Normalized defining polynomial
\( x^{20} - 4 x^{19} - 32 x^{18} - 152 x^{17} + 662 x^{16} + 12176 x^{15} + 16832 x^{14} + \cdots + 17862528270 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(4, 8)$ |
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| Discriminant: |
\(1026012039426577859100909389414445029369055466909335552\)
\(\medspace = 2^{64}\cdot 3^{15}\cdot 53^{16}\)
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| |
| Root discriminant: | \(501.83\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(53\)
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| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{8}+\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}$, $\frac{1}{8}a^{13}+\frac{1}{4}a^{9}+\frac{1}{8}a^{5}-\frac{1}{4}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}-\frac{1}{4}a^{11}-\frac{3}{8}a^{10}-\frac{1}{2}a^{9}-\frac{1}{8}a^{8}-\frac{7}{16}a^{6}-\frac{1}{2}a^{5}-\frac{1}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{8}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{3}{8}a^{11}-\frac{1}{2}a^{10}-\frac{1}{8}a^{9}-\frac{1}{2}a^{8}-\frac{7}{16}a^{7}-\frac{1}{2}a^{6}-\frac{1}{16}a^{5}-\frac{1}{8}a^{3}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{16}-\frac{1}{16}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}+\frac{3}{16}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{3}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{16}a^{17}-\frac{1}{16}a^{13}-\frac{1}{2}a^{11}+\frac{3}{16}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{3}{16}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{424304}a^{18}-\frac{2533}{106076}a^{17}-\frac{8241}{424304}a^{16}-\frac{5845}{424304}a^{15}-\frac{795}{424304}a^{14}-\frac{21639}{424304}a^{13}+\frac{5671}{424304}a^{12}-\frac{78171}{212152}a^{11}-\frac{105073}{424304}a^{10}+\frac{30951}{212152}a^{9}-\frac{179895}{424304}a^{8}-\frac{12509}{424304}a^{7}+\frac{42825}{424304}a^{6}-\frac{179447}{424304}a^{5}-\frac{116501}{424304}a^{4}-\frac{49373}{212152}a^{3}+\frac{79429}{212152}a^{2}-\frac{2993}{212152}a-\frac{82053}{212152}$, $\frac{1}{35\cdots 28}a^{19}+\frac{27\cdots 41}{35\cdots 28}a^{18}+\frac{44\cdots 29}{17\cdots 64}a^{17}+\frac{58\cdots 89}{38\cdots 12}a^{16}-\frac{45\cdots 51}{35\cdots 28}a^{15}+\frac{68\cdots 31}{35\cdots 28}a^{14}-\frac{56\cdots 09}{17\cdots 64}a^{13}+\frac{10\cdots 05}{89\cdots 32}a^{12}+\frac{31\cdots 75}{35\cdots 28}a^{11}-\frac{10\cdots 09}{35\cdots 28}a^{10}+\frac{96\cdots 25}{38\cdots 12}a^{9}-\frac{17\cdots 23}{78\cdots 68}a^{8}+\frac{10\cdots 17}{35\cdots 28}a^{7}-\frac{60\cdots 57}{35\cdots 28}a^{6}+\frac{60\cdots 99}{17\cdots 64}a^{5}+\frac{11\cdots 15}{89\cdots 32}a^{4}-\frac{61\cdots 39}{17\cdots 64}a^{3}+\frac{60\cdots 03}{17\cdots 64}a^{2}-\frac{33\cdots 95}{89\cdots 32}a-\frac{22\cdots 18}{22\cdots 33}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{20\cdots 69}{67\cdots 56}a^{19}+\frac{72\cdots 25}{67\cdots 56}a^{18}+\frac{70\cdots 87}{67\cdots 56}a^{17}+\frac{74\cdots 63}{14\cdots 48}a^{16}-\frac{15\cdots 30}{84\cdots 07}a^{15}-\frac{65\cdots 29}{16\cdots 14}a^{14}-\frac{24\cdots 57}{33\cdots 28}a^{13}-\frac{27\cdots 46}{84\cdots 07}a^{12}+\frac{12\cdots 27}{67\cdots 56}a^{11}-\frac{78\cdots 55}{67\cdots 56}a^{10}-\frac{17\cdots 95}{14\cdots 48}a^{9}-\frac{32\cdots 07}{67\cdots 56}a^{8}-\frac{40\cdots 43}{16\cdots 14}a^{7}+\frac{57\cdots 35}{84\cdots 07}a^{6}+\frac{37\cdots 25}{33\cdots 28}a^{5}+\frac{82\cdots 07}{16\cdots 14}a^{4}+\frac{90\cdots 49}{16\cdots 14}a^{3}-\frac{10\cdots 37}{16\cdots 14}a^{2}-\frac{70\cdots 62}{84\cdots 07}a+\frac{15\cdots 43}{16\cdots 14}$, $\frac{35\cdots 65}{67\cdots 56}a^{19}-\frac{74\cdots 79}{67\cdots 56}a^{18}-\frac{12\cdots 95}{67\cdots 56}a^{17}-\frac{16\cdots 51}{14\cdots 48}a^{16}+\frac{10\cdots 64}{84\cdots 07}a^{15}+\frac{22\cdots 91}{33\cdots 28}a^{14}+\frac{71\cdots 91}{33\cdots 28}a^{13}+\frac{15\cdots 09}{16\cdots 14}a^{12}-\frac{10\cdots 15}{67\cdots 56}a^{11}+\frac{10\cdots 01}{67\cdots 56}a^{10}+\frac{34\cdots 23}{14\cdots 48}a^{9}+\frac{77\cdots 87}{67\cdots 56}a^{8}+\frac{10\cdots 43}{16\cdots 14}a^{7}-\frac{43\cdots 59}{33\cdots 28}a^{6}-\frac{36\cdots 75}{33\cdots 28}a^{5}-\frac{72\cdots 33}{84\cdots 07}a^{4}-\frac{35\cdots 33}{16\cdots 14}a^{3}+\frac{48\cdots 68}{84\cdots 07}a^{2}+\frac{32\cdots 41}{84\cdots 07}a-\frac{14\cdots 77}{16\cdots 14}$, $\frac{35\cdots 65}{67\cdots 56}a^{19}-\frac{74\cdots 79}{67\cdots 56}a^{18}-\frac{12\cdots 95}{67\cdots 56}a^{17}-\frac{16\cdots 51}{14\cdots 48}a^{16}+\frac{10\cdots 64}{84\cdots 07}a^{15}+\frac{22\cdots 91}{33\cdots 28}a^{14}+\frac{71\cdots 91}{33\cdots 28}a^{13}+\frac{15\cdots 09}{16\cdots 14}a^{12}-\frac{10\cdots 15}{67\cdots 56}a^{11}+\frac{10\cdots 01}{67\cdots 56}a^{10}+\frac{34\cdots 23}{14\cdots 48}a^{9}+\frac{77\cdots 87}{67\cdots 56}a^{8}+\frac{10\cdots 43}{16\cdots 14}a^{7}-\frac{43\cdots 59}{33\cdots 28}a^{6}-\frac{36\cdots 75}{33\cdots 28}a^{5}-\frac{72\cdots 33}{84\cdots 07}a^{4}-\frac{35\cdots 33}{16\cdots 14}a^{3}+\frac{48\cdots 68}{84\cdots 07}a^{2}+\frac{32\cdots 41}{84\cdots 07}a-\frac{11\cdots 49}{16\cdots 14}$, $\frac{14\cdots 01}{58\cdots 48}a^{19}-\frac{74\cdots 59}{58\cdots 48}a^{18}-\frac{36\cdots 25}{58\cdots 48}a^{17}-\frac{37\cdots 65}{12\cdots 84}a^{16}+\frac{11\cdots 99}{58\cdots 48}a^{15}+\frac{16\cdots 83}{58\cdots 48}a^{14}+\frac{47\cdots 79}{58\cdots 48}a^{13}+\frac{13\cdots 41}{58\cdots 48}a^{12}-\frac{10\cdots 33}{58\cdots 48}a^{11}+\frac{24\cdots 87}{58\cdots 48}a^{10}+\frac{49\cdots 51}{12\cdots 84}a^{9}+\frac{75\cdots 93}{25\cdots 76}a^{8}+\frac{81\cdots 11}{58\cdots 48}a^{7}-\frac{44\cdots 25}{58\cdots 48}a^{6}+\frac{26\cdots 47}{58\cdots 48}a^{5}-\frac{23\cdots 83}{58\cdots 48}a^{4}+\frac{10\cdots 69}{29\cdots 24}a^{3}+\frac{12\cdots 83}{29\cdots 24}a^{2}-\frac{23\cdots 05}{29\cdots 24}a+\frac{10\cdots 77}{29\cdots 24}$, $\frac{66\cdots 11}{35\cdots 28}a^{19}-\frac{20\cdots 91}{35\cdots 28}a^{18}-\frac{23\cdots 27}{35\cdots 28}a^{17}-\frac{13\cdots 41}{38\cdots 12}a^{16}+\frac{34\cdots 91}{35\cdots 28}a^{15}+\frac{21\cdots 07}{89\cdots 32}a^{14}+\frac{19\cdots 99}{35\cdots 28}a^{13}+\frac{77\cdots 61}{35\cdots 28}a^{12}-\frac{38\cdots 23}{35\cdots 28}a^{11}-\frac{13\cdots 31}{35\cdots 28}a^{10}+\frac{50\cdots 73}{76\cdots 24}a^{9}+\frac{27\cdots 79}{89\cdots 32}a^{8}+\frac{55\cdots 39}{35\cdots 28}a^{7}-\frac{15\cdots 65}{44\cdots 66}a^{6}-\frac{15\cdots 71}{15\cdots 36}a^{5}-\frac{14\cdots 51}{35\cdots 28}a^{4}-\frac{10\cdots 75}{17\cdots 64}a^{3}+\frac{28\cdots 87}{89\cdots 32}a^{2}+\frac{39\cdots 39}{17\cdots 64}a-\frac{46\cdots 89}{17\cdots 64}$, $\frac{48\cdots 69}{17\cdots 64}a^{19}+\frac{10\cdots 71}{35\cdots 28}a^{18}+\frac{35\cdots 27}{35\cdots 28}a^{17}+\frac{29\cdots 71}{38\cdots 12}a^{16}-\frac{63\cdots 57}{17\cdots 64}a^{15}-\frac{12\cdots 29}{35\cdots 28}a^{14}-\frac{49\cdots 17}{35\cdots 28}a^{13}-\frac{11\cdots 05}{17\cdots 64}a^{12}+\frac{12\cdots 69}{17\cdots 64}a^{11}-\frac{35\cdots 27}{35\cdots 28}a^{10}-\frac{93\cdots 81}{76\cdots 24}a^{9}-\frac{12\cdots 33}{17\cdots 64}a^{8}-\frac{69\cdots 45}{17\cdots 64}a^{7}-\frac{25\cdots 29}{35\cdots 28}a^{6}+\frac{13\cdots 67}{35\cdots 28}a^{5}+\frac{96\cdots 47}{17\cdots 64}a^{4}+\frac{13\cdots 03}{89\cdots 32}a^{3}-\frac{22\cdots 79}{78\cdots 68}a^{2}-\frac{51\cdots 13}{17\cdots 64}a+\frac{23\cdots 01}{89\cdots 32}$, $\frac{26\cdots 11}{22\cdots 33}a^{19}+\frac{38\cdots 27}{17\cdots 64}a^{18}+\frac{15\cdots 15}{35\cdots 28}a^{17}+\frac{25\cdots 97}{95\cdots 78}a^{16}-\frac{70\cdots 33}{35\cdots 28}a^{15}-\frac{11\cdots 81}{78\cdots 68}a^{14}-\frac{91\cdots 31}{17\cdots 64}a^{13}-\frac{40\cdots 37}{17\cdots 64}a^{12}+\frac{19\cdots 21}{78\cdots 68}a^{11}-\frac{65\cdots 31}{17\cdots 64}a^{10}-\frac{39\cdots 23}{76\cdots 24}a^{9}-\frac{24\cdots 61}{89\cdots 32}a^{8}-\frac{50\cdots 33}{35\cdots 28}a^{7}-\frac{42\cdots 31}{17\cdots 64}a^{6}+\frac{33\cdots 67}{17\cdots 64}a^{5}+\frac{15\cdots 33}{78\cdots 68}a^{4}+\frac{93\cdots 41}{17\cdots 64}a^{3}-\frac{94\cdots 25}{89\cdots 32}a^{2}-\frac{46\cdots 47}{44\cdots 66}a+\frac{87\cdots 17}{89\cdots 32}$, $\frac{30\cdots 51}{35\cdots 28}a^{19}-\frac{27\cdots 27}{89\cdots 32}a^{18}-\frac{17\cdots 61}{35\cdots 28}a^{17}-\frac{19\cdots 41}{76\cdots 24}a^{16}+\frac{48\cdots 61}{35\cdots 28}a^{15}+\frac{39\cdots 53}{44\cdots 66}a^{14}-\frac{33\cdots 67}{35\cdots 28}a^{13}-\frac{26\cdots 03}{35\cdots 28}a^{12}+\frac{18\cdots 49}{35\cdots 28}a^{11}+\frac{22\cdots 71}{89\cdots 32}a^{10}+\frac{36\cdots 19}{76\cdots 24}a^{9}+\frac{68\cdots 79}{35\cdots 28}a^{8}-\frac{84\cdots 63}{35\cdots 28}a^{7}-\frac{94\cdots 81}{22\cdots 33}a^{6}-\frac{12\cdots 99}{35\cdots 28}a^{5}+\frac{24\cdots 13}{35\cdots 28}a^{4}+\frac{46\cdots 67}{17\cdots 64}a^{3}-\frac{91\cdots 69}{22\cdots 33}a^{2}-\frac{25\cdots 41}{78\cdots 68}a+\frac{91\cdots 29}{17\cdots 64}$, $\frac{11\cdots 45}{17\cdots 64}a^{19}+\frac{15\cdots 59}{35\cdots 28}a^{18}+\frac{72\cdots 45}{78\cdots 68}a^{17}+\frac{55\cdots 01}{76\cdots 24}a^{16}-\frac{21\cdots 93}{35\cdots 28}a^{15}-\frac{11\cdots 87}{17\cdots 64}a^{14}+\frac{20\cdots 73}{35\cdots 28}a^{13}-\frac{34\cdots 99}{44\cdots 66}a^{12}+\frac{54\cdots 17}{89\cdots 32}a^{11}-\frac{74\cdots 09}{35\cdots 28}a^{10}+\frac{62\cdots 17}{20\cdots 93}a^{9}-\frac{58\cdots 01}{35\cdots 28}a^{8}-\frac{14\cdots 25}{35\cdots 28}a^{7}+\frac{30\cdots 91}{17\cdots 64}a^{6}-\frac{12\cdots 19}{35\cdots 28}a^{5}+\frac{16\cdots 59}{89\cdots 32}a^{4}-\frac{76\cdots 83}{17\cdots 64}a^{3}-\frac{24\cdots 99}{44\cdots 66}a^{2}+\frac{14\cdots 31}{17\cdots 64}a-\frac{38\cdots 99}{89\cdots 32}$, $\frac{23\cdots 77}{35\cdots 28}a^{19}-\frac{76\cdots 35}{35\cdots 28}a^{18}-\frac{78\cdots 75}{35\cdots 28}a^{17}-\frac{42\cdots 45}{38\cdots 12}a^{16}+\frac{78\cdots 89}{22\cdots 33}a^{15}+\frac{18\cdots 60}{22\cdots 33}a^{14}+\frac{28\cdots 65}{17\cdots 64}a^{13}+\frac{26\cdots 15}{35\cdots 28}a^{12}-\frac{12\cdots 67}{35\cdots 28}a^{11}+\frac{94\cdots 69}{35\cdots 28}a^{10}+\frac{19\cdots 67}{76\cdots 24}a^{9}+\frac{46\cdots 53}{44\cdots 66}a^{8}+\frac{47\cdots 27}{89\cdots 32}a^{7}-\frac{10\cdots 15}{89\cdots 32}a^{6}-\frac{37\cdots 89}{17\cdots 64}a^{5}-\frac{36\cdots 13}{35\cdots 28}a^{4}-\frac{53\cdots 99}{44\cdots 66}a^{3}+\frac{43\cdots 79}{39\cdots 84}a^{2}+\frac{33\cdots 75}{44\cdots 66}a-\frac{30\cdots 47}{17\cdots 64}$, $\frac{63\cdots 47}{44\cdots 66}a^{19}+\frac{50\cdots 25}{44\cdots 66}a^{18}+\frac{11\cdots 07}{35\cdots 28}a^{17}-\frac{15\cdots 31}{76\cdots 24}a^{16}-\frac{59\cdots 63}{17\cdots 64}a^{15}-\frac{41\cdots 91}{35\cdots 28}a^{14}+\frac{31\cdots 07}{15\cdots 36}a^{13}+\frac{48\cdots 67}{17\cdots 64}a^{12}-\frac{58\cdots 13}{89\cdots 32}a^{11}+\frac{66\cdots 77}{17\cdots 64}a^{10}-\frac{43\cdots 81}{76\cdots 24}a^{9}+\frac{85\cdots 47}{35\cdots 28}a^{8}+\frac{19\cdots 01}{17\cdots 64}a^{7}+\frac{14\cdots 69}{35\cdots 28}a^{6}+\frac{12\cdots 33}{35\cdots 28}a^{5}-\frac{11\cdots 83}{17\cdots 64}a^{4}-\frac{13\cdots 97}{89\cdots 32}a^{3}+\frac{35\cdots 83}{17\cdots 64}a^{2}+\frac{32\cdots 61}{17\cdots 64}a-\frac{69\cdots 57}{44\cdots 66}$
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| Regulator: | \( 42907053192269430000 \) (assuming GRH) |
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| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 42907053192269430000 \cdot 2}{2\cdot\sqrt{1026012039426577859100909389414445029369055466909335552}}\cr\approx \mathstrut & 1.64630914174129 \end{aligned}\] (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
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3 + \sqrt{6}})\), 5.1.145437345792.1, 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: | 20.4.513006019713288929550454694707222514684527733454667776.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.10.5 | $x^{4} + 4 x^{3} + 4 x^{2} + 10$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.16.54.4 | $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ | $16$ | $1$ | $54$ | $C_4:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(3\)
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(53\)
| 53.5.4.1 | $x^{5} + 53$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 53.5.4.1 | $x^{5} + 53$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
| 53.10.8.1 | $x^{10} + 245 x^{9} + 24020 x^{8} + 1178450 x^{7} + 28968105 x^{6} + 287187089 x^{5} + 57936210 x^{4} + 4713800 x^{3} + 192160 x^{2} + 3920 x + 85$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |