71\cdots 84}{29\cdots 25}a^{9}+\frac{22\cdots 13}{29\cdots 25}a^{8}-\frac{45\cdots 26}{42\cdots 75}a^{7}+\frac{32\cdots 85}{24\cdots 09}a^{6}+\frac{18\cdots 73}{29\cdots 25}a^{5}-\frac{93\cdots 68}{29\cdots 25}a^{4}+\frac{37\cdots 62}{29\cdots 25}a^{3}+\frac{11\cdots 89}{29\cdots 25}a^{2}+\frac{32\cdots 66}{29\cdots 25}a+\frac{21\cdots 29}{42\cdots 75}$
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| Regulator: | \( 179235004.46434572 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 179235004.46434572 \cdot 32800}{2\cdot\sqrt{10644112278586200359531250000000000000000}}\cr\approx \mathstrut & 2.73218837477612 \end{aligned}\] (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
| \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-21 +2 \sqrt{105}})\), 5.5.2450000.1, deg 10 |
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: | 20.0.2128822455717240071906250000000000000000.2 |
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 | $20$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 2.5.4.1 | $x^{5} + 2$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
|
\(3\)
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(5\)
| 5.10.11.1 | $x^{10} + 20 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.10.11.1 | $x^{10} + 20 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
|
\(7\)
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 7.16.12.1 | $x^{16} + 20 x^{14} + 16 x^{13} + 162 x^{12} + 240 x^{11} + 776 x^{10} + 1344 x^{9} + 2539 x^{8} + 3696 x^{7} + 5016 x^{6} + 5312 x^{5} + 4594 x^{4} + 2928 x^{3} + 1404 x^{2} + 432 x + 88$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ |