Normalized defining polynomial
\( x^{20} - 3 x^{19} - 109 x^{18} + 874 x^{17} - 759 x^{16} - 15157 x^{15} - 135227 x^{14} + \cdots + 6774058131520 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(4, 8)$ |
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| Discriminant: |
\(5117380860656444079832785566505699574289404854272\)
\(\medspace = 2^{16}\cdot 7^{15}\cdot 11^{15}\cdot 13^{14}\)
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| Root discriminant: | \(272.55\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\), \(13\)
|
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| Discriminant root field: | \(\Q(\sqrt{77}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{14}+\frac{1}{16}a^{13}-\frac{1}{4}a^{12}+\frac{1}{16}a^{11}+\frac{5}{16}a^{10}+\frac{7}{16}a^{9}+\frac{3}{8}a^{8}+\frac{3}{16}a^{7}+\frac{1}{16}a^{6}-\frac{1}{16}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{32}a^{16}-\frac{1}{32}a^{15}+\frac{1}{32}a^{14}-\frac{1}{8}a^{13}+\frac{1}{32}a^{12}-\frac{11}{32}a^{11}-\frac{9}{32}a^{10}-\frac{5}{16}a^{9}-\frac{13}{32}a^{8}-\frac{15}{32}a^{7}+\frac{15}{32}a^{6}+\frac{1}{16}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{17}-\frac{3}{32}a^{14}-\frac{3}{32}a^{13}+\frac{3}{16}a^{12}-\frac{1}{8}a^{11}-\frac{3}{32}a^{10}+\frac{9}{32}a^{9}-\frac{3}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{32}a^{6}+\frac{1}{16}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{64}a^{18}-\frac{1}{64}a^{17}-\frac{1}{64}a^{16}-\frac{1}{32}a^{15}-\frac{1}{64}a^{14}-\frac{3}{64}a^{13}-\frac{11}{64}a^{12}+\frac{3}{16}a^{11}+\frac{5}{64}a^{10}-\frac{27}{64}a^{9}-\frac{23}{64}a^{8}-\frac{1}{2}a^{7}-\frac{15}{32}a^{6}+\frac{5}{16}a^{5}-\frac{1}{8}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{11\cdots 88}a^{19}-\frac{15\cdots 99}{11\cdots 88}a^{18}+\frac{10\cdots 65}{11\cdots 88}a^{17}+\frac{92\cdots 57}{14\cdots 36}a^{16}+\frac{14\cdots 79}{11\cdots 88}a^{15}+\frac{28\cdots 15}{31\cdots 24}a^{14}-\frac{21\cdots 57}{11\cdots 88}a^{13}+\frac{93\cdots 69}{15\cdots 12}a^{12}+\frac{46\cdots 45}{11\cdots 88}a^{11}-\frac{39\cdots 13}{11\cdots 88}a^{10}-\frac{52\cdots 73}{11\cdots 88}a^{9}-\frac{42\cdots 93}{58\cdots 44}a^{8}+\frac{19\cdots 33}{58\cdots 44}a^{7}+\frac{24\cdots 33}{14\cdots 36}a^{6}-\frac{49\cdots 39}{14\cdots 36}a^{5}+\frac{63\cdots 29}{14\cdots 36}a^{4}+\frac{54\cdots 93}{36\cdots 34}a^{3}+\frac{23\cdots 17}{73\cdots 68}a^{2}+\frac{66\cdots 15}{36\cdots 34}a+\frac{83\cdots 25}{18\cdots 17}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{43\cdots 39}{28\cdots 12}a^{19}+\frac{12\cdots 93}{14\cdots 56}a^{18}+\frac{21\cdots 87}{14\cdots 56}a^{17}-\frac{50\cdots 69}{28\cdots 12}a^{16}+\frac{15\cdots 37}{28\cdots 12}a^{15}+\frac{55\cdots 85}{35\cdots 14}a^{14}+\frac{16\cdots 11}{14\cdots 56}a^{13}-\frac{70\cdots 05}{28\cdots 12}a^{12}+\frac{11\cdots 85}{28\cdots 12}a^{11}+\frac{63\cdots 21}{14\cdots 56}a^{10}-\frac{29\cdots 35}{14\cdots 56}a^{9}-\frac{20\cdots 01}{28\cdots 12}a^{8}+\frac{11\cdots 65}{14\cdots 56}a^{7}+\frac{63\cdots 35}{35\cdots 14}a^{6}-\frac{49\cdots 69}{35\cdots 14}a^{5}-\frac{27\cdots 91}{70\cdots 28}a^{4}+\frac{88\cdots 39}{17\cdots 57}a^{3}+\frac{16\cdots 93}{35\cdots 14}a^{2}+\frac{73\cdots 56}{17\cdots 57}a+\frac{78\cdots 83}{17\cdots 57}$, $\frac{26\cdots 02}{17\cdots 57}a^{19}-\frac{65\cdots 97}{14\cdots 56}a^{18}-\frac{12\cdots 19}{70\cdots 28}a^{17}+\frac{10\cdots 69}{70\cdots 28}a^{16}+\frac{33\cdots 89}{14\cdots 56}a^{15}-\frac{65\cdots 85}{14\cdots 56}a^{14}-\frac{18\cdots 74}{17\cdots 57}a^{13}+\frac{16\cdots 87}{70\cdots 28}a^{12}+\frac{34\cdots 05}{14\cdots 56}a^{11}-\frac{10\cdots 45}{14\cdots 56}a^{10}+\frac{10\cdots 71}{70\cdots 28}a^{9}+\frac{11\cdots 61}{70\cdots 28}a^{8}-\frac{11\cdots 91}{14\cdots 56}a^{7}-\frac{31\cdots 87}{70\cdots 28}a^{6}+\frac{67\cdots 01}{35\cdots 14}a^{5}+\frac{28\cdots 85}{35\cdots 14}a^{4}-\frac{19\cdots 29}{17\cdots 57}a^{3}-\frac{27\cdots 35}{35\cdots 14}a^{2}-\frac{10\cdots 68}{17\cdots 57}a+\frac{82\cdots 61}{17\cdots 57}$, $\frac{26\cdots 27}{92\cdots 14}a^{19}-\frac{24\cdots 31}{36\cdots 56}a^{18}+\frac{10\cdots 47}{36\cdots 56}a^{17}-\frac{35\cdots 85}{36\cdots 56}a^{16}-\frac{64\cdots 23}{18\cdots 28}a^{15}+\frac{25\cdots 79}{99\cdots 88}a^{14}+\frac{20\cdots 21}{36\cdots 56}a^{13}-\frac{15\cdots 03}{99\cdots 88}a^{12}-\frac{55\cdots 33}{46\cdots 07}a^{11}+\frac{89\cdots 77}{36\cdots 56}a^{10}+\frac{32\cdots 77}{36\cdots 56}a^{9}-\frac{81\cdots 07}{36\cdots 56}a^{8}-\frac{62\cdots 36}{46\cdots 07}a^{7}+\frac{12\cdots 77}{18\cdots 28}a^{6}+\frac{20\cdots 91}{92\cdots 14}a^{5}-\frac{23\cdots 89}{46\cdots 07}a^{4}-\frac{19\cdots 81}{46\cdots 07}a^{3}-\frac{41\cdots 02}{46\cdots 07}a^{2}-\frac{19\cdots 18}{46\cdots 07}a+\frac{16\cdots 89}{46\cdots 07}$, $\frac{38\cdots 99}{36\cdots 56}a^{19}+\frac{30\cdots 11}{14\cdots 24}a^{18}+\frac{18\cdots 45}{14\cdots 24}a^{17}-\frac{12\cdots 99}{14\cdots 24}a^{16}-\frac{83\cdots 77}{73\cdots 12}a^{15}+\frac{12\cdots 77}{39\cdots 52}a^{14}+\frac{12\cdots 67}{14\cdots 24}a^{13}-\frac{56\cdots 65}{39\cdots 52}a^{12}-\frac{17\cdots 05}{46\cdots 07}a^{11}+\frac{77\cdots 43}{14\cdots 24}a^{10}-\frac{86\cdots 85}{14\cdots 24}a^{9}-\frac{16\cdots 01}{14\cdots 24}a^{8}+\frac{20\cdots 28}{46\cdots 07}a^{7}+\frac{24\cdots 55}{73\cdots 12}a^{6}-\frac{32\cdots 35}{36\cdots 56}a^{5}-\frac{12\cdots 79}{18\cdots 28}a^{4}+\frac{19\cdots 29}{18\cdots 28}a^{3}+\frac{61\cdots 29}{92\cdots 14}a^{2}+\frac{12\cdots 09}{92\cdots 14}a-\frac{30\cdots 39}{46\cdots 07}$, $\frac{83\cdots 75}{11\cdots 88}a^{19}+\frac{58\cdots 89}{11\cdots 88}a^{18}+\frac{86\cdots 01}{11\cdots 88}a^{17}-\frac{14\cdots 09}{14\cdots 36}a^{16}+\frac{34\cdots 03}{11\cdots 88}a^{15}+\frac{48\cdots 99}{31\cdots 24}a^{14}-\frac{23\cdots 73}{11\cdots 88}a^{13}-\frac{17\cdots 39}{15\cdots 12}a^{12}+\frac{19\cdots 77}{11\cdots 88}a^{11}+\frac{44\cdots 91}{11\cdots 88}a^{10}-\frac{22\cdots 01}{11\cdots 88}a^{9}-\frac{11\cdots 01}{58\cdots 44}a^{8}+\frac{29\cdots 71}{58\cdots 44}a^{7}+\frac{99\cdots 81}{18\cdots 17}a^{6}-\frac{30\cdots 45}{29\cdots 72}a^{5}-\frac{37\cdots 21}{18\cdots 17}a^{4}+\frac{15\cdots 36}{18\cdots 17}a^{3}+\frac{27\cdots 41}{73\cdots 68}a^{2}+\frac{76\cdots 05}{36\cdots 34}a-\frac{10\cdots 71}{36\cdots 34}$, $\frac{67\cdots 01}{11\cdots 88}a^{19}-\frac{43\cdots 79}{11\cdots 88}a^{18}-\frac{59\cdots 47}{11\cdots 88}a^{17}+\frac{10\cdots 29}{14\cdots 36}a^{16}-\frac{32\cdots 37}{11\cdots 88}a^{15}-\frac{26\cdots 01}{31\cdots 24}a^{14}-\frac{74\cdots 69}{11\cdots 88}a^{13}+\frac{16\cdots 81}{15\cdots 12}a^{12}-\frac{34\cdots 43}{11\cdots 88}a^{11}-\frac{11\cdots 85}{11\cdots 88}a^{10}+\frac{69\cdots 99}{11\cdots 88}a^{9}+\frac{17\cdots 75}{58\cdots 44}a^{8}-\frac{18\cdots 41}{58\cdots 44}a^{7}-\frac{25\cdots 59}{73\cdots 68}a^{6}+\frac{13\cdots 55}{29\cdots 72}a^{5}+\frac{10\cdots 31}{73\cdots 68}a^{4}-\frac{58\cdots 06}{18\cdots 17}a^{3}-\frac{13\cdots 91}{73\cdots 68}a^{2}-\frac{55\cdots 11}{36\cdots 34}a+\frac{14\cdots 21}{36\cdots 34}$, $\frac{10\cdots 73}{46\cdots 07}a^{19}+\frac{74\cdots 29}{18\cdots 28}a^{18}+\frac{59\cdots 39}{18\cdots 28}a^{17}-\frac{41\cdots 89}{18\cdots 28}a^{16}-\frac{43\cdots 95}{92\cdots 14}a^{15}+\frac{61\cdots 11}{49\cdots 44}a^{14}-\frac{35\cdots 15}{18\cdots 28}a^{13}-\frac{88\cdots 07}{49\cdots 44}a^{12}-\frac{71\cdots 04}{46\cdots 07}a^{11}+\frac{35\cdots 85}{18\cdots 28}a^{10}-\frac{10\cdots 55}{18\cdots 28}a^{9}-\frac{36\cdots 35}{18\cdots 28}a^{8}+\frac{67\cdots 20}{46\cdots 07}a^{7}+\frac{54\cdots 37}{92\cdots 14}a^{6}-\frac{14\cdots 61}{46\cdots 07}a^{5}-\frac{52\cdots 98}{46\cdots 07}a^{4}+\frac{59\cdots 74}{46\cdots 07}a^{3}+\frac{62\cdots 92}{46\cdots 07}a^{2}+\frac{56\cdots 24}{46\cdots 07}a-\frac{33\cdots 31}{46\cdots 07}$, $\frac{87\cdots 99}{58\cdots 44}a^{19}+\frac{56\cdots 99}{11\cdots 88}a^{18}-\frac{16\cdots 89}{11\cdots 88}a^{17}+\frac{47\cdots 83}{11\cdots 88}a^{16}+\frac{25\cdots 85}{14\cdots 36}a^{15}-\frac{37\cdots 69}{31\cdots 24}a^{14}-\frac{33\cdots 15}{11\cdots 88}a^{13}+\frac{17\cdots 41}{31\cdots 24}a^{12}+\frac{34\cdots 53}{58\cdots 44}a^{11}-\frac{67\cdots 59}{11\cdots 88}a^{10}-\frac{29\cdots 15}{11\cdots 88}a^{9}+\frac{13\cdots 93}{11\cdots 88}a^{8}+\frac{55\cdots 89}{29\cdots 72}a^{7}-\frac{89\cdots 89}{29\cdots 72}a^{6}-\frac{41\cdots 31}{29\cdots 72}a^{5}+\frac{62\cdots 23}{73\cdots 68}a^{4}+\frac{20\cdots 75}{14\cdots 36}a^{3}+\frac{25\cdots 11}{73\cdots 68}a^{2}+\frac{33\cdots 66}{18\cdots 17}a-\frac{51\cdots 31}{36\cdots 34}$, $\frac{84\cdots 07}{69\cdots 64}a^{19}-\frac{26\cdots 85}{69\cdots 64}a^{18}-\frac{87\cdots 23}{69\cdots 64}a^{17}+\frac{36\cdots 11}{34\cdots 32}a^{16}-\frac{12\cdots 73}{69\cdots 64}a^{15}-\frac{19\cdots 47}{18\cdots 72}a^{14}-\frac{11\cdots 05}{69\cdots 64}a^{13}+\frac{37\cdots 71}{23\cdots 84}a^{12}+\frac{14\cdots 69}{69\cdots 64}a^{11}-\frac{20\cdots 07}{69\cdots 64}a^{10}-\frac{13\cdots 05}{69\cdots 64}a^{9}+\frac{18\cdots 77}{17\cdots 16}a^{8}-\frac{14\cdots 53}{43\cdots 04}a^{7}-\frac{53\cdots 99}{17\cdots 16}a^{6}+\frac{15\cdots 65}{43\cdots 04}a^{5}+\frac{60\cdots 27}{86\cdots 08}a^{4}+\frac{39\cdots 15}{43\cdots 04}a^{3}-\frac{11\cdots 09}{21\cdots 02}a^{2}-\frac{45\cdots 05}{21\cdots 02}a-\frac{22\cdots 47}{10\cdots 01}$, $\frac{65\cdots 21}{11\cdots 88}a^{19}-\frac{87\cdots 95}{29\cdots 72}a^{18}-\frac{36\cdots 29}{73\cdots 68}a^{17}+\frac{79\cdots 89}{11\cdots 88}a^{16}-\frac{24\cdots 35}{11\cdots 88}a^{15}-\frac{14\cdots 91}{15\cdots 12}a^{14}-\frac{18\cdots 83}{29\cdots 72}a^{13}+\frac{35\cdots 81}{31\cdots 24}a^{12}-\frac{18\cdots 47}{11\cdots 88}a^{11}-\frac{28\cdots 62}{18\cdots 17}a^{10}+\frac{42\cdots 43}{73\cdots 68}a^{9}+\frac{57\cdots 89}{11\cdots 88}a^{8}-\frac{21\cdots 81}{58\cdots 44}a^{7}-\frac{16\cdots 37}{14\cdots 36}a^{6}+\frac{99\cdots 84}{18\cdots 17}a^{5}+\frac{22\cdots 97}{73\cdots 68}a^{4}-\frac{31\cdots 27}{14\cdots 36}a^{3}-\frac{55\cdots 69}{18\cdots 17}a^{2}-\frac{98\cdots 27}{18\cdots 17}a+\frac{49\cdots 03}{18\cdots 17}$, $\frac{26\cdots 17}{11\cdots 88}a^{19}-\frac{23\cdots 07}{29\cdots 72}a^{18}+\frac{27\cdots 81}{14\cdots 36}a^{17}+\frac{11\cdots 47}{11\cdots 88}a^{16}-\frac{33\cdots 37}{11\cdots 88}a^{15}-\frac{58\cdots 05}{15\cdots 12}a^{14}+\frac{25\cdots 21}{29\cdots 72}a^{13}+\frac{73\cdots 95}{31\cdots 24}a^{12}-\frac{57\cdots 69}{11\cdots 88}a^{11}-\frac{73\cdots 77}{73\cdots 68}a^{10}-\frac{52\cdots 89}{14\cdots 36}a^{9}+\frac{33\cdots 47}{11\cdots 88}a^{8}+\frac{39\cdots 21}{58\cdots 44}a^{7}-\frac{63\cdots 28}{18\cdots 17}a^{6}-\frac{66\cdots 81}{36\cdots 34}a^{5}-\frac{42\cdots 00}{18\cdots 17}a^{4}+\frac{22\cdots 43}{14\cdots 36}a^{3}+\frac{13\cdots 19}{36\cdots 34}a^{2}+\frac{33\cdots 61}{18\cdots 17}a-\frac{26\cdots 21}{18\cdots 17}$
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| Regulator: | \( 33260752823462220 \) (assuming GRH) |
| |
| Unit signature rank: | \( 2 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 33260752823462220 \cdot 8}{2\cdot\sqrt{5117380860656444079832785566505699574289404854272}}\cr\approx \mathstrut & 2.28574156303384 \end{aligned}\] (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
| \(\Q(\sqrt{1001}) \), \(\Q(\sqrt{154 +2 \sqrt{1001}})\), 5.1.35152.1, 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: | 20.4.393644681588957236910214274346592274945338834944.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | R | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | $20$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 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 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
|
\(7\)
| 7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 7.16.12.1 | $x^{16} + 20 x^{14} + 16 x^{13} + 162 x^{12} + 240 x^{11} + 776 x^{10} + 1344 x^{9} + 2539 x^{8} + 3696 x^{7} + 5016 x^{6} + 5312 x^{5} + 4594 x^{4} + 2928 x^{3} + 1404 x^{2} + 432 x + 88$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(11\)
| 11.4.3.2 | $x^{4} + 22$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 11.16.12.1 | $x^{16} + 32 x^{14} + 40 x^{13} + 392 x^{12} + 960 x^{11} + 2840 x^{10} + 7920 x^{9} + 15256 x^{8} + 28320 x^{7} + 45280 x^{6} + 47840 x^{5} + 30768 x^{4} + 11840 x^{3} + 2656 x^{2} + 320 x + 27$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(13\)
| 13.4.2.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 13.8.6.1 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 384 x + 29$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
| 13.8.6.1 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 384 x + 29$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |