Normalized defining polynomial
\( x^{20} - 6 x^{19} - 169 x^{18} + 986 x^{17} + 11703 x^{16} - 65220 x^{15} - 419598 x^{14} + \cdots + 3619949909 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(4, 8)$ |
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| Discriminant: |
\(393644681588957236910214274346592274945338834944\)
\(\medspace = 2^{16}\cdot 7^{15}\cdot 11^{15}\cdot 13^{13}\)
|
| |
| Root discriminant: | \(239.75\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{1001}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{11}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{3}{8}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{3}{8}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}+\frac{3}{8}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}+\frac{3}{8}a^{8}+\frac{1}{8}a^{6}+\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{11}+\frac{3}{8}a^{9}+\frac{1}{8}a^{7}+\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{8}a$, $\frac{1}{104}a^{16}-\frac{1}{52}a^{15}-\frac{1}{104}a^{14}-\frac{3}{104}a^{13}+\frac{1}{104}a^{12}-\frac{5}{104}a^{11}-\frac{1}{13}a^{10}-\frac{11}{26}a^{9}+\frac{11}{52}a^{8}-\frac{25}{104}a^{7}-\frac{3}{26}a^{5}+\frac{9}{104}a^{4}+\frac{21}{104}a^{3}+\frac{23}{104}a^{2}+\frac{45}{104}a-\frac{9}{104}$, $\frac{1}{104}a^{17}-\frac{5}{104}a^{15}-\frac{5}{104}a^{14}-\frac{5}{104}a^{13}-\frac{3}{104}a^{12}+\frac{1}{13}a^{11}-\frac{1}{13}a^{10}-\frac{7}{52}a^{9}-\frac{7}{104}a^{8}-\frac{3}{13}a^{7}-\frac{3}{26}a^{6}-\frac{15}{104}a^{5}+\frac{1}{8}a^{4}-\frac{1}{8}a^{3}+\frac{3}{8}a^{2}+\frac{29}{104}a+\frac{1}{13}$, $\frac{1}{13\cdots 24}a^{18}+\frac{18\cdots 05}{33\cdots 06}a^{17}+\frac{10\cdots 49}{33\cdots 06}a^{16}+\frac{37\cdots 71}{67\cdots 12}a^{15}-\frac{68\cdots 65}{13\cdots 24}a^{14}+\frac{83\cdots 49}{13\cdots 24}a^{13}-\frac{82\cdots 40}{16\cdots 53}a^{12}-\frac{90\cdots 71}{13\cdots 24}a^{11}-\frac{20\cdots 29}{33\cdots 06}a^{10}+\frac{20\cdots 17}{67\cdots 12}a^{9}+\frac{65\cdots 93}{13\cdots 24}a^{8}+\frac{39\cdots 83}{13\cdots 24}a^{7}+\frac{10\cdots 01}{13\cdots 24}a^{6}+\frac{16\cdots 25}{67\cdots 12}a^{5}-\frac{60\cdots 47}{13\cdots 24}a^{4}+\frac{58\cdots 35}{13\cdots 24}a^{3}+\frac{39\cdots 25}{13\cdots 24}a^{2}+\frac{61\cdots 55}{13\cdots 24}a-\frac{38\cdots 09}{13\cdots 24}$, $\frac{1}{76\cdots 28}a^{19}-\frac{16\cdots 57}{76\cdots 28}a^{18}-\frac{44\cdots 98}{95\cdots 41}a^{17}-\frac{28\cdots 33}{76\cdots 28}a^{16}-\frac{15\cdots 25}{76\cdots 28}a^{15}-\frac{45\cdots 93}{95\cdots 41}a^{14}+\frac{13\cdots 85}{58\cdots 56}a^{13}+\frac{58\cdots 39}{95\cdots 41}a^{12}-\frac{46\cdots 11}{76\cdots 28}a^{11}+\frac{35\cdots 89}{76\cdots 28}a^{10}+\frac{37\cdots 39}{76\cdots 28}a^{9}-\frac{24\cdots 23}{76\cdots 28}a^{8}+\frac{28\cdots 19}{19\cdots 82}a^{7}+\frac{13\cdots 35}{38\cdots 64}a^{6}+\frac{33\cdots 37}{76\cdots 28}a^{5}+\frac{69\cdots 83}{38\cdots 64}a^{4}+\frac{26\cdots 47}{19\cdots 82}a^{3}+\frac{93\cdots 59}{76\cdots 28}a^{2}+\frac{17\cdots 21}{38\cdots 64}a-\frac{37\cdots 81}{76\cdots 28}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH) |
|
Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{95\cdots 60}{73\cdots 57}a^{19}-\frac{58\cdots 13}{73\cdots 57}a^{18}-\frac{16\cdots 01}{73\cdots 57}a^{17}+\frac{96\cdots 29}{73\cdots 57}a^{16}+\frac{11\cdots 76}{73\cdots 57}a^{15}-\frac{63\cdots 19}{73\cdots 57}a^{14}-\frac{39\cdots 77}{73\cdots 57}a^{13}+\frac{43\cdots 57}{14\cdots 14}a^{12}+\frac{73\cdots 82}{73\cdots 57}a^{11}-\frac{80\cdots 79}{14\cdots 14}a^{10}-\frac{56\cdots 08}{73\cdots 57}a^{9}+\frac{36\cdots 39}{73\cdots 57}a^{8}+\frac{57\cdots 10}{73\cdots 57}a^{7}-\frac{23\cdots 93}{14\cdots 14}a^{6}-\frac{40\cdots 87}{73\cdots 57}a^{5}+\frac{63\cdots 72}{73\cdots 57}a^{4}+\frac{69\cdots 22}{73\cdots 57}a^{3}-\frac{23\cdots 01}{14\cdots 14}a^{2}-\frac{80\cdots 81}{73\cdots 57}a-\frac{15\cdots 81}{14\cdots 14}$, $\frac{46\cdots 12}{73\cdots 57}a^{19}+\frac{27\cdots 29}{73\cdots 57}a^{18}+\frac{77\cdots 37}{73\cdots 57}a^{17}-\frac{68\cdots 73}{11\cdots 78}a^{16}-\frac{41\cdots 04}{56\cdots 89}a^{15}+\frac{29\cdots 45}{73\cdots 57}a^{14}+\frac{18\cdots 89}{73\cdots 57}a^{13}-\frac{19\cdots 03}{14\cdots 14}a^{12}-\frac{33\cdots 10}{73\cdots 57}a^{11}+\frac{34\cdots 27}{14\cdots 14}a^{10}+\frac{22\cdots 80}{73\cdots 57}a^{9}-\frac{27\cdots 03}{14\cdots 14}a^{8}+\frac{83\cdots 78}{73\cdots 57}a^{7}+\frac{33\cdots 01}{14\cdots 14}a^{6}-\frac{38\cdots 09}{56\cdots 89}a^{5}+\frac{14\cdots 65}{56\cdots 89}a^{4}-\frac{83\cdots 22}{73\cdots 57}a^{3}-\frac{86\cdots 35}{14\cdots 14}a^{2}+\frac{14\cdots 05}{73\cdots 57}a-\frac{46\cdots 61}{56\cdots 89}$, $\frac{64\cdots 83}{26\cdots 62}a^{19}-\frac{16\cdots 45}{52\cdots 24}a^{18}+\frac{33\cdots 37}{52\cdots 24}a^{17}+\frac{24\cdots 21}{52\cdots 24}a^{16}-\frac{78\cdots 06}{13\cdots 81}a^{15}-\frac{36\cdots 36}{13\cdots 81}a^{14}+\frac{14\cdots 65}{52\cdots 24}a^{13}+\frac{81\cdots 69}{10\cdots 48}a^{12}-\frac{80\cdots 15}{13\cdots 81}a^{11}-\frac{89\cdots 57}{10\cdots 48}a^{10}+\frac{83\cdots 10}{13\cdots 81}a^{9}-\frac{54\cdots 57}{52\cdots 24}a^{8}-\frac{57\cdots 81}{26\cdots 62}a^{7}-\frac{43\cdots 71}{10\cdots 48}a^{6}+\frac{42\cdots 85}{52\cdots 24}a^{5}+\frac{65\cdots 39}{52\cdots 24}a^{4}-\frac{13\cdots 47}{13\cdots 81}a^{3}-\frac{20\cdots 37}{10\cdots 48}a^{2}-\frac{16\cdots 12}{13\cdots 81}a-\frac{11\cdots 47}{10\cdots 48}$, $\frac{43\cdots 61}{14\cdots 14}a^{19}-\frac{26\cdots 43}{14\cdots 14}a^{18}-\frac{73\cdots 11}{14\cdots 14}a^{17}+\frac{87\cdots 35}{29\cdots 28}a^{16}+\frac{25\cdots 38}{73\cdots 57}a^{15}-\frac{58\cdots 21}{29\cdots 28}a^{14}-\frac{18\cdots 93}{14\cdots 14}a^{13}+\frac{99\cdots 53}{14\cdots 14}a^{12}+\frac{33\cdots 85}{14\cdots 14}a^{11}-\frac{36\cdots 99}{29\cdots 28}a^{10}-\frac{12\cdots 96}{73\cdots 57}a^{9}+\frac{84\cdots 41}{73\cdots 57}a^{8}+\frac{14\cdots 14}{73\cdots 57}a^{7}-\frac{10\cdots 97}{29\cdots 28}a^{6}-\frac{92\cdots 54}{73\cdots 57}a^{5}+\frac{57\cdots 87}{29\cdots 28}a^{4}+\frac{31\cdots 17}{14\cdots 14}a^{3}-\frac{53\cdots 95}{14\cdots 14}a^{2}-\frac{70\cdots 29}{14\cdots 14}a-\frac{31\cdots 75}{73\cdots 57}$, $\frac{40\cdots 00}{73\cdots 57}a^{19}-\frac{24\cdots 42}{73\cdots 57}a^{18}-\frac{67\cdots 90}{73\cdots 57}a^{17}+\frac{40\cdots 60}{73\cdots 57}a^{16}+\frac{46\cdots 56}{73\cdots 57}a^{15}-\frac{26\cdots 62}{73\cdots 57}a^{14}-\frac{16\cdots 30}{73\cdots 57}a^{13}+\frac{91\cdots 97}{73\cdots 57}a^{12}+\frac{30\cdots 04}{73\cdots 57}a^{11}-\frac{16\cdots 15}{73\cdots 57}a^{10}-\frac{23\cdots 00}{73\cdots 57}a^{9}+\frac{15\cdots 38}{73\cdots 57}a^{8}+\frac{26\cdots 92}{73\cdots 57}a^{7}-\frac{49\cdots 61}{73\cdots 57}a^{6}-\frac{17\cdots 90}{73\cdots 57}a^{5}+\frac{26\cdots 64}{73\cdots 57}a^{4}+\frac{29\cdots 52}{73\cdots 57}a^{3}-\frac{49\cdots 33}{73\cdots 57}a^{2}-\frac{64\cdots 40}{73\cdots 57}a-\frac{57\cdots 28}{73\cdots 57}$, $\frac{13\cdots 63}{76\cdots 28}a^{19}+\frac{62\cdots 27}{95\cdots 41}a^{18}+\frac{23\cdots 23}{76\cdots 28}a^{17}-\frac{19\cdots 27}{19\cdots 82}a^{16}-\frac{21\cdots 63}{95\cdots 41}a^{15}+\frac{60\cdots 91}{95\cdots 41}a^{14}+\frac{12\cdots 11}{14\cdots 14}a^{13}-\frac{18\cdots 47}{95\cdots 41}a^{12}-\frac{13\cdots 13}{76\cdots 28}a^{11}+\frac{28\cdots 70}{95\cdots 41}a^{10}+\frac{13\cdots 35}{76\cdots 28}a^{9}-\frac{22\cdots 76}{95\cdots 41}a^{8}-\frac{12\cdots 77}{19\cdots 82}a^{7}+\frac{46\cdots 24}{95\cdots 41}a^{6}+\frac{84\cdots 09}{38\cdots 64}a^{5}+\frac{18\cdots 95}{38\cdots 64}a^{4}-\frac{11\cdots 71}{76\cdots 28}a^{3}-\frac{29\cdots 01}{38\cdots 64}a^{2}-\frac{29\cdots 49}{76\cdots 28}a+\frac{90\cdots 49}{38\cdots 64}$, $\frac{34\cdots 69}{76\cdots 28}a^{19}-\frac{10\cdots 33}{38\cdots 64}a^{18}-\frac{59\cdots 33}{76\cdots 28}a^{17}+\frac{82\cdots 61}{19\cdots 82}a^{16}+\frac{41\cdots 77}{76\cdots 28}a^{15}-\frac{27\cdots 82}{95\cdots 41}a^{14}-\frac{11\cdots 79}{58\cdots 56}a^{13}+\frac{73\cdots 61}{76\cdots 28}a^{12}+\frac{28\cdots 49}{76\cdots 28}a^{11}-\frac{13\cdots 41}{76\cdots 28}a^{10}-\frac{29\cdots 00}{95\cdots 41}a^{9}+\frac{58\cdots 91}{38\cdots 64}a^{8}+\frac{79\cdots 83}{19\cdots 82}a^{7}-\frac{34\cdots 79}{76\cdots 28}a^{6}-\frac{26\cdots 29}{76\cdots 28}a^{5}-\frac{15\cdots 57}{95\cdots 41}a^{4}+\frac{11\cdots 33}{38\cdots 64}a^{3}+\frac{62\cdots 01}{76\cdots 28}a^{2}-\frac{28\cdots 07}{76\cdots 28}a-\frac{31\cdots 61}{76\cdots 28}$, $\frac{30\cdots 61}{76\cdots 28}a^{19}+\frac{16\cdots 15}{76\cdots 28}a^{18}+\frac{25\cdots 03}{38\cdots 64}a^{17}-\frac{67\cdots 55}{19\cdots 82}a^{16}-\frac{34\cdots 21}{76\cdots 28}a^{15}+\frac{16\cdots 13}{76\cdots 28}a^{14}+\frac{45\cdots 01}{29\cdots 28}a^{13}-\frac{52\cdots 91}{76\cdots 28}a^{12}-\frac{10\cdots 35}{38\cdots 64}a^{11}+\frac{83\cdots 55}{76\cdots 28}a^{10}+\frac{13\cdots 07}{76\cdots 28}a^{9}-\frac{54\cdots 73}{76\cdots 28}a^{8}+\frac{15\cdots 59}{76\cdots 28}a^{7}+\frac{88\cdots 99}{19\cdots 82}a^{6}+\frac{32\cdots 03}{76\cdots 28}a^{5}+\frac{19\cdots 63}{76\cdots 28}a^{4}-\frac{25\cdots 15}{76\cdots 28}a^{3}-\frac{15\cdots 91}{38\cdots 64}a^{2}+\frac{10\cdots 27}{76\cdots 28}a+\frac{43\cdots 89}{38\cdots 64}$, $\frac{21\cdots 21}{76\cdots 28}a^{19}-\frac{21\cdots 69}{76\cdots 28}a^{18}-\frac{26\cdots 17}{76\cdots 28}a^{17}+\frac{32\cdots 69}{76\cdots 28}a^{16}+\frac{10\cdots 93}{76\cdots 28}a^{15}-\frac{46\cdots 41}{19\cdots 82}a^{14}-\frac{18\cdots 37}{29\cdots 28}a^{13}+\frac{25\cdots 41}{38\cdots 64}a^{12}-\frac{78\cdots 65}{95\cdots 41}a^{11}-\frac{58\cdots 65}{76\cdots 28}a^{10}+\frac{35\cdots 53}{19\cdots 82}a^{9}+\frac{12\cdots 23}{76\cdots 28}a^{8}-\frac{50\cdots 67}{76\cdots 28}a^{7}-\frac{35\cdots 99}{38\cdots 64}a^{6}-\frac{61\cdots 31}{76\cdots 28}a^{5}+\frac{72\cdots 35}{19\cdots 82}a^{4}+\frac{97\cdots 01}{76\cdots 28}a^{3}-\frac{38\cdots 87}{76\cdots 28}a^{2}-\frac{43\cdots 20}{95\cdots 41}a+\frac{17\cdots 87}{76\cdots 28}$, $\frac{56\cdots 41}{38\cdots 64}a^{19}+\frac{69\cdots 97}{76\cdots 28}a^{18}+\frac{24\cdots 15}{95\cdots 41}a^{17}-\frac{57\cdots 31}{38\cdots 64}a^{16}-\frac{33\cdots 39}{19\cdots 82}a^{15}+\frac{38\cdots 17}{38\cdots 64}a^{14}+\frac{36\cdots 61}{58\cdots 56}a^{13}-\frac{67\cdots 47}{19\cdots 82}a^{12}-\frac{89\cdots 07}{76\cdots 28}a^{11}+\frac{50\cdots 13}{76\cdots 28}a^{10}+\frac{85\cdots 97}{95\cdots 41}a^{9}-\frac{60\cdots 28}{95\cdots 41}a^{8}+\frac{55\cdots 53}{76\cdots 28}a^{7}+\frac{22\cdots 96}{95\cdots 41}a^{6}-\frac{64\cdots 63}{95\cdots 41}a^{5}-\frac{67\cdots 79}{95\cdots 41}a^{4}-\frac{65\cdots 97}{76\cdots 28}a^{3}-\frac{93\cdots 41}{76\cdots 28}a^{2}+\frac{21\cdots 27}{76\cdots 28}a-\frac{49\cdots 41}{38\cdots 64}$, $\frac{35\cdots 73}{76\cdots 28}a^{19}-\frac{50\cdots 65}{95\cdots 41}a^{18}-\frac{61\cdots 01}{76\cdots 28}a^{17}+\frac{46\cdots 13}{76\cdots 28}a^{16}+\frac{20\cdots 73}{38\cdots 64}a^{15}-\frac{12\cdots 87}{38\cdots 64}a^{14}-\frac{13\cdots 17}{73\cdots 57}a^{13}+\frac{13\cdots 29}{95\cdots 41}a^{12}+\frac{25\cdots 93}{76\cdots 28}a^{11}-\frac{80\cdots 85}{19\cdots 82}a^{10}-\frac{23\cdots 11}{76\cdots 28}a^{9}+\frac{40\cdots 67}{76\cdots 28}a^{8}+\frac{54\cdots 05}{38\cdots 64}a^{7}-\frac{14\cdots 79}{38\cdots 64}a^{6}-\frac{11\cdots 29}{38\cdots 64}a^{5}-\frac{71\cdots 70}{95\cdots 41}a^{4}+\frac{37\cdots 59}{76\cdots 28}a^{3}+\frac{29\cdots 17}{19\cdots 82}a^{2}+\frac{55\cdots 83}{76\cdots 28}a-\frac{36\cdots 97}{76\cdots 28}$
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| Regulator: | \( 4987918329401424.0 \) (assuming GRH) |
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| Unit signature rank: | \( 2 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 4987918329401424.0 \cdot 4}{2\cdot\sqrt{393644681588957236910214274346592274945338834944}}\cr\approx \mathstrut & 0.617953882170120 \end{aligned}\] (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
| \(\Q(\sqrt{77}) \), \(\Q(\sqrt{77 +8 \sqrt{77}})\), 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: | 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 | $20$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | R | R | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | $20$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 51 x^{5} + 45 x^{4} + 30 x^{3} + 15 x^{2} + 5 x + 3$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
|
\(7\)
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 7.16.12.1 | $x^{16} + 20 x^{14} + 16 x^{13} + 162 x^{12} + 240 x^{11} + 776 x^{10} + 1344 x^{9} + 2539 x^{8} + 3696 x^{7} + 5016 x^{6} + 5312 x^{5} + 4594 x^{4} + 2928 x^{3} + 1404 x^{2} + 432 x + 88$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(11\)
| 11.4.3.1 | $x^{4} + 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 11.16.12.1 | $x^{16} + 32 x^{14} + 40 x^{13} + 392 x^{12} + 960 x^{11} + 2840 x^{10} + 7920 x^{9} + 15256 x^{8} + 28320 x^{7} + 45280 x^{6} + 47840 x^{5} + 30768 x^{4} + 11840 x^{3} + 2656 x^{2} + 320 x + 27$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 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 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 384 x + 29$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ |