Normalized defining polynomial
\( x^{20} - 8 x^{19} + 170 x^{18} - 264 x^{17} + 5814 x^{16} + 83776 x^{15} - 182240 x^{14} + \cdots + 38\!\cdots\!76 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(4, 8)$ |
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| Discriminant: |
\(513006019713288929550454694707222514684527733454667776\)
\(\medspace = 2^{63}\cdot 3^{15}\cdot 53^{16}\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{6}$, $\frac{1}{8}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{64}a^{12}-\frac{1}{32}a^{10}-\frac{1}{8}a^{9}+\frac{1}{32}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{128}a^{13}-\frac{1}{64}a^{11}-\frac{1}{16}a^{10}+\frac{1}{64}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{8}a$, $\frac{1}{128}a^{14}-\frac{1}{16}a^{11}-\frac{1}{64}a^{10}-\frac{1}{8}a^{9}-\frac{3}{32}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}+\frac{1}{4}$, $\frac{1}{128}a^{15}-\frac{1}{64}a^{11}-\frac{3}{32}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{896768}a^{16}+\frac{67}{224192}a^{15}-\frac{433}{112096}a^{14}+\frac{763}{224192}a^{13}+\frac{1111}{448384}a^{12}+\frac{1303}{56048}a^{11}-\frac{8703}{224192}a^{10}+\frac{4395}{112096}a^{9}+\frac{3087}{28024}a^{8}+\frac{1823}{28024}a^{7}+\frac{10313}{56048}a^{6}+\frac{758}{3503}a^{5}+\frac{3267}{56048}a^{4}+\frac{2115}{7006}a^{3}-\frac{5731}{28024}a^{2}-\frac{1193}{14012}a-\frac{1438}{3503}$, $\frac{1}{896768}a^{17}+\frac{889}{448384}a^{15}-\frac{197}{448384}a^{14}-\frac{1509}{448384}a^{13}-\frac{39}{224192}a^{12}+\frac{3299}{56048}a^{11}-\frac{2313}{224192}a^{10}-\frac{13025}{112096}a^{9}-\frac{286}{3503}a^{8}+\frac{13}{56048}a^{7}+\frac{538}{3503}a^{6}+\frac{10753}{56048}a^{5}+\frac{1551}{28024}a^{4}-\frac{1697}{3503}a^{3}+\frac{2711}{28024}a^{2}-\frac{4801}{14012}a+\frac{54}{3503}$, $\frac{1}{25\cdots 32}a^{18}+\frac{58\cdots 17}{12\cdots 16}a^{17}-\frac{19\cdots 73}{12\cdots 16}a^{16}+\frac{17\cdots 39}{32\cdots 04}a^{15}-\frac{94\cdots 85}{12\cdots 16}a^{14}-\frac{74\cdots 39}{20\cdots 44}a^{13}-\frac{40\cdots 81}{80\cdots 76}a^{12}-\frac{18\cdots 33}{40\cdots 88}a^{11}+\frac{32\cdots 01}{32\cdots 04}a^{10}-\frac{19\cdots 27}{32\cdots 04}a^{9}-\frac{10\cdots 61}{20\cdots 44}a^{8}-\frac{70\cdots 41}{80\cdots 76}a^{7}-\frac{11\cdots 51}{16\cdots 52}a^{6}-\frac{14\cdots 69}{80\cdots 76}a^{5}-\frac{84\cdots 53}{40\cdots 88}a^{4}-\frac{45\cdots 01}{20\cdots 44}a^{3}+\frac{30\cdots 25}{40\cdots 88}a^{2}-\frac{46\cdots 47}{40\cdots 88}a-\frac{73\cdots 33}{20\cdots 44}$, $\frac{1}{58\cdots 04}a^{19}+\frac{15\cdots 51}{29\cdots 52}a^{18}-\frac{74\cdots 63}{14\cdots 76}a^{17}+\frac{11\cdots 69}{29\cdots 52}a^{16}-\frac{29\cdots 73}{29\cdots 52}a^{15}-\frac{28\cdots 99}{14\cdots 76}a^{14}+\frac{27\cdots 91}{14\cdots 76}a^{13}-\frac{99\cdots 59}{14\cdots 76}a^{12}+\frac{15\cdots 63}{36\cdots 44}a^{11}-\frac{19\cdots 39}{73\cdots 88}a^{10}-\frac{14\cdots 55}{36\cdots 44}a^{9}-\frac{29\cdots 29}{18\cdots 72}a^{8}-\frac{12\cdots 41}{36\cdots 44}a^{7}+\frac{63\cdots 89}{91\cdots 36}a^{6}-\frac{81\cdots 25}{18\cdots 72}a^{5}-\frac{38\cdots 19}{18\cdots 72}a^{4}+\frac{30\cdots 29}{45\cdots 68}a^{3}-\frac{14\cdots 07}{91\cdots 36}a^{2}-\frac{13\cdots 92}{11\cdots 17}a+\frac{54\cdots 99}{11\cdots 17}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
| |
| Narrow class group: | $C_{6}$, which has order $6$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{13\cdots 75}{10\cdots 48}a^{19}+\frac{26\cdots 05}{20\cdots 96}a^{18}-\frac{69\cdots 53}{41\cdots 92}a^{17}+\frac{55\cdots 95}{20\cdots 96}a^{16}+\frac{19\cdots 31}{41\cdots 92}a^{15}-\frac{18\cdots 15}{20\cdots 96}a^{14}+\frac{11\cdots 55}{20\cdots 96}a^{13}-\frac{46\cdots 99}{10\cdots 48}a^{12}+\frac{78\cdots 87}{20\cdots 96}a^{11}+\frac{48\cdots 27}{10\cdots 48}a^{10}-\frac{10\cdots 39}{10\cdots 48}a^{9}+\frac{29\cdots 83}{52\cdots 24}a^{8}-\frac{54\cdots 25}{26\cdots 12}a^{7}+\frac{77\cdots 77}{32\cdots 39}a^{6}+\frac{16\cdots 97}{32\cdots 39}a^{5}-\frac{66\cdots 31}{32\cdots 39}a^{4}+\frac{32\cdots 15}{26\cdots 12}a^{3}+\frac{22\cdots 43}{13\cdots 56}a^{2}-\frac{47\cdots 27}{13\cdots 56}a+\frac{25\cdots 69}{65\cdots 78}$, $\frac{13\cdots 75}{10\cdots 48}a^{19}+\frac{26\cdots 05}{20\cdots 96}a^{18}-\frac{69\cdots 53}{41\cdots 92}a^{17}+\frac{55\cdots 95}{20\cdots 96}a^{16}+\frac{19\cdots 31}{41\cdots 92}a^{15}-\frac{18\cdots 15}{20\cdots 96}a^{14}+\frac{11\cdots 55}{20\cdots 96}a^{13}-\frac{46\cdots 99}{10\cdots 48}a^{12}+\frac{78\cdots 87}{20\cdots 96}a^{11}+\frac{48\cdots 27}{10\cdots 48}a^{10}-\frac{10\cdots 39}{10\cdots 48}a^{9}+\frac{29\cdots 83}{52\cdots 24}a^{8}-\frac{54\cdots 25}{26\cdots 12}a^{7}+\frac{77\cdots 77}{32\cdots 39}a^{6}+\frac{16\cdots 97}{32\cdots 39}a^{5}-\frac{66\cdots 31}{32\cdots 39}a^{4}+\frac{32\cdots 15}{26\cdots 12}a^{3}+\frac{22\cdots 43}{13\cdots 56}a^{2}-\frac{47\cdots 27}{13\cdots 56}a+\frac{38\cdots 25}{65\cdots 78}$, $\frac{73\cdots 13}{52\cdots 24}a^{19}-\frac{23\cdots 09}{10\cdots 48}a^{18}-\frac{18\cdots 51}{20\cdots 96}a^{17}-\frac{40\cdots 15}{10\cdots 48}a^{16}-\frac{37\cdots 67}{20\cdots 96}a^{15}-\frac{23\cdots 89}{10\cdots 48}a^{14}-\frac{27\cdots 75}{10\cdots 48}a^{13}-\frac{53\cdots 47}{52\cdots 24}a^{12}-\frac{17\cdots 51}{10\cdots 48}a^{11}-\frac{42\cdots 99}{52\cdots 24}a^{10}-\frac{28\cdots 85}{52\cdots 24}a^{9}-\frac{12\cdots 29}{26\cdots 12}a^{8}-\frac{21\cdots 11}{13\cdots 56}a^{7}-\frac{46\cdots 87}{32\cdots 39}a^{6}-\frac{20\cdots 42}{32\cdots 39}a^{5}-\frac{76\cdots 79}{32\cdots 39}a^{4}-\frac{19\cdots 79}{13\cdots 56}a^{3}-\frac{23\cdots 39}{65\cdots 78}a^{2}-\frac{83\cdots 09}{65\cdots 78}a-\frac{14\cdots 02}{32\cdots 39}$, $\frac{39\cdots 81}{14\cdots 76}a^{19}-\frac{34\cdots 63}{14\cdots 76}a^{18}+\frac{50\cdots 27}{29\cdots 52}a^{17}-\frac{41\cdots 69}{14\cdots 76}a^{16}-\frac{67\cdots 63}{36\cdots 44}a^{15}-\frac{22\cdots 25}{73\cdots 88}a^{14}-\frac{26\cdots 11}{14\cdots 76}a^{13}+\frac{27\cdots 41}{73\cdots 88}a^{12}-\frac{72\cdots 35}{73\cdots 88}a^{11}-\frac{85\cdots 97}{36\cdots 44}a^{10}-\frac{56\cdots 41}{45\cdots 68}a^{9}-\frac{52\cdots 03}{18\cdots 72}a^{8}-\frac{56\cdots 59}{18\cdots 72}a^{7}-\frac{36\cdots 95}{45\cdots 68}a^{6}-\frac{44\cdots 43}{18\cdots 72}a^{5}-\frac{20\cdots 55}{29\cdots 56}a^{4}-\frac{81\cdots 31}{91\cdots 36}a^{3}-\frac{45\cdots 09}{45\cdots 68}a^{2}-\frac{78\cdots 65}{11\cdots 17}a-\frac{68\cdots 25}{22\cdots 34}$, $\frac{18\cdots 03}{29\cdots 52}a^{19}-\frac{32\cdots 35}{73\cdots 88}a^{18}+\frac{14\cdots 75}{18\cdots 72}a^{17}-\frac{25\cdots 59}{73\cdots 88}a^{16}+\frac{11\cdots 07}{14\cdots 76}a^{15}+\frac{15\cdots 39}{36\cdots 44}a^{14}-\frac{12\cdots 07}{73\cdots 88}a^{13}+\frac{10\cdots 39}{45\cdots 68}a^{12}+\frac{31\cdots 67}{36\cdots 44}a^{11}+\frac{68\cdots 78}{11\cdots 17}a^{10}+\frac{97\cdots 49}{18\cdots 72}a^{9}-\frac{30\cdots 71}{18\cdots 72}a^{8}+\frac{24\cdots 25}{18\cdots 72}a^{7}-\frac{99\cdots 51}{45\cdots 68}a^{6}-\frac{79\cdots 75}{91\cdots 36}a^{5}+\frac{44\cdots 37}{45\cdots 68}a^{4}-\frac{20\cdots 81}{45\cdots 68}a^{3}-\frac{33\cdots 72}{11\cdots 17}a^{2}-\frac{79\cdots 89}{11\cdots 17}a-\frac{46\cdots 89}{22\cdots 34}$, $\frac{83\cdots 69}{73\cdots 88}a^{19}-\frac{82\cdots 67}{45\cdots 68}a^{18}+\frac{15\cdots 51}{14\cdots 76}a^{17}+\frac{13\cdots 35}{18\cdots 72}a^{16}+\frac{43\cdots 11}{14\cdots 76}a^{15}+\frac{66\cdots 11}{73\cdots 88}a^{14}+\frac{17\cdots 63}{73\cdots 88}a^{13}+\frac{36\cdots 97}{91\cdots 36}a^{12}+\frac{21\cdots 15}{73\cdots 88}a^{11}+\frac{40\cdots 17}{36\cdots 44}a^{10}+\frac{50\cdots 85}{36\cdots 44}a^{9}+\frac{91\cdots 05}{18\cdots 72}a^{8}+\frac{29\cdots 45}{91\cdots 36}a^{7}+\frac{22\cdots 90}{11\cdots 17}a^{6}+\frac{68\cdots 53}{11\cdots 17}a^{5}+\frac{15\cdots 67}{45\cdots 68}a^{4}+\frac{11\cdots 87}{91\cdots 36}a^{3}+\frac{10\cdots 61}{45\cdots 68}a^{2}+\frac{34\cdots 13}{45\cdots 68}a+\frac{98\cdots 05}{73\cdots 14}$, $\frac{66\cdots 35}{58\cdots 04}a^{19}+\frac{46\cdots 39}{58\cdots 04}a^{18}-\frac{45\cdots 91}{29\cdots 52}a^{17}+\frac{24\cdots 21}{29\cdots 52}a^{16}-\frac{82\cdots 29}{29\cdots 52}a^{15}-\frac{25\cdots 55}{29\cdots 52}a^{14}+\frac{80\cdots 03}{36\cdots 44}a^{13}-\frac{39\cdots 37}{73\cdots 88}a^{12}-\frac{43\cdots 37}{73\cdots 88}a^{11}-\frac{48\cdots 05}{73\cdots 88}a^{10}-\frac{47\cdots 03}{36\cdots 44}a^{9}+\frac{53\cdots 19}{36\cdots 44}a^{8}-\frac{14\cdots 07}{36\cdots 44}a^{7}-\frac{16\cdots 05}{36\cdots 44}a^{6}-\frac{17\cdots 98}{11\cdots 17}a^{5}-\frac{30\cdots 33}{91\cdots 36}a^{4}+\frac{42\cdots 51}{91\cdots 36}a^{3}-\frac{16\cdots 69}{91\cdots 36}a^{2}-\frac{54\cdots 49}{22\cdots 34}a+\frac{46\cdots 93}{11\cdots 17}$, $\frac{47\cdots 87}{58\cdots 04}a^{19}+\frac{13\cdots 31}{58\cdots 04}a^{18}-\frac{30\cdots 25}{29\cdots 52}a^{17}-\frac{48\cdots 17}{18\cdots 72}a^{16}-\frac{89\cdots 53}{29\cdots 52}a^{15}-\frac{17\cdots 11}{29\cdots 52}a^{14}+\frac{61\cdots 47}{14\cdots 76}a^{13}-\frac{13\cdots 59}{47\cdots 96}a^{12}-\frac{12\cdots 89}{36\cdots 44}a^{11}-\frac{15\cdots 57}{91\cdots 36}a^{10}-\frac{34\cdots 25}{73\cdots 88}a^{9}+\frac{82\cdots 33}{36\cdots 44}a^{8}-\frac{10\cdots 49}{11\cdots 24}a^{7}+\frac{53\cdots 59}{11\cdots 24}a^{6}+\frac{25\cdots 47}{91\cdots 36}a^{5}+\frac{20\cdots 15}{18\cdots 72}a^{4}+\frac{21\cdots 33}{22\cdots 34}a^{3}+\frac{29\cdots 05}{22\cdots 34}a^{2}+\frac{51\cdots 45}{91\cdots 36}a+\frac{70\cdots 63}{22\cdots 34}$, $\frac{34\cdots 55}{58\cdots 04}a^{19}-\frac{71\cdots 97}{58\cdots 04}a^{18}+\frac{89\cdots 39}{14\cdots 76}a^{17}+\frac{12\cdots 05}{29\cdots 52}a^{16}+\frac{73\cdots 29}{29\cdots 52}a^{15}+\frac{15\cdots 57}{29\cdots 52}a^{14}+\frac{62\cdots 43}{36\cdots 44}a^{13}+\frac{11\cdots 75}{45\cdots 68}a^{12}+\frac{10\cdots 47}{64\cdots 76}a^{11}+\frac{59\cdots 61}{73\cdots 88}a^{10}+\frac{61\cdots 49}{73\cdots 88}a^{9}+\frac{71\cdots 09}{22\cdots 34}a^{8}+\frac{82\cdots 73}{36\cdots 44}a^{7}+\frac{43\cdots 95}{36\cdots 44}a^{6}+\frac{76\cdots 93}{18\cdots 72}a^{5}+\frac{22\cdots 37}{91\cdots 36}a^{4}+\frac{69\cdots 73}{91\cdots 36}a^{3}+\frac{19\cdots 49}{91\cdots 36}a^{2}+\frac{72\cdots 03}{91\cdots 36}a+\frac{48\cdots 81}{45\cdots 68}$, $\frac{74\cdots 83}{58\cdots 04}a^{19}-\frac{17\cdots 73}{29\cdots 52}a^{18}+\frac{19\cdots 19}{29\cdots 52}a^{17}+\frac{41\cdots 51}{36\cdots 44}a^{16}-\frac{11\cdots 21}{94\cdots 92}a^{15}+\frac{15\cdots 91}{91\cdots 36}a^{14}-\frac{66\cdots 93}{73\cdots 88}a^{13}+\frac{35\cdots 63}{73\cdots 88}a^{12}+\frac{57\cdots 95}{73\cdots 88}a^{11}-\frac{15\cdots 43}{73\cdots 88}a^{10}+\frac{92\cdots 19}{45\cdots 68}a^{9}-\frac{46\cdots 53}{36\cdots 44}a^{8}+\frac{11\cdots 63}{36\cdots 44}a^{7}-\frac{55\cdots 69}{59\cdots 12}a^{6}-\frac{12\cdots 27}{11\cdots 17}a^{5}+\frac{28\cdots 76}{11\cdots 17}a^{4}-\frac{28\cdots 65}{91\cdots 36}a^{3}-\frac{26\cdots 55}{91\cdots 36}a^{2}-\frac{15\cdots 45}{14\cdots 28}a-\frac{54\cdots 03}{45\cdots 68}$, $\frac{82\cdots 75}{29\cdots 52}a^{19}+\frac{36\cdots 75}{14\cdots 76}a^{18}-\frac{31\cdots 43}{73\cdots 88}a^{17}+\frac{16\cdots 41}{18\cdots 72}a^{16}-\frac{10\cdots 19}{11\cdots 17}a^{15}-\frac{71\cdots 37}{36\cdots 44}a^{14}+\frac{64\cdots 65}{73\cdots 88}a^{13}-\frac{56\cdots 85}{36\cdots 44}a^{12}+\frac{12\cdots 83}{73\cdots 88}a^{11}-\frac{90\cdots 13}{36\cdots 44}a^{10}-\frac{10\cdots 07}{36\cdots 44}a^{9}+\frac{14\cdots 39}{18\cdots 72}a^{8}-\frac{23\cdots 93}{18\cdots 72}a^{7}+\frac{11\cdots 77}{91\cdots 36}a^{6}-\frac{14\cdots 37}{11\cdots 17}a^{5}-\frac{93\cdots 65}{11\cdots 17}a^{4}+\frac{21\cdots 83}{91\cdots 36}a^{3}-\frac{10\cdots 01}{45\cdots 68}a^{2}+\frac{19\cdots 59}{45\cdots 68}a-\frac{11\cdots 27}{73\cdots 14}$
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| Regulator: | \( 17127663543868910000 \) (assuming GRH) |
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| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 17127663543868910000 \cdot 3}{2\cdot\sqrt{513006019713288929550454694707222514684527733454667776}}\cr\approx \mathstrut & 1.39407802376883 \end{aligned}\] (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
| \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3 + \sqrt{3}})\), 5.1.16159705088.1, 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 | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $20$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | 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 | $x^{10} + 245 x^{9} + 24020 x^{8} + 1178450 x^{7} + 28968105 x^{6} + 287187089 x^{5} + 57936210 x^{4} + 4713800 x^{3} + 192160 x^{2} + 3920 x + 85$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |