Normalized defining polynomial
\( x^{20} - 60 x^{18} - 160 x^{17} + 870 x^{16} + 4320 x^{15} - 3520 x^{14} - 54960 x^{13} + \cdots - 39496248 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(4, 8)$ |
| |
| Discriminant: |
\(50617612400283069972480000000000000000000000\)
\(\medspace = 2^{55}\cdot 3^{20}\cdot 5^{22}\cdot 13^{2}\)
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| |
| Root discriminant: | \(153.18\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(13\)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{8}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{4}$, $\frac{1}{8}a^{13}-\frac{3}{8}a^{9}-\frac{1}{2}a^{8}+\frac{3}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{14}+\frac{1}{8}a^{10}-\frac{1}{2}a^{9}-\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{24}a^{15}-\frac{1}{24}a^{14}+\frac{1}{24}a^{13}+\frac{1}{24}a^{12}-\frac{1}{24}a^{11}+\frac{1}{24}a^{10}+\frac{1}{24}a^{9}+\frac{5}{24}a^{8}+\frac{1}{24}a^{7}-\frac{3}{8}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{24}a^{16}-\frac{1}{24}a^{13}+\frac{1}{12}a^{10}-\frac{3}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{3}a^{7}-\frac{1}{8}a^{5}-\frac{11}{24}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{72}a^{17}-\frac{1}{18}a^{14}+\frac{1}{24}a^{13}-\frac{1}{24}a^{12}+\frac{1}{36}a^{11}-\frac{1}{4}a^{10}+\frac{11}{24}a^{9}+\frac{25}{72}a^{8}-\frac{1}{2}a^{7}-\frac{5}{12}a^{6}-\frac{7}{36}a^{5}-\frac{1}{8}a^{4}-\frac{5}{12}a^{3}+\frac{7}{18}a^{2}-\frac{1}{3}a$, $\frac{1}{432}a^{18}-\frac{1}{144}a^{17}-\frac{1}{72}a^{16}-\frac{7}{432}a^{15}-\frac{1}{16}a^{14}-\frac{1}{24}a^{13}+\frac{17}{432}a^{12}+\frac{5}{144}a^{11}-\frac{11}{48}a^{10}+\frac{37}{108}a^{9}-\frac{5}{48}a^{8}+\frac{65}{144}a^{7}+\frac{47}{216}a^{6}+\frac{11}{144}a^{5}+\frac{17}{48}a^{4}-\frac{5}{108}a^{3}+\frac{4}{9}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{17\cdots 16}a^{19}+\frac{99\cdots 07}{48\cdots 56}a^{18}-\frac{37\cdots 55}{58\cdots 72}a^{17}+\frac{27\cdots 33}{17\cdots 16}a^{16}+\frac{30\cdots 51}{14\cdots 68}a^{15}-\frac{10\cdots 89}{19\cdots 24}a^{14}-\frac{50\cdots 95}{17\cdots 16}a^{13}-\frac{54\cdots 75}{29\cdots 36}a^{12}-\frac{66\cdots 31}{10\cdots 68}a^{11}+\frac{14\cdots 25}{17\cdots 16}a^{10}+\frac{41\cdots 31}{58\cdots 72}a^{9}+\frac{14\cdots 29}{14\cdots 68}a^{8}+\frac{21\cdots 63}{17\cdots 16}a^{7}+\frac{86\cdots 97}{21\cdots 36}a^{6}+\frac{60\cdots 79}{48\cdots 56}a^{5}-\frac{69\cdots 11}{17\cdots 16}a^{4}-\frac{15\cdots 31}{73\cdots 34}a^{3}-\frac{28\cdots 35}{12\cdots 39}a^{2}+\frac{13\cdots 08}{40\cdots 13}a-\frac{40\cdots 59}{27\cdots 42}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{32\cdots 12}{10\cdots 51}a^{19}-\frac{55\cdots 37}{14\cdots 68}a^{18}-\frac{69\cdots 84}{36\cdots 17}a^{17}-\frac{74\cdots 49}{43\cdots 04}a^{16}+\frac{13\cdots 03}{36\cdots 17}a^{15}+\frac{62\cdots 31}{81\cdots 26}a^{14}-\frac{43\cdots 84}{10\cdots 51}a^{13}-\frac{22\cdots 45}{16\cdots 52}a^{12}+\frac{22\cdots 79}{12\cdots 39}a^{11}+\frac{13\cdots 43}{10\cdots 51}a^{10}+\frac{10\cdots 90}{36\cdots 17}a^{9}-\frac{10\cdots 65}{14\cdots 68}a^{8}-\frac{73\cdots 55}{10\cdots 51}a^{7}+\frac{38\cdots 51}{14\cdots 68}a^{6}+\frac{37\cdots 74}{12\cdots 39}a^{5}-\frac{94\cdots 87}{10\cdots 51}a^{4}-\frac{42\cdots 52}{40\cdots 13}a^{3}+\frac{25\cdots 94}{12\cdots 39}a^{2}+\frac{15\cdots 08}{13\cdots 71}a-\frac{45\cdots 05}{13\cdots 71}$, $\frac{15\cdots 27}{10\cdots 51}a^{19}-\frac{47\cdots 95}{14\cdots 68}a^{18}-\frac{29\cdots 62}{36\cdots 17}a^{17}-\frac{11\cdots 37}{21\cdots 02}a^{16}+\frac{16\cdots 70}{12\cdots 39}a^{15}+\frac{51\cdots 27}{16\cdots 52}a^{14}-\frac{13\cdots 30}{10\cdots 51}a^{13}-\frac{76\cdots 15}{14\cdots 68}a^{12}+\frac{23\cdots 46}{13\cdots 71}a^{11}+\frac{17\cdots 49}{43\cdots 04}a^{10}+\frac{63\cdots 80}{13\cdots 71}a^{9}-\frac{21\cdots 97}{14\cdots 68}a^{8}-\frac{37\cdots 63}{10\cdots 51}a^{7}+\frac{60\cdots 19}{36\cdots 17}a^{6}+\frac{10\cdots 00}{13\cdots 71}a^{5}-\frac{22\cdots 21}{43\cdots 04}a^{4}-\frac{54\cdots 92}{36\cdots 17}a^{3}+\frac{12\cdots 66}{12\cdots 39}a^{2}-\frac{17\cdots 52}{40\cdots 13}a-\frac{34\cdots 29}{13\cdots 71}$, $\frac{89\cdots 47}{87\cdots 08}a^{19}-\frac{86\cdots 11}{58\cdots 72}a^{18}-\frac{34\cdots 53}{58\cdots 72}a^{17}-\frac{84\cdots 03}{10\cdots 51}a^{16}+\frac{18\cdots 99}{19\cdots 24}a^{15}+\frac{56\cdots 69}{19\cdots 24}a^{14}-\frac{80\cdots 89}{10\cdots 51}a^{13}-\frac{25\cdots 89}{58\cdots 72}a^{12}-\frac{17\cdots 55}{19\cdots 24}a^{11}+\frac{53\cdots 81}{17\cdots 16}a^{10}+\frac{48\cdots 97}{97\cdots 12}a^{9}-\frac{53\cdots 03}{58\cdots 72}a^{8}-\frac{53\cdots 05}{17\cdots 16}a^{7}+\frac{21\cdots 53}{73\cdots 34}a^{6}+\frac{13\cdots 07}{19\cdots 24}a^{5}-\frac{26\cdots 09}{17\cdots 16}a^{4}-\frac{22\cdots 09}{14\cdots 68}a^{3}+\frac{37\cdots 27}{81\cdots 26}a^{2}+\frac{55\cdots 25}{81\cdots 26}a-\frac{78\cdots 47}{27\cdots 42}$, $\frac{98\cdots 29}{87\cdots 08}a^{19}-\frac{13\cdots 09}{29\cdots 36}a^{18}-\frac{42\cdots 21}{73\cdots 34}a^{17}-\frac{73\cdots 57}{43\cdots 04}a^{16}+\frac{23\cdots 00}{40\cdots 13}a^{15}+\frac{19\cdots 79}{48\cdots 56}a^{14}+\frac{19\cdots 81}{87\cdots 08}a^{13}-\frac{55\cdots 69}{14\cdots 68}a^{12}-\frac{14\cdots 86}{12\cdots 39}a^{11}+\frac{71\cdots 15}{21\cdots 02}a^{10}+\frac{89\cdots 79}{97\cdots 12}a^{9}+\frac{13\cdots 31}{73\cdots 34}a^{8}-\frac{10\cdots 97}{87\cdots 08}a^{7}-\frac{31\cdots 47}{29\cdots 36}a^{6}-\frac{15\cdots 19}{97\cdots 12}a^{5}+\frac{30\cdots 95}{43\cdots 04}a^{4}+\frac{30\cdots 35}{73\cdots 34}a^{3}+\frac{62\cdots 16}{12\cdots 39}a^{2}-\frac{40\cdots 45}{40\cdots 13}a-\frac{11\cdots 39}{13\cdots 71}$, $\frac{10\cdots 61}{97\cdots 12}a^{19}-\frac{10\cdots 12}{12\cdots 39}a^{18}-\frac{20\cdots 77}{32\cdots 04}a^{17}-\frac{11\cdots 15}{97\cdots 12}a^{16}+\frac{46\cdots 13}{48\cdots 56}a^{15}+\frac{15\cdots 74}{40\cdots 13}a^{14}-\frac{52\cdots 99}{97\cdots 12}a^{13}-\frac{12\cdots 41}{24\cdots 78}a^{12}-\frac{57\cdots 18}{13\cdots 71}a^{11}+\frac{14\cdots 63}{48\cdots 56}a^{10}+\frac{72\cdots 81}{97\cdots 12}a^{9}-\frac{72\cdots 61}{16\cdots 52}a^{8}-\frac{35\cdots 11}{97\cdots 12}a^{7}-\frac{64\cdots 83}{24\cdots 78}a^{6}+\frac{26\cdots 07}{54\cdots 84}a^{5}+\frac{30\cdots 21}{97\cdots 12}a^{4}-\frac{10\cdots 12}{12\cdots 39}a^{3}-\frac{10\cdots 71}{81\cdots 26}a^{2}-\frac{47\cdots 64}{40\cdots 13}a-\frac{49\cdots 12}{13\cdots 71}$, $\frac{16\cdots 69}{87\cdots 08}a^{19}-\frac{13\cdots 83}{32\cdots 04}a^{18}-\frac{30\cdots 75}{29\cdots 36}a^{17}-\frac{70\cdots 07}{87\cdots 08}a^{16}+\frac{13\cdots 81}{73\cdots 34}a^{15}+\frac{35\cdots 25}{81\cdots 26}a^{14}-\frac{17\cdots 59}{10\cdots 51}a^{13}-\frac{20\cdots 63}{29\cdots 36}a^{12}+\frac{49\cdots 65}{24\cdots 78}a^{11}+\frac{23\cdots 31}{43\cdots 04}a^{10}+\frac{23\cdots 99}{36\cdots 17}a^{9}-\frac{57\cdots 35}{29\cdots 36}a^{8}-\frac{40\cdots 99}{87\cdots 08}a^{7}+\frac{22\cdots 69}{97\cdots 12}a^{6}+\frac{10\cdots 69}{97\cdots 12}a^{5}-\frac{82\cdots 30}{10\cdots 51}a^{4}-\frac{78\cdots 16}{36\cdots 17}a^{3}+\frac{12\cdots 53}{81\cdots 26}a^{2}-\frac{58\cdots 65}{13\cdots 71}a-\frac{47\cdots 61}{13\cdots 71}$, $\frac{44\cdots 37}{14\cdots 68}a^{19}-\frac{38\cdots 75}{65\cdots 08}a^{18}-\frac{32\cdots 45}{19\cdots 24}a^{17}-\frac{51\cdots 77}{29\cdots 36}a^{16}+\frac{53\cdots 17}{19\cdots 24}a^{15}+\frac{14\cdots 43}{19\cdots 24}a^{14}-\frac{78\cdots 90}{36\cdots 17}a^{13}-\frac{22\cdots 59}{19\cdots 24}a^{12}-\frac{31\cdots 41}{19\cdots 24}a^{11}+\frac{45\cdots 33}{58\cdots 72}a^{10}+\frac{12\cdots 29}{97\cdots 12}a^{9}-\frac{13\cdots 79}{65\cdots 08}a^{8}-\frac{40\cdots 19}{58\cdots 72}a^{7}-\frac{10\cdots 35}{10\cdots 68}a^{6}+\frac{19\cdots 73}{19\cdots 24}a^{5}-\frac{43\cdots 49}{58\cdots 72}a^{4}-\frac{10\cdots 97}{48\cdots 56}a^{3}+\frac{40\cdots 31}{24\cdots 78}a^{2}+\frac{68\cdots 81}{81\cdots 26}a-\frac{78\cdots 49}{27\cdots 42}$, $\frac{76\cdots 53}{17\cdots 16}a^{19}-\frac{28\cdots 79}{19\cdots 24}a^{18}-\frac{17\cdots 59}{73\cdots 34}a^{17}+\frac{20\cdots 13}{17\cdots 16}a^{16}+\frac{27\cdots 77}{58\cdots 72}a^{15}+\frac{65\cdots 93}{12\cdots 39}a^{14}-\frac{95\cdots 49}{17\cdots 16}a^{13}-\frac{80\cdots 53}{58\cdots 72}a^{12}+\frac{57\cdots 47}{19\cdots 24}a^{11}+\frac{12\cdots 17}{87\cdots 08}a^{10}+\frac{14\cdots 19}{58\cdots 72}a^{9}-\frac{49\cdots 57}{58\cdots 72}a^{8}-\frac{96\cdots 09}{10\cdots 51}a^{7}+\frac{50\cdots 39}{19\cdots 24}a^{6}+\frac{28\cdots 07}{65\cdots 08}a^{5}-\frac{44\cdots 17}{87\cdots 08}a^{4}-\frac{15\cdots 55}{14\cdots 68}a^{3}+\frac{18\cdots 41}{24\cdots 78}a^{2}+\frac{28\cdots 09}{27\cdots 42}a-\frac{36\cdots 68}{13\cdots 71}$, $\frac{16\cdots 99}{21\cdots 36}a^{19}-\frac{22\cdots 11}{58\cdots 72}a^{18}-\frac{33\cdots 57}{97\cdots 12}a^{17}+\frac{16\cdots 31}{19\cdots 24}a^{16}+\frac{34\cdots 59}{58\cdots 72}a^{15}-\frac{15\cdots 06}{40\cdots 13}a^{14}-\frac{44\cdots 63}{65\cdots 08}a^{13}-\frac{36\cdots 21}{58\cdots 72}a^{12}+\frac{91\cdots 65}{19\cdots 24}a^{11}+\frac{29\cdots 55}{32\cdots 04}a^{10}-\frac{89\cdots 49}{58\cdots 72}a^{9}-\frac{12\cdots 07}{21\cdots 36}a^{8}+\frac{10\cdots 13}{48\cdots 56}a^{7}+\frac{98\cdots 79}{58\cdots 72}a^{6}-\frac{22\cdots 37}{19\cdots 24}a^{5}-\frac{15\cdots 69}{54\cdots 84}a^{4}+\frac{48\cdots 87}{14\cdots 68}a^{3}-\frac{15\cdots 25}{24\cdots 78}a^{2}-\frac{12\cdots 33}{27\cdots 42}a+\frac{24\cdots 63}{13\cdots 71}$, $\frac{65\cdots 87}{17\cdots 16}a^{19}+\frac{52\cdots 57}{58\cdots 72}a^{18}-\frac{31\cdots 95}{14\cdots 68}a^{17}-\frac{19\cdots 95}{17\cdots 16}a^{16}+\frac{21\cdots 95}{19\cdots 24}a^{15}+\frac{34\cdots 37}{16\cdots 52}a^{14}+\frac{58\cdots 81}{17\cdots 16}a^{13}-\frac{99\cdots 09}{58\cdots 72}a^{12}-\frac{15\cdots 65}{21\cdots 36}a^{11}-\frac{14\cdots 91}{87\cdots 08}a^{10}+\frac{29\cdots 57}{65\cdots 08}a^{9}+\frac{55\cdots 07}{58\cdots 72}a^{8}-\frac{30\cdots 05}{87\cdots 08}a^{7}-\frac{23\cdots 93}{58\cdots 72}a^{6}-\frac{36\cdots 51}{65\cdots 08}a^{5}-\frac{29\cdots 51}{87\cdots 08}a^{4}-\frac{72\cdots 69}{14\cdots 68}a^{3}-\frac{10\cdots 59}{12\cdots 39}a^{2}-\frac{53\cdots 55}{27\cdots 42}a+\frac{36\cdots 06}{13\cdots 71}$, $\frac{26\cdots 21}{21\cdots 02}a^{19}-\frac{44\cdots 05}{14\cdots 68}a^{18}-\frac{19\cdots 01}{29\cdots 36}a^{17}-\frac{27\cdots 07}{87\cdots 08}a^{16}+\frac{33\cdots 43}{29\cdots 36}a^{15}+\frac{13\cdots 69}{54\cdots 84}a^{14}-\frac{90\cdots 89}{87\cdots 08}a^{13}-\frac{99\cdots 37}{24\cdots 78}a^{12}+\frac{19\cdots 53}{97\cdots 12}a^{11}+\frac{35\cdots 13}{10\cdots 51}a^{10}+\frac{98\cdots 99}{29\cdots 36}a^{9}-\frac{17\cdots 89}{14\cdots 68}a^{8}-\frac{22\cdots 75}{87\cdots 08}a^{7}+\frac{58\cdots 15}{36\cdots 17}a^{6}+\frac{14\cdots 23}{24\cdots 78}a^{5}-\frac{43\cdots 39}{87\cdots 08}a^{4}-\frac{27\cdots 73}{24\cdots 78}a^{3}+\frac{20\cdots 35}{24\cdots 78}a^{2}-\frac{14\cdots 26}{40\cdots 13}a-\frac{25\cdots 26}{13\cdots 71}$
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| Regulator: | \( 222617324287000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 222617324287000 \cdot 4}{2\cdot\sqrt{50617612400283069972480000000000000000000000}}\cr\approx \mathstrut & 2.43218752022106 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.(F_5\times S_4)$ (as 20T811):
| A solvable group of order 122880 |
| The 80 conjugacy class representatives for $C_2^8.(F_5\times S_4)$ |
| Character table for $C_2^8.(F_5\times S_4)$ |
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 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 | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.9 | $x^{4} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ |
| 2.2.8.44d13.54 | $x^{16} + 12 x^{15} + 64 x^{14} + 224 x^{13} + 580 x^{12} + 1188 x^{11} + 2000 x^{10} + 2836 x^{9} + 3437 x^{8} + 3580 x^{7} + 3204 x^{6} + 2444 x^{5} + 1572 x^{4} + 840 x^{3} + 368 x^{2} + 132 x + 31$ | $8$ | $2$ | $44$ | 16T236 | $$[2, 2, 2, 3, \frac{7}{2}]^{4}$$ | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.3.4a1.1 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $$[2]^{2}$$ | |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 3.4.3.16a1.1 | $x^{12} + 6 x^{11} + 12 x^{10} + 8 x^{9} + 9 x^{8} + 36 x^{7} + 36 x^{6} + 24 x^{4} + 48 x^{3} + 23$ | $3$ | $4$ | $16$ | $C_3 : C_4$ | $$[2]^{4}$$ | |
|
\(5\)
| 5.1.5.5a1.4 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.1.15.17a1.4 | $x^{15} + 20 x^{3} + 5$ | $15$ | $1$ | $17$ | $F_5 \times S_3$ | $$[\frac{5}{4}]_{12}^{2}$$ | |
|
\(13\)
| 13.2.2.2a1.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 13.8.1.0a1.1 | $x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
| 13.8.1.0a1.1 | $x^{8} + 8 x^{4} + 12 x^{3} + 2 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ |