Normalized defining polynomial
\( x^{20} - 5 x^{19} - 25 x^{18} + 35 x^{17} + 1000 x^{16} - 2478 x^{15} - 12250 x^{14} + 74190 x^{13} + \cdots + 8792410 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(4, 8)$ |
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| Discriminant: |
\(24264439127689134080000000000000000000000\)
\(\medspace = 2^{55}\cdot 5^{22}\cdot 7^{10}\)
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| Root discriminant: | \(104.53\) |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{20}a^{17}-\frac{1}{20}a^{16}-\frac{1}{20}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{20}a^{18}-\frac{1}{10}a^{16}-\frac{1}{20}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}$, $\frac{1}{32\cdots 00}a^{19}+\frac{99\cdots 97}{80\cdots 75}a^{18}-\frac{33\cdots 24}{80\cdots 75}a^{17}+\frac{19\cdots 09}{10\cdots 00}a^{16}-\frac{32\cdots 99}{32\cdots 00}a^{15}-\frac{77\cdots 93}{64\cdots 26}a^{14}-\frac{30\cdots 01}{43\cdots 84}a^{13}-\frac{30\cdots 19}{21\cdots 20}a^{12}+\frac{91\cdots 67}{64\cdots 60}a^{11}-\frac{21\cdots 81}{12\cdots 52}a^{10}+\frac{71\cdots 31}{21\cdots 20}a^{9}-\frac{97\cdots 01}{21\cdots 20}a^{8}-\frac{48\cdots 08}{16\cdots 15}a^{7}+\frac{63\cdots 57}{21\cdots 20}a^{6}+\frac{29\cdots 51}{12\cdots 52}a^{5}-\frac{41\cdots 52}{10\cdots 21}a^{4}-\frac{23\cdots 74}{53\cdots 05}a^{3}+\frac{27\cdots 59}{64\cdots 60}a^{2}+\frac{68\cdots 63}{16\cdots 15}a-\frac{57\cdots 61}{16\cdots 15}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| 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: |
$\frac{52\cdots 71}{61\cdots 50}a^{19}-\frac{14\cdots 31}{30\cdots 75}a^{18}-\frac{12\cdots 11}{61\cdots 50}a^{17}+\frac{25\cdots 07}{61\cdots 50}a^{16}+\frac{26\cdots 38}{30\cdots 75}a^{15}-\frac{63\cdots 41}{24\cdots 78}a^{14}-\frac{12\cdots 64}{12\cdots 39}a^{13}+\frac{87\cdots 73}{12\cdots 90}a^{12}-\frac{15\cdots 23}{12\cdots 90}a^{11}+\frac{25\cdots 21}{24\cdots 78}a^{10}-\frac{36\cdots 47}{12\cdots 90}a^{9}+\frac{10\cdots 57}{12\cdots 90}a^{8}+\frac{11\cdots 19}{61\cdots 95}a^{7}-\frac{13\cdots 59}{12\cdots 90}a^{6}+\frac{22\cdots 62}{12\cdots 39}a^{5}-\frac{11\cdots 68}{12\cdots 39}a^{4}-\frac{16\cdots 83}{12\cdots 90}a^{3}+\frac{46\cdots 92}{61\cdots 95}a^{2}+\frac{62\cdots 97}{12\cdots 90}a-\frac{39\cdots 02}{61\cdots 95}$, $\frac{33\cdots 16}{30\cdots 75}a^{19}-\frac{15\cdots 92}{30\cdots 75}a^{18}-\frac{89\cdots 21}{30\cdots 75}a^{17}+\frac{18\cdots 09}{61\cdots 50}a^{16}+\frac{33\cdots 76}{30\cdots 75}a^{15}-\frac{59\cdots 15}{24\cdots 78}a^{14}-\frac{17\cdots 25}{12\cdots 39}a^{13}+\frac{47\cdots 13}{61\cdots 95}a^{12}-\frac{73\cdots 03}{61\cdots 95}a^{11}+\frac{12\cdots 02}{12\cdots 39}a^{10}-\frac{21\cdots 47}{61\cdots 95}a^{9}+\frac{56\cdots 67}{61\cdots 95}a^{8}+\frac{14\cdots 53}{61\cdots 95}a^{7}-\frac{74\cdots 94}{61\cdots 95}a^{6}+\frac{24\cdots 58}{12\cdots 39}a^{5}-\frac{24\cdots 83}{24\cdots 78}a^{4}-\frac{87\cdots 63}{61\cdots 95}a^{3}-\frac{17\cdots 27}{12\cdots 90}a^{2}+\frac{33\cdots 82}{61\cdots 95}a+\frac{24\cdots 61}{61\cdots 95}$, $\frac{14\cdots 69}{16\cdots 50}a^{19}-\frac{85\cdots 51}{32\cdots 00}a^{18}-\frac{11\cdots 83}{32\cdots 00}a^{17}+\frac{99\cdots 67}{10\cdots 00}a^{16}+\frac{32\cdots 43}{32\cdots 00}a^{15}-\frac{81\cdots 37}{12\cdots 52}a^{14}-\frac{80\cdots 45}{43\cdots 84}a^{13}+\frac{62\cdots 49}{10\cdots 10}a^{12}+\frac{73\cdots 94}{16\cdots 15}a^{11}-\frac{27\cdots 43}{64\cdots 26}a^{10}+\frac{85\cdots 59}{10\cdots 10}a^{9}-\frac{13\cdots 29}{10\cdots 10}a^{8}+\frac{91\cdots 01}{16\cdots 15}a^{7}-\frac{26\cdots 99}{21\cdots 20}a^{6}-\frac{31\cdots 01}{12\cdots 52}a^{5}+\frac{30\cdots 15}{43\cdots 84}a^{4}-\frac{39\cdots 63}{21\cdots 20}a^{3}+\frac{13\cdots 47}{64\cdots 60}a^{2}-\frac{55\cdots 59}{64\cdots 60}a+\frac{68\cdots 69}{32\cdots 30}$, $\frac{21\cdots 47}{32\cdots 00}a^{19}-\frac{25\cdots 56}{80\cdots 75}a^{18}-\frac{56\cdots 77}{32\cdots 00}a^{17}+\frac{16\cdots 43}{10\cdots 00}a^{16}+\frac{21\cdots 97}{32\cdots 00}a^{15}-\frac{92\cdots 93}{64\cdots 26}a^{14}-\frac{35\cdots 09}{43\cdots 84}a^{13}+\frac{24\cdots 83}{53\cdots 05}a^{12}-\frac{12\cdots 89}{16\cdots 15}a^{11}+\frac{66\cdots 51}{64\cdots 26}a^{10}-\frac{14\cdots 92}{53\cdots 05}a^{9}+\frac{32\cdots 57}{53\cdots 05}a^{8}+\frac{72\cdots 11}{64\cdots 60}a^{7}-\frac{71\cdots 13}{10\cdots 10}a^{6}+\frac{17\cdots 19}{12\cdots 52}a^{5}-\frac{54\cdots 39}{43\cdots 84}a^{4}-\frac{34\cdots 87}{21\cdots 20}a^{3}-\frac{14\cdots 68}{16\cdots 15}a^{2}+\frac{32\cdots 19}{64\cdots 60}a-\frac{51\cdots 79}{32\cdots 30}$, $\frac{10\cdots 43}{16\cdots 50}a^{19}+\frac{21\cdots 33}{32\cdots 00}a^{18}-\frac{17\cdots 11}{32\cdots 00}a^{17}-\frac{16\cdots 63}{53\cdots 50}a^{16}+\frac{22\cdots 51}{32\cdots 00}a^{15}+\frac{12\cdots 31}{12\cdots 52}a^{14}-\frac{41\cdots 49}{21\cdots 42}a^{13}-\frac{13\cdots 37}{10\cdots 10}a^{12}+\frac{15\cdots 61}{32\cdots 30}a^{11}-\frac{46\cdots 21}{32\cdots 63}a^{10}-\frac{35\cdots 41}{53\cdots 05}a^{9}-\frac{40\cdots 53}{10\cdots 10}a^{8}+\frac{23\cdots 89}{32\cdots 30}a^{7}+\frac{67\cdots 27}{21\cdots 20}a^{6}-\frac{91\cdots 53}{12\cdots 52}a^{5}-\frac{22\cdots 58}{10\cdots 21}a^{4}+\frac{13\cdots 99}{21\cdots 20}a^{3}+\frac{69\cdots 09}{64\cdots 60}a^{2}-\frac{10\cdots 92}{16\cdots 15}a+\frac{36\cdots 44}{16\cdots 15}$, $\frac{41\cdots 63}{32\cdots 00}a^{19}-\frac{19\cdots 31}{32\cdots 00}a^{18}-\frac{11\cdots 63}{32\cdots 00}a^{17}+\frac{35\cdots 97}{10\cdots 00}a^{16}+\frac{10\cdots 32}{80\cdots 75}a^{15}-\frac{35\cdots 37}{12\cdots 52}a^{14}-\frac{72\cdots 01}{43\cdots 84}a^{13}+\frac{19\cdots 93}{21\cdots 20}a^{12}-\frac{86\cdots 59}{64\cdots 60}a^{11}+\frac{15\cdots 29}{12\cdots 52}a^{10}-\frac{95\cdots 57}{21\cdots 20}a^{9}+\frac{23\cdots 87}{21\cdots 20}a^{8}+\frac{88\cdots 57}{32\cdots 30}a^{7}-\frac{14\cdots 57}{10\cdots 10}a^{6}+\frac{14\cdots 63}{64\cdots 26}a^{5}-\frac{12\cdots 03}{10\cdots 21}a^{4}-\frac{30\cdots 33}{21\cdots 20}a^{3}-\frac{15\cdots 42}{16\cdots 15}a^{2}+\frac{22\cdots 53}{32\cdots 30}a-\frac{13\cdots 78}{16\cdots 15}$, $\frac{78\cdots 96}{16\cdots 15}a^{19}+\frac{18\cdots 04}{16\cdots 15}a^{18}-\frac{62\cdots 49}{64\cdots 60}a^{17}-\frac{35\cdots 38}{10\cdots 21}a^{16}+\frac{50\cdots 75}{32\cdots 63}a^{15}+\frac{95\cdots 15}{64\cdots 26}a^{14}-\frac{11\cdots 19}{21\cdots 42}a^{13}-\frac{74\cdots 51}{43\cdots 84}a^{12}+\frac{15\cdots 09}{12\cdots 52}a^{11}-\frac{23\cdots 47}{12\cdots 52}a^{10}+\frac{38\cdots 93}{43\cdots 84}a^{9}-\frac{26\cdots 61}{43\cdots 84}a^{8}+\frac{19\cdots 77}{12\cdots 52}a^{7}+\frac{10\cdots 19}{43\cdots 84}a^{6}-\frac{60\cdots 16}{32\cdots 63}a^{5}+\frac{12\cdots 85}{43\cdots 84}a^{4}-\frac{70\cdots 19}{43\cdots 84}a^{3}-\frac{27\cdots 39}{12\cdots 52}a^{2}-\frac{11\cdots 67}{12\cdots 52}a+\frac{72\cdots 13}{64\cdots 26}$, $\frac{55\cdots 57}{64\cdots 60}a^{19}-\frac{87\cdots 03}{64\cdots 60}a^{18}+\frac{70\cdots 41}{32\cdots 30}a^{17}+\frac{10\cdots 29}{53\cdots 05}a^{16}+\frac{64\cdots 37}{64\cdots 60}a^{15}-\frac{12\cdots 61}{12\cdots 52}a^{14}+\frac{40\cdots 87}{43\cdots 84}a^{13}+\frac{44\cdots 59}{43\cdots 84}a^{12}-\frac{78\cdots 53}{12\cdots 52}a^{11}+\frac{25\cdots 95}{12\cdots 52}a^{10}-\frac{26\cdots 49}{43\cdots 84}a^{9}+\frac{64\cdots 15}{43\cdots 84}a^{8}-\frac{11\cdots 14}{32\cdots 63}a^{7}+\frac{63\cdots 25}{10\cdots 21}a^{6}-\frac{11\cdots 67}{12\cdots 52}a^{5}+\frac{11\cdots 89}{43\cdots 84}a^{4}-\frac{76\cdots 17}{10\cdots 21}a^{3}+\frac{77\cdots 39}{64\cdots 26}a^{2}-\frac{16\cdots 08}{32\cdots 63}a+\frac{36\cdots 27}{32\cdots 63}$, $\frac{42\cdots 31}{10\cdots 00}a^{19}-\frac{17\cdots 07}{10\cdots 00}a^{18}-\frac{12\cdots 51}{10\cdots 00}a^{17}+\frac{28\cdots 91}{53\cdots 50}a^{16}+\frac{21\cdots 33}{53\cdots 50}a^{15}-\frac{27\cdots 23}{43\cdots 84}a^{14}-\frac{11\cdots 57}{21\cdots 42}a^{13}+\frac{53\cdots 83}{21\cdots 20}a^{12}-\frac{61\cdots 93}{21\cdots 20}a^{11}+\frac{78\cdots 83}{43\cdots 84}a^{10}-\frac{26\cdots 37}{21\cdots 20}a^{9}+\frac{57\cdots 97}{21\cdots 20}a^{8}+\frac{53\cdots 67}{53\cdots 05}a^{7}-\frac{20\cdots 31}{53\cdots 05}a^{6}+\frac{10\cdots 49}{21\cdots 42}a^{5}-\frac{26\cdots 87}{43\cdots 84}a^{4}-\frac{12\cdots 63}{21\cdots 20}a^{3}-\frac{27\cdots 49}{53\cdots 05}a^{2}+\frac{43\cdots 37}{21\cdots 20}a+\frac{80\cdots 63}{10\cdots 10}$, $\frac{10\cdots 83}{26\cdots 25}a^{19}-\frac{29\cdots 99}{10\cdots 00}a^{18}-\frac{42\cdots 37}{10\cdots 00}a^{17}+\frac{22\cdots 59}{10\cdots 00}a^{16}+\frac{36\cdots 37}{10\cdots 00}a^{15}-\frac{70\cdots 97}{43\cdots 84}a^{14}-\frac{53\cdots 27}{43\cdots 84}a^{13}+\frac{32\cdots 23}{10\cdots 10}a^{12}-\frac{60\cdots 74}{53\cdots 05}a^{11}+\frac{62\cdots 69}{21\cdots 42}a^{10}-\frac{83\cdots 37}{10\cdots 10}a^{9}+\frac{22\cdots 27}{10\cdots 10}a^{8}-\frac{40\cdots 17}{10\cdots 10}a^{7}+\frac{87\cdots 67}{21\cdots 20}a^{6}-\frac{72\cdots 27}{43\cdots 84}a^{5}-\frac{18\cdots 37}{43\cdots 84}a^{4}-\frac{96\cdots 31}{21\cdots 20}a^{3}+\frac{24\cdots 63}{21\cdots 20}a^{2}-\frac{10\cdots 31}{21\cdots 20}a+\frac{15\cdots 21}{10\cdots 10}$, $\frac{35\cdots 01}{32\cdots 00}a^{19}+\frac{11\cdots 42}{80\cdots 75}a^{18}-\frac{93\cdots 69}{80\cdots 75}a^{17}-\frac{25\cdots 13}{53\cdots 50}a^{16}+\frac{12\cdots 69}{80\cdots 75}a^{15}+\frac{54\cdots 36}{32\cdots 63}a^{14}-\frac{22\cdots 17}{43\cdots 84}a^{13}-\frac{92\cdots 51}{53\cdots 05}a^{12}+\frac{19\cdots 48}{16\cdots 15}a^{11}-\frac{11\cdots 39}{64\cdots 26}a^{10}+\frac{15\cdots 03}{10\cdots 10}a^{9}-\frac{65\cdots 53}{10\cdots 10}a^{8}+\frac{12\cdots 83}{64\cdots 60}a^{7}+\frac{27\cdots 81}{10\cdots 10}a^{6}-\frac{55\cdots 10}{32\cdots 63}a^{5}+\frac{30\cdots 61}{10\cdots 21}a^{4}-\frac{15\cdots 23}{10\cdots 10}a^{3}-\frac{74\cdots 53}{32\cdots 30}a^{2}-\frac{59\cdots 53}{64\cdots 60}a+\frac{31\cdots 73}{32\cdots 30}$
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| Regulator: | \( 93637547912100 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 93637547912100 \cdot 1}{2\cdot\sqrt{24264439127689134080000000000000000000000}}\cr\approx \mathstrut & 11.6813715459254 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.(S_3\times F_5)$ (as 20T559):
| A solvable group of order 30720 |
| The 37 conjugacy class representatives for $C_2^8.(S_3\times F_5)$ |
| Character table for $C_2^8.(S_3\times F_5)$ |
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 32 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 | ${\href{/padicField/3.8.0.1}{8} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | $15{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.2.8.52c1.71 | $x^{16} + 8 x^{15} + 44 x^{14} + 168 x^{13} + 494 x^{12} + 1152 x^{11} + 2196 x^{10} + 3464 x^{9} + 4563 x^{8} + 5024 x^{7} + 4620 x^{6} + 3520 x^{5} + 2202 x^{4} + 1104 x^{3} + 440 x^{2} + 136 x + 31$ | $8$ | $2$ | $52$ | 16T863 | $$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]^{4}$$ | |
|
\(5\)
| 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 | $x^{12} + 15 x^{10} + 12 x^{9} + 84 x^{8} + 120 x^{7} + 263 x^{6} + 372 x^{5} + 492 x^{4} + 424 x^{3} + 279 x^{2} + 115 x + 27$ | $3$ | $4$ | $8$ | $C_{12}$ | $$[\ ]_{3}^{4}$$ |