Normalized defining polynomial
\( x^{20} - 8 x^{19} + 170 x^{18} - 264 x^{17} + 9206 x^{16} + 39680 x^{15} + 448672 x^{14} + \cdots + 284662974336 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(4, 8)$ |
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| Discriminant: |
\(513006019713288929550454694707222514684527733454667776\)
\(\medspace = 2^{63}\cdot 3^{15}\cdot 53^{16}\)
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| |
| Root discriminant: | \(484.74\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(53\)
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| Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{192}a^{12}+\frac{1}{24}a^{11}-\frac{5}{96}a^{10}+\frac{1}{24}a^{9}-\frac{7}{96}a^{8}+\frac{1}{12}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{384}a^{13}-\frac{1}{192}a^{11}-\frac{1}{48}a^{10}-\frac{5}{64}a^{9}+\frac{1}{12}a^{8}-\frac{1}{8}a^{7}-\frac{1}{6}a^{5}-\frac{1}{12}a^{4}-\frac{1}{8}a^{3}+\frac{3}{8}a$, $\frac{1}{384}a^{14}+\frac{1}{48}a^{11}-\frac{1}{192}a^{10}-\frac{1}{8}a^{9}+\frac{5}{96}a^{8}+\frac{1}{12}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{5}{24}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}$, $\frac{1}{384}a^{15}-\frac{3}{64}a^{11}-\frac{1}{24}a^{10}-\frac{11}{96}a^{9}-\frac{1}{8}a^{8}-\frac{1}{12}a^{7}-\frac{1}{6}a^{6}-\frac{1}{8}a^{5}+\frac{1}{6}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a$, $\frac{1}{281856}a^{16}+\frac{15}{23488}a^{15}+\frac{35}{46976}a^{14}-\frac{23}{23488}a^{13}-\frac{203}{140928}a^{12}-\frac{343}{5872}a^{11}-\frac{1949}{35232}a^{10}-\frac{2257}{35232}a^{9}+\frac{319}{2936}a^{8}+\frac{1043}{8808}a^{7}+\frac{2953}{17616}a^{6}+\frac{315}{1468}a^{5}+\frac{725}{17616}a^{4}+\frac{179}{734}a^{3}-\frac{37}{1468}a^{2}-\frac{119}{1468}a-\frac{130}{367}$, $\frac{1}{281856}a^{17}+\frac{53}{140928}a^{15}+\frac{23}{70464}a^{14}+\frac{1}{2936}a^{13}-\frac{53}{23488}a^{12}-\frac{3257}{70464}a^{11}-\frac{99}{2936}a^{10}-\frac{3005}{70464}a^{9}+\frac{1055}{35232}a^{8}-\frac{619}{5872}a^{7}+\frac{457}{2202}a^{6}+\frac{1477}{17616}a^{5}+\frac{63}{734}a^{4}-\frac{137}{2936}a^{3}-\frac{108}{367}a^{2}+\frac{329}{2936}a-\frac{719}{1468}$, $\frac{1}{98\cdots 72}a^{18}+\frac{72\cdots 17}{61\cdots 92}a^{17}+\frac{31\cdots 99}{24\cdots 68}a^{16}+\frac{12\cdots 69}{12\cdots 84}a^{15}+\frac{58\cdots 79}{49\cdots 36}a^{14}-\frac{75\cdots 19}{24\cdots 68}a^{13}-\frac{55\cdots 55}{24\cdots 68}a^{12}+\frac{74\cdots 91}{12\cdots 84}a^{11}-\frac{11\cdots 09}{61\cdots 92}a^{10}-\frac{12\cdots 79}{12\cdots 84}a^{9}+\frac{88\cdots 27}{30\cdots 96}a^{8}-\frac{50\cdots 15}{38\cdots 62}a^{7}+\frac{79\cdots 99}{61\cdots 92}a^{6}-\frac{92\cdots 79}{77\cdots 24}a^{5}-\frac{60\cdots 17}{30\cdots 96}a^{4}+\frac{11\cdots 63}{51\cdots 16}a^{3}-\frac{80\cdots 45}{25\cdots 08}a^{2}+\frac{15\cdots 39}{51\cdots 16}a+\frac{98\cdots 59}{25\cdots 08}$, $\frac{1}{14\cdots 76}a^{19}-\frac{29\cdots 07}{40\cdots 28}a^{18}+\frac{80\cdots 53}{24\cdots 96}a^{17}-\frac{11\cdots 81}{74\cdots 88}a^{16}-\frac{75\cdots 65}{74\cdots 88}a^{15}+\frac{19\cdots 79}{82\cdots 32}a^{14}-\frac{10\cdots 05}{41\cdots 16}a^{13}-\frac{21\cdots 49}{93\cdots 36}a^{12}+\frac{12\cdots 13}{64\cdots 94}a^{11}+\frac{20\cdots 77}{93\cdots 36}a^{10}+\frac{18\cdots 27}{18\cdots 72}a^{9}+\frac{19\cdots 59}{32\cdots 97}a^{8}-\frac{35\cdots 69}{31\cdots 12}a^{7}-\frac{86\cdots 01}{93\cdots 36}a^{6}+\frac{83\cdots 65}{77\cdots 28}a^{5}+\frac{32\cdots 29}{23\cdots 84}a^{4}+\frac{11\cdots 67}{38\cdots 64}a^{3}+\frac{68\cdots 67}{12\cdots 88}a^{2}+\frac{10\cdots 89}{25\cdots 76}a-\frac{10\cdots 81}{38\cdots 64}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{39\cdots 95}{33\cdots 44}a^{19}-\frac{19\cdots 95}{33\cdots 44}a^{18}+\frac{17\cdots 49}{11\cdots 48}a^{17}+\frac{66\cdots 03}{16\cdots 72}a^{16}+\frac{27\cdots 87}{33\cdots 44}a^{15}+\frac{14\cdots 19}{16\cdots 72}a^{14}+\frac{96\cdots 39}{16\cdots 72}a^{13}+\frac{10\cdots 49}{21\cdots 84}a^{12}+\frac{59\cdots 45}{16\cdots 72}a^{11}+\frac{10\cdots 27}{42\cdots 68}a^{10}+\frac{39\cdots 19}{28\cdots 12}a^{9}+\frac{16\cdots 79}{21\cdots 84}a^{8}+\frac{20\cdots 25}{52\cdots 46}a^{7}+\frac{10\cdots 95}{70\cdots 28}a^{6}+\frac{10\cdots 73}{17\cdots 82}a^{5}+\frac{46\cdots 17}{26\cdots 73}a^{4}+\frac{32\cdots 31}{70\cdots 28}a^{3}+\frac{15\cdots 25}{17\cdots 82}a^{2}+\frac{56\cdots 99}{35\cdots 64}a+\frac{28\cdots 79}{88\cdots 91}$, $\frac{39\cdots 95}{33\cdots 44}a^{19}-\frac{19\cdots 95}{33\cdots 44}a^{18}+\frac{17\cdots 49}{11\cdots 48}a^{17}+\frac{66\cdots 03}{16\cdots 72}a^{16}+\frac{27\cdots 87}{33\cdots 44}a^{15}+\frac{14\cdots 19}{16\cdots 72}a^{14}+\frac{96\cdots 39}{16\cdots 72}a^{13}+\frac{10\cdots 49}{21\cdots 84}a^{12}+\frac{59\cdots 45}{16\cdots 72}a^{11}+\frac{10\cdots 27}{42\cdots 68}a^{10}+\frac{39\cdots 19}{28\cdots 12}a^{9}+\frac{16\cdots 79}{21\cdots 84}a^{8}+\frac{20\cdots 25}{52\cdots 46}a^{7}+\frac{10\cdots 95}{70\cdots 28}a^{6}+\frac{10\cdots 73}{17\cdots 82}a^{5}+\frac{46\cdots 17}{26\cdots 73}a^{4}+\frac{32\cdots 31}{70\cdots 28}a^{3}+\frac{15\cdots 25}{17\cdots 82}a^{2}+\frac{56\cdots 99}{35\cdots 64}a+\frac{10\cdots 97}{88\cdots 91}$, $\frac{40\cdots 11}{56\cdots 24}a^{19}+\frac{31\cdots 55}{56\cdots 24}a^{18}-\frac{20\cdots 97}{16\cdots 72}a^{17}+\frac{16\cdots 51}{84\cdots 36}a^{16}-\frac{11\cdots 81}{16\cdots 72}a^{15}-\frac{23\cdots 09}{84\cdots 36}a^{14}-\frac{29\cdots 25}{84\cdots 36}a^{13}-\frac{16\cdots 41}{70\cdots 28}a^{12}-\frac{15\cdots 71}{84\cdots 36}a^{11}-\frac{24\cdots 45}{21\cdots 84}a^{10}-\frac{28\cdots 39}{42\cdots 68}a^{9}-\frac{20\cdots 69}{52\cdots 46}a^{8}-\frac{15\cdots 37}{88\cdots 91}a^{7}-\frac{81\cdots 43}{10\cdots 92}a^{6}-\frac{69\cdots 05}{26\cdots 73}a^{5}-\frac{21\cdots 97}{26\cdots 73}a^{4}-\frac{74\cdots 45}{35\cdots 64}a^{3}-\frac{35\cdots 99}{88\cdots 91}a^{2}-\frac{12\cdots 93}{17\cdots 82}a-\frac{32\cdots 35}{88\cdots 91}$, $\frac{67\cdots 19}{82\cdots 32}a^{19}-\frac{15\cdots 45}{24\cdots 96}a^{18}+\frac{70\cdots 83}{51\cdots 52}a^{17}-\frac{15\cdots 99}{82\cdots 32}a^{16}+\frac{15\cdots 31}{20\cdots 08}a^{15}+\frac{67\cdots 81}{20\cdots 08}a^{14}+\frac{56\cdots 47}{15\cdots 56}a^{13}+\frac{32\cdots 03}{12\cdots 48}a^{12}+\frac{11\cdots 67}{62\cdots 24}a^{11}+\frac{13\cdots 01}{10\cdots 04}a^{10}+\frac{11\cdots 07}{15\cdots 56}a^{9}+\frac{12\cdots 53}{31\cdots 12}a^{8}+\frac{28\cdots 93}{15\cdots 56}a^{7}+\frac{77\cdots 43}{96\cdots 91}a^{6}+\frac{26\cdots 12}{96\cdots 91}a^{5}+\frac{13\cdots 93}{15\cdots 56}a^{4}+\frac{59\cdots 01}{25\cdots 76}a^{3}+\frac{59\cdots 13}{12\cdots 88}a^{2}+\frac{28\cdots 32}{32\cdots 97}a+\frac{81\cdots 17}{12\cdots 88}$, $\frac{70\cdots 75}{24\cdots 96}a^{19}-\frac{43\cdots 15}{12\cdots 88}a^{18}-\frac{71\cdots 77}{12\cdots 48}a^{17}-\frac{99\cdots 95}{15\cdots 56}a^{16}-\frac{99\cdots 89}{41\cdots 16}a^{15}-\frac{10\cdots 73}{31\cdots 12}a^{14}-\frac{29\cdots 89}{10\cdots 04}a^{13}-\frac{11\cdots 77}{51\cdots 52}a^{12}-\frac{11\cdots 61}{77\cdots 28}a^{11}-\frac{14\cdots 87}{15\cdots 56}a^{10}-\frac{75\cdots 13}{12\cdots 88}a^{9}-\frac{22\cdots 75}{77\cdots 28}a^{8}-\frac{64\cdots 85}{51\cdots 52}a^{7}-\frac{14\cdots 16}{32\cdots 97}a^{6}-\frac{28\cdots 13}{19\cdots 82}a^{5}-\frac{74\cdots 91}{19\cdots 82}a^{4}-\frac{25\cdots 92}{32\cdots 97}a^{3}-\frac{92\cdots 29}{64\cdots 94}a^{2}-\frac{63\cdots 35}{64\cdots 94}a-\frac{10\cdots 54}{32\cdots 97}$, $\frac{38\cdots 77}{24\cdots 96}a^{19}+\frac{11\cdots 99}{62\cdots 24}a^{18}-\frac{24\cdots 39}{82\cdots 32}a^{17}+\frac{38\cdots 29}{31\cdots 12}a^{16}-\frac{42\cdots 05}{31\cdots 12}a^{15}-\frac{21\cdots 37}{10\cdots 04}a^{14}-\frac{15\cdots 73}{41\cdots 16}a^{13}-\frac{97\cdots 67}{38\cdots 64}a^{12}-\frac{17\cdots 31}{10\cdots 04}a^{11}-\frac{15\cdots 37}{15\cdots 56}a^{10}-\frac{65\cdots 57}{15\cdots 56}a^{9}-\frac{68\cdots 69}{25\cdots 76}a^{8}-\frac{43\cdots 87}{64\cdots 94}a^{7}-\frac{73\cdots 19}{38\cdots 64}a^{6}-\frac{39\cdots 21}{51\cdots 52}a^{5}+\frac{17\cdots 11}{19\cdots 82}a^{4}+\frac{52\cdots 57}{12\cdots 88}a^{3}+\frac{20\cdots 63}{64\cdots 94}a^{2}+\frac{10\cdots 14}{32\cdots 97}a+\frac{36\cdots 53}{32\cdots 97}$, $\frac{11\cdots 51}{16\cdots 64}a^{19}+\frac{32\cdots 29}{49\cdots 92}a^{18}-\frac{16\cdots 11}{12\cdots 48}a^{17}+\frac{20\cdots 57}{62\cdots 24}a^{16}-\frac{17\cdots 79}{24\cdots 96}a^{15}-\frac{52\cdots 69}{24\cdots 96}a^{14}-\frac{62\cdots 03}{20\cdots 08}a^{13}-\frac{24\cdots 11}{12\cdots 48}a^{12}-\frac{93\cdots 75}{62\cdots 24}a^{11}-\frac{99\cdots 83}{10\cdots 04}a^{10}-\frac{32\cdots 41}{62\cdots 24}a^{9}-\frac{96\cdots 31}{31\cdots 12}a^{8}-\frac{13\cdots 93}{10\cdots 04}a^{7}-\frac{17\cdots 53}{31\cdots 12}a^{6}-\frac{29\cdots 23}{15\cdots 56}a^{5}-\frac{92\cdots 71}{15\cdots 56}a^{4}-\frac{38\cdots 35}{25\cdots 76}a^{3}-\frac{34\cdots 41}{12\cdots 88}a^{2}-\frac{14\cdots 45}{25\cdots 76}a-\frac{60\cdots 69}{32\cdots 97}$, $\frac{14\cdots 87}{49\cdots 92}a^{19}+\frac{13\cdots 21}{49\cdots 92}a^{18}-\frac{15\cdots 61}{31\cdots 12}a^{17}+\frac{61\cdots 97}{82\cdots 32}a^{16}-\frac{49\cdots 61}{24\cdots 96}a^{15}-\frac{34\cdots 79}{24\cdots 96}a^{14}-\frac{63\cdots 43}{77\cdots 28}a^{13}-\frac{13\cdots 85}{15\cdots 56}a^{12}-\frac{33\cdots 65}{62\cdots 24}a^{11}-\frac{10\cdots 91}{31\cdots 12}a^{10}-\frac{39\cdots 97}{20\cdots 08}a^{9}-\frac{11\cdots 93}{10\cdots 04}a^{8}-\frac{47\cdots 43}{10\cdots 04}a^{7}-\frac{52\cdots 47}{31\cdots 12}a^{6}-\frac{96\cdots 63}{15\cdots 56}a^{5}-\frac{10\cdots 95}{64\cdots 94}a^{4}-\frac{92\cdots 61}{25\cdots 76}a^{3}-\frac{96\cdots 25}{12\cdots 88}a^{2}-\frac{13\cdots 31}{25\cdots 76}a-\frac{58\cdots 55}{32\cdots 97}$, $\frac{33\cdots 11}{16\cdots 64}a^{19}+\frac{28\cdots 63}{16\cdots 64}a^{18}-\frac{21\cdots 43}{82\cdots 32}a^{17}+\frac{58\cdots 91}{15\cdots 56}a^{16}-\frac{19\cdots 63}{82\cdots 32}a^{15}+\frac{48\cdots 45}{24\cdots 96}a^{14}-\frac{52\cdots 13}{20\cdots 08}a^{13}+\frac{39\cdots 53}{12\cdots 48}a^{12}+\frac{82\cdots 83}{31\cdots 12}a^{11}+\frac{44\cdots 87}{62\cdots 24}a^{10}+\frac{32\cdots 33}{31\cdots 12}a^{9}+\frac{53\cdots 25}{15\cdots 56}a^{8}+\frac{76\cdots 71}{31\cdots 12}a^{7}+\frac{25\cdots 67}{31\cdots 12}a^{6}+\frac{78\cdots 07}{25\cdots 76}a^{5}+\frac{56\cdots 69}{51\cdots 52}a^{4}+\frac{23\cdots 89}{12\cdots 88}a^{3}+\frac{14\cdots 01}{25\cdots 76}a^{2}+\frac{17\cdots 74}{32\cdots 97}a+\frac{23\cdots 11}{12\cdots 88}$, $\frac{26\cdots 61}{49\cdots 92}a^{19}-\frac{18\cdots 21}{49\cdots 92}a^{18}+\frac{10\cdots 85}{12\cdots 48}a^{17}-\frac{82\cdots 63}{24\cdots 96}a^{16}+\frac{98\cdots 21}{24\cdots 96}a^{15}+\frac{59\cdots 97}{24\cdots 96}a^{14}+\frac{26\cdots 25}{12\cdots 48}a^{13}+\frac{32\cdots 91}{20\cdots 08}a^{12}+\frac{23\cdots 45}{20\cdots 08}a^{11}+\frac{44\cdots 91}{62\cdots 24}a^{10}+\frac{11\cdots 25}{31\cdots 12}a^{9}+\frac{63\cdots 33}{31\cdots 12}a^{8}+\frac{87\cdots 77}{10\cdots 04}a^{7}+\frac{92\cdots 23}{31\cdots 12}a^{6}+\frac{11\cdots 77}{15\cdots 56}a^{5}+\frac{10\cdots 87}{77\cdots 28}a^{4}+\frac{29\cdots 33}{25\cdots 76}a^{3}-\frac{10\cdots 31}{25\cdots 76}a^{2}-\frac{30\cdots 93}{64\cdots 94}a-\frac{10\cdots 31}{64\cdots 94}$, $\frac{12\cdots 23}{16\cdots 64}a^{19}+\frac{95\cdots 83}{49\cdots 92}a^{18}-\frac{20\cdots 67}{24\cdots 96}a^{17}-\frac{56\cdots 33}{24\cdots 96}a^{16}-\frac{10\cdots 71}{24\cdots 96}a^{15}-\frac{24\cdots 81}{82\cdots 32}a^{14}-\frac{28\cdots 87}{12\cdots 48}a^{13}-\frac{43\cdots 47}{31\cdots 12}a^{12}-\frac{25\cdots 32}{32\cdots 97}a^{11}-\frac{24\cdots 31}{62\cdots 24}a^{10}-\frac{80\cdots 35}{62\cdots 24}a^{9}-\frac{41\cdots 87}{10\cdots 04}a^{8}-\frac{53\cdots 83}{10\cdots 04}a^{7}+\frac{23\cdots 43}{31\cdots 12}a^{6}+\frac{93\cdots 11}{77\cdots 28}a^{5}+\frac{10\cdots 57}{19\cdots 82}a^{4}+\frac{17\cdots 93}{12\cdots 88}a^{3}+\frac{97\cdots 91}{25\cdots 76}a^{2}+\frac{81\cdots 25}{25\cdots 76}a+\frac{34\cdots 29}{32\cdots 97}$
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| Regulator: | \( 24649276698462750000 \) (assuming GRH) |
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| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 24649276698462750000 \cdot 2}{2\cdot\sqrt{513006019713288929550454694707222514684527733454667776}}\cr\approx \mathstrut & 1.33752490949745 \end{aligned}\] (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
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3 + \sqrt{3}})\), 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: | 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 | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.9.8 | $x^{4} + 10 x^{2} + 6$ | $4$ | $1$ | $9$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.16.54.4 | $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ | $16$ | $1$ | $54$ | $C_4:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(3\)
| 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(53\)
| 53.10.8.1 | $x^{10} + 245 x^{9} + 24020 x^{8} + 1178450 x^{7} + 28968105 x^{6} + 287187089 x^{5} + 57936210 x^{4} + 4713800 x^{3} + 192160 x^{2} + 3920 x + 85$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 53.10.8.1 | $x^{10} + 245 x^{9} + 24020 x^{8} + 1178450 x^{7} + 28968105 x^{6} + 287187089 x^{5} + 57936210 x^{4} + 4713800 x^{3} + 192160 x^{2} + 3920 x + 85$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |