Normalized defining polynomial
\( x^{20} - x^{19} - 57 x^{18} + 66 x^{17} + 1198 x^{16} - 1526 x^{15} - 12200 x^{14} + 17240 x^{13} + \cdots - 4645 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(20, 0)$ |
| |
| Discriminant: |
\(21333423461884919389012763702392578125\)
\(\medspace = 5^{15}\cdot 31^{18}\)
|
| |
| Root discriminant: | \(73.53\) |
| |
| Galois root discriminant: | $5^{3/4}31^{9/10}\approx 73.52804649223957$ | ||
| Ramified primes: |
\(5\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(155=5\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(66,·)$, $\chi_{155}(4,·)$, $\chi_{155}(77,·)$, $\chi_{155}(16,·)$, $\chi_{155}(147,·)$, $\chi_{155}(23,·)$, $\chi_{155}(153,·)$, $\chi_{155}(27,·)$, $\chi_{155}(92,·)$, $\chi_{155}(58,·)$, $\chi_{155}(94,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(108,·)$, $\chi_{155}(109,·)$, $\chi_{155}(122,·)$, $\chi_{155}(123,·)$, $\chi_{155}(126,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{15}+\frac{1}{5}a^{12}+\frac{1}{5}a^{10}+\frac{1}{5}a^{8}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}$, $\frac{1}{5}a^{17}+\frac{1}{5}a^{15}+\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}a^{4}$, $\frac{1}{3095}a^{18}-\frac{168}{3095}a^{17}+\frac{57}{3095}a^{16}+\frac{1449}{3095}a^{15}-\frac{1239}{3095}a^{14}+\frac{266}{619}a^{13}+\frac{378}{3095}a^{12}-\frac{255}{619}a^{11}+\frac{1408}{3095}a^{10}-\frac{56}{619}a^{9}-\frac{1448}{3095}a^{8}+\frac{368}{3095}a^{7}-\frac{1}{5}a^{6}-\frac{1209}{3095}a^{5}+\frac{152}{3095}a^{4}+\frac{216}{619}a^{3}+\frac{302}{619}a^{2}-\frac{198}{619}a+\frac{217}{619}$, $\frac{1}{11\cdots 25}a^{19}-\frac{14\cdots 04}{11\cdots 25}a^{18}+\frac{41\cdots 46}{45\cdots 49}a^{17}-\frac{73\cdots 34}{11\cdots 25}a^{16}-\frac{10\cdots 57}{22\cdots 45}a^{15}-\frac{18\cdots 81}{11\cdots 25}a^{14}-\frac{12\cdots 12}{11\cdots 25}a^{13}+\frac{15\cdots 11}{11\cdots 25}a^{12}-\frac{39\cdots 07}{11\cdots 25}a^{11}+\frac{52\cdots 01}{11\cdots 25}a^{10}-\frac{30\cdots 13}{11\cdots 25}a^{9}+\frac{12\cdots 51}{22\cdots 45}a^{8}+\frac{21\cdots 88}{11\cdots 25}a^{7}+\frac{97\cdots 97}{22\cdots 45}a^{6}-\frac{50\cdots 42}{11\cdots 25}a^{5}-\frac{44\cdots 98}{11\cdots 25}a^{4}-\frac{13\cdots 76}{22\cdots 45}a^{3}+\frac{50\cdots 32}{22\cdots 45}a^{2}-\frac{43\cdots 67}{22\cdots 45}a-\frac{60\cdots 73}{22\cdots 45}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $64$ (assuming GRH) |
|
Unit group
| Rank: | $19$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{36\cdots 53}{53\cdots 05}a^{19}+\frac{11\cdots 03}{53\cdots 05}a^{18}-\frac{20\cdots 86}{53\cdots 05}a^{17}-\frac{30\cdots 67}{53\cdots 05}a^{16}+\frac{42\cdots 59}{53\cdots 05}a^{15}+\frac{15\cdots 57}{53\cdots 05}a^{14}-\frac{43\cdots 27}{53\cdots 05}a^{13}+\frac{79\cdots 66}{10\cdots 81}a^{12}+\frac{23\cdots 18}{53\cdots 05}a^{11}-\frac{11\cdots 25}{10\cdots 81}a^{10}-\frac{14\cdots 66}{10\cdots 81}a^{9}+\frac{26\cdots 47}{53\cdots 05}a^{8}+\frac{11\cdots 74}{53\cdots 05}a^{7}-\frac{56\cdots 78}{53\cdots 05}a^{6}-\frac{85\cdots 91}{53\cdots 05}a^{5}+\frac{52\cdots 63}{53\cdots 05}a^{4}+\frac{47\cdots 40}{10\cdots 81}a^{3}-\frac{32\cdots 84}{10\cdots 81}a^{2}-\frac{27\cdots 96}{10\cdots 81}a+\frac{22\cdots 95}{10\cdots 81}$, $\frac{12\cdots 21}{11\cdots 25}a^{19}-\frac{29\cdots 64}{11\cdots 25}a^{18}-\frac{13\cdots 14}{22\cdots 45}a^{17}+\frac{27\cdots 91}{11\cdots 25}a^{16}+\frac{29\cdots 43}{22\cdots 45}a^{15}-\frac{71\cdots 81}{11\cdots 25}a^{14}-\frac{15\cdots 37}{11\cdots 25}a^{13}+\frac{89\cdots 51}{11\cdots 25}a^{12}+\frac{85\cdots 68}{11\cdots 25}a^{11}-\frac{59\cdots 34}{11\cdots 25}a^{10}-\frac{25\cdots 08}{11\cdots 25}a^{9}+\frac{41\cdots 86}{22\cdots 45}a^{8}+\frac{41\cdots 83}{11\cdots 25}a^{7}-\frac{14\cdots 13}{45\cdots 49}a^{6}-\frac{31\cdots 67}{11\cdots 25}a^{5}+\frac{31\cdots 92}{11\cdots 25}a^{4}+\frac{16\cdots 39}{22\cdots 45}a^{3}-\frac{19\cdots 98}{22\cdots 45}a^{2}-\frac{70\cdots 82}{22\cdots 45}a+\frac{13\cdots 97}{22\cdots 45}$, $\frac{20\cdots 42}{11\cdots 25}a^{19}-\frac{44\cdots 28}{11\cdots 25}a^{18}-\frac{23\cdots 33}{22\cdots 45}a^{17}+\frac{43\cdots 32}{11\cdots 25}a^{16}+\frac{49\cdots 86}{22\cdots 45}a^{15}-\frac{11\cdots 87}{11\cdots 25}a^{14}-\frac{25\cdots 49}{11\cdots 25}a^{13}+\frac{14\cdots 77}{11\cdots 25}a^{12}+\frac{14\cdots 61}{11\cdots 25}a^{11}-\frac{96\cdots 93}{11\cdots 25}a^{10}-\frac{43\cdots 66}{11\cdots 25}a^{9}+\frac{67\cdots 07}{22\cdots 45}a^{8}+\frac{69\cdots 91}{11\cdots 25}a^{7}-\frac{24\cdots 04}{45\cdots 49}a^{6}-\frac{53\cdots 59}{11\cdots 25}a^{5}+\frac{51\cdots 09}{11\cdots 25}a^{4}+\frac{27\cdots 03}{22\cdots 45}a^{3}-\frac{31\cdots 71}{22\cdots 45}a^{2}-\frac{11\cdots 14}{22\cdots 45}a+\frac{21\cdots 44}{22\cdots 45}$, $\frac{12\cdots 31}{22\cdots 45}a^{19}-\frac{39\cdots 69}{22\cdots 45}a^{18}-\frac{13\cdots 71}{45\cdots 49}a^{17}+\frac{13\cdots 61}{22\cdots 45}a^{16}+\frac{29\cdots 87}{45\cdots 49}a^{15}-\frac{46\cdots 81}{22\cdots 45}a^{14}-\frac{14\cdots 87}{22\cdots 45}a^{13}+\frac{66\cdots 91}{22\cdots 45}a^{12}+\frac{81\cdots 18}{22\cdots 45}a^{11}-\frac{49\cdots 14}{22\cdots 45}a^{10}-\frac{24\cdots 68}{22\cdots 45}a^{9}+\frac{36\cdots 60}{45\cdots 49}a^{8}+\frac{37\cdots 18}{22\cdots 45}a^{7}-\frac{69\cdots 71}{45\cdots 49}a^{6}-\frac{26\cdots 22}{22\cdots 45}a^{5}+\frac{30\cdots 57}{22\cdots 45}a^{4}+\frac{10\cdots 54}{45\cdots 49}a^{3}-\frac{19\cdots 13}{45\cdots 49}a^{2}+\frac{52\cdots 63}{45\cdots 49}a+\frac{15\cdots 60}{45\cdots 49}$, $\frac{75\cdots 43}{11\cdots 25}a^{19}+\frac{39\cdots 88}{11\cdots 25}a^{18}-\frac{84\cdots 47}{22\cdots 45}a^{17}-\frac{13\cdots 22}{11\cdots 25}a^{16}+\frac{17\cdots 84}{22\cdots 45}a^{15}+\frac{10\cdots 27}{11\cdots 25}a^{14}-\frac{88\cdots 71}{11\cdots 25}a^{13}+\frac{11\cdots 58}{11\cdots 25}a^{12}+\frac{47\cdots 69}{11\cdots 25}a^{11}-\frac{21\cdots 97}{11\cdots 25}a^{10}-\frac{14\cdots 64}{11\cdots 25}a^{9}+\frac{23\cdots 83}{22\cdots 45}a^{8}+\frac{21\cdots 14}{11\cdots 25}a^{7}-\frac{11\cdots 40}{45\cdots 49}a^{6}-\frac{14\cdots 36}{11\cdots 25}a^{5}+\frac{30\cdots 11}{11\cdots 25}a^{4}+\frac{19\cdots 87}{22\cdots 45}a^{3}-\frac{23\cdots 84}{22\cdots 45}a^{2}+\frac{16\cdots 69}{22\cdots 45}a+\frac{27\cdots 51}{22\cdots 45}$, $\frac{12\cdots 54}{22\cdots 45}a^{19}+\frac{22\cdots 28}{22\cdots 45}a^{18}-\frac{69\cdots 34}{22\cdots 45}a^{17}-\frac{16\cdots 97}{22\cdots 45}a^{16}+\frac{14\cdots 81}{22\cdots 45}a^{15}-\frac{25\cdots 59}{45\cdots 49}a^{14}-\frac{30\cdots 48}{45\cdots 49}a^{13}+\frac{31\cdots 56}{22\cdots 45}a^{12}+\frac{16\cdots 91}{45\cdots 49}a^{11}-\frac{29\cdots 89}{22\cdots 45}a^{10}-\frac{24\cdots 79}{22\cdots 45}a^{9}+\frac{12\cdots 33}{22\cdots 45}a^{8}+\frac{39\cdots 84}{22\cdots 45}a^{7}-\frac{24\cdots 77}{22\cdots 45}a^{6}-\frac{31\cdots 71}{22\cdots 45}a^{5}+\frac{22\cdots 21}{22\cdots 45}a^{4}+\frac{17\cdots 72}{45\cdots 49}a^{3}-\frac{13\cdots 30}{45\cdots 49}a^{2}-\frac{10\cdots 66}{45\cdots 49}a+\frac{99\cdots 80}{45\cdots 49}$, $\frac{15\cdots 22}{22\cdots 45}a^{19}+\frac{12\cdots 99}{22\cdots 45}a^{18}-\frac{90\cdots 79}{22\cdots 45}a^{17}+\frac{68\cdots 64}{22\cdots 45}a^{16}+\frac{19\cdots 26}{22\cdots 45}a^{15}-\frac{69\cdots 11}{45\cdots 49}a^{14}-\frac{19\cdots 32}{22\cdots 45}a^{13}+\frac{57\cdots 12}{22\cdots 45}a^{12}+\frac{10\cdots 78}{22\cdots 45}a^{11}-\frac{45\cdots 53}{22\cdots 45}a^{10}-\frac{32\cdots 59}{22\cdots 45}a^{9}+\frac{17\cdots 73}{22\cdots 45}a^{8}+\frac{52\cdots 52}{22\cdots 45}a^{7}-\frac{33\cdots 72}{22\cdots 45}a^{6}-\frac{40\cdots 43}{22\cdots 45}a^{5}+\frac{28\cdots 97}{22\cdots 45}a^{4}+\frac{23\cdots 87}{45\cdots 49}a^{3}-\frac{17\cdots 32}{45\cdots 49}a^{2}-\frac{15\cdots 68}{45\cdots 49}a+\frac{10\cdots 61}{45\cdots 49}$, $\frac{26\cdots 74}{11\cdots 25}a^{19}+\frac{54\cdots 84}{11\cdots 25}a^{18}-\frac{30\cdots 77}{22\cdots 45}a^{17}+\frac{20\cdots 04}{11\cdots 25}a^{16}+\frac{63\cdots 06}{22\cdots 45}a^{15}-\frac{77\cdots 39}{11\cdots 25}a^{14}-\frac{32\cdots 08}{11\cdots 25}a^{13}+\frac{11\cdots 29}{11\cdots 25}a^{12}+\frac{18\cdots 87}{11\cdots 25}a^{11}-\frac{88\cdots 61}{11\cdots 25}a^{10}-\frac{54\cdots 82}{11\cdots 25}a^{9}+\frac{66\cdots 36}{22\cdots 45}a^{8}+\frac{87\cdots 02}{11\cdots 25}a^{7}-\frac{12\cdots 23}{22\cdots 45}a^{6}-\frac{67\cdots 73}{11\cdots 25}a^{5}+\frac{55\cdots 83}{11\cdots 25}a^{4}+\frac{36\cdots 81}{22\cdots 45}a^{3}-\frac{33\cdots 77}{22\cdots 45}a^{2}-\frac{18\cdots 08}{22\cdots 45}a+\frac{23\cdots 43}{22\cdots 45}$, $\frac{21\cdots 49}{11\cdots 25}a^{19}+\frac{37\cdots 54}{11\cdots 25}a^{18}-\frac{24\cdots 39}{22\cdots 45}a^{17}-\frac{17\cdots 26}{11\cdots 25}a^{16}+\frac{51\cdots 34}{22\cdots 45}a^{15}-\frac{24\cdots 44}{11\cdots 25}a^{14}-\frac{26\cdots 58}{11\cdots 25}a^{13}+\frac{56\cdots 69}{11\cdots 25}a^{12}+\frac{14\cdots 62}{11\cdots 25}a^{11}-\frac{51\cdots 96}{11\cdots 25}a^{10}-\frac{43\cdots 32}{11\cdots 25}a^{9}+\frac{85\cdots 47}{45\cdots 49}a^{8}+\frac{70\cdots 87}{11\cdots 25}a^{7}-\frac{84\cdots 84}{22\cdots 45}a^{6}-\frac{54\cdots 63}{11\cdots 25}a^{5}+\frac{38\cdots 28}{11\cdots 25}a^{4}+\frac{31\cdots 06}{22\cdots 45}a^{3}-\frac{24\cdots 47}{22\cdots 45}a^{2}-\frac{19\cdots 23}{22\cdots 45}a+\frac{17\cdots 03}{22\cdots 45}$, $\frac{10\cdots 62}{11\cdots 25}a^{19}-\frac{60\cdots 58}{11\cdots 25}a^{18}-\frac{12\cdots 32}{22\cdots 45}a^{17}+\frac{43\cdots 07}{11\cdots 25}a^{16}+\frac{26\cdots 24}{22\cdots 45}a^{15}-\frac{10\cdots 82}{11\cdots 25}a^{14}-\frac{13\cdots 84}{11\cdots 25}a^{13}+\frac{12\cdots 92}{11\cdots 25}a^{12}+\frac{74\cdots 01}{11\cdots 25}a^{11}-\frac{76\cdots 78}{11\cdots 25}a^{10}-\frac{21\cdots 96}{11\cdots 25}a^{9}+\frac{51\cdots 11}{22\cdots 45}a^{8}+\frac{33\cdots 26}{11\cdots 25}a^{7}-\frac{89\cdots 32}{22\cdots 45}a^{6}-\frac{22\cdots 09}{11\cdots 25}a^{5}+\frac{37\cdots 44}{11\cdots 25}a^{4}+\frac{56\cdots 48}{22\cdots 45}a^{3}-\frac{23\cdots 56}{22\cdots 45}a^{2}+\frac{14\cdots 16}{22\cdots 45}a+\frac{21\cdots 99}{22\cdots 45}$, $\frac{28\cdots 71}{22\cdots 45}a^{19}-\frac{11\cdots 49}{45\cdots 49}a^{18}-\frac{16\cdots 06}{22\cdots 45}a^{17}+\frac{11\cdots 92}{45\cdots 49}a^{16}+\frac{34\cdots 74}{22\cdots 45}a^{15}-\frac{15\cdots 52}{22\cdots 45}a^{14}-\frac{36\cdots 96}{22\cdots 45}a^{13}+\frac{20\cdots 12}{22\cdots 45}a^{12}+\frac{20\cdots 29}{22\cdots 45}a^{11}-\frac{13\cdots 33}{22\cdots 45}a^{10}-\frac{60\cdots 22}{22\cdots 45}a^{9}+\frac{47\cdots 37}{22\cdots 45}a^{8}+\frac{19\cdots 54}{45\cdots 49}a^{7}-\frac{84\cdots 23}{22\cdots 45}a^{6}-\frac{14\cdots 74}{45\cdots 49}a^{5}+\frac{71\cdots 34}{22\cdots 45}a^{4}+\frac{37\cdots 77}{45\cdots 49}a^{3}-\frac{43\cdots 25}{45\cdots 49}a^{2}-\frac{16\cdots 52}{45\cdots 49}a+\frac{32\cdots 81}{45\cdots 49}$, $\frac{37\cdots 79}{45\cdots 49}a^{19}-\frac{32\cdots 58}{45\cdots 49}a^{18}-\frac{21\cdots 37}{45\cdots 49}a^{17}+\frac{25\cdots 26}{22\cdots 45}a^{16}+\frac{22\cdots 37}{22\cdots 45}a^{15}-\frac{15\cdots 12}{45\cdots 49}a^{14}-\frac{46\cdots 45}{45\cdots 49}a^{13}+\frac{10\cdots 11}{22\cdots 45}a^{12}+\frac{25\cdots 10}{45\cdots 49}a^{11}-\frac{71\cdots 79}{22\cdots 45}a^{10}-\frac{75\cdots 64}{45\cdots 49}a^{9}+\frac{25\cdots 61}{22\cdots 45}a^{8}+\frac{11\cdots 56}{45\cdots 49}a^{7}-\frac{89\cdots 61}{45\cdots 49}a^{6}-\frac{44\cdots 47}{22\cdots 45}a^{5}+\frac{37\cdots 01}{22\cdots 45}a^{4}+\frac{24\cdots 51}{45\cdots 49}a^{3}-\frac{22\cdots 49}{45\cdots 49}a^{2}-\frac{12\cdots 66}{45\cdots 49}a+\frac{15\cdots 22}{45\cdots 49}$, $\frac{20\cdots 26}{11\cdots 25}a^{19}+\frac{17\cdots 11}{11\cdots 25}a^{18}-\frac{22\cdots 93}{22\cdots 45}a^{17}-\frac{78\cdots 24}{11\cdots 25}a^{16}+\frac{46\cdots 48}{22\cdots 45}a^{15}+\frac{13\cdots 59}{11\cdots 25}a^{14}-\frac{23\cdots 57}{11\cdots 25}a^{13}-\frac{10\cdots 99}{11\cdots 25}a^{12}+\frac{12\cdots 98}{11\cdots 25}a^{11}+\frac{37\cdots 41}{11\cdots 25}a^{10}-\frac{38\cdots 58}{11\cdots 25}a^{9}-\frac{10\cdots 74}{22\cdots 45}a^{8}+\frac{63\cdots 83}{11\cdots 25}a^{7}+\frac{23\cdots 98}{22\cdots 45}a^{6}-\frac{54\cdots 12}{11\cdots 25}a^{5}+\frac{22\cdots 77}{11\cdots 25}a^{4}+\frac{41\cdots 64}{22\cdots 45}a^{3}+\frac{49\cdots 57}{22\cdots 45}a^{2}-\frac{56\cdots 72}{22\cdots 45}a-\frac{92\cdots 48}{22\cdots 45}$, $\frac{31\cdots 62}{11\cdots 25}a^{19}+\frac{15\cdots 12}{11\cdots 25}a^{18}-\frac{36\cdots 54}{22\cdots 45}a^{17}+\frac{20\cdots 52}{11\cdots 25}a^{16}+\frac{76\cdots 57}{22\cdots 45}a^{15}-\frac{84\cdots 87}{11\cdots 25}a^{14}-\frac{39\cdots 59}{11\cdots 25}a^{13}+\frac{13\cdots 07}{11\cdots 25}a^{12}+\frac{21\cdots 26}{11\cdots 25}a^{11}-\frac{10\cdots 13}{11\cdots 25}a^{10}-\frac{65\cdots 46}{11\cdots 25}a^{9}+\frac{15\cdots 76}{45\cdots 49}a^{8}+\frac{10\cdots 86}{11\cdots 25}a^{7}-\frac{14\cdots 48}{22\cdots 45}a^{6}-\frac{81\cdots 24}{11\cdots 25}a^{5}+\frac{65\cdots 64}{11\cdots 25}a^{4}+\frac{44\cdots 63}{22\cdots 45}a^{3}-\frac{40\cdots 11}{22\cdots 45}a^{2}-\frac{24\cdots 94}{22\cdots 45}a+\frac{28\cdots 09}{22\cdots 45}$, $\frac{27\cdots 51}{11\cdots 25}a^{19}+\frac{76\cdots 61}{11\cdots 25}a^{18}-\frac{30\cdots 78}{22\cdots 45}a^{17}-\frac{41\cdots 49}{11\cdots 25}a^{16}+\frac{64\cdots 08}{22\cdots 45}a^{15}+\frac{84\cdots 09}{11\cdots 25}a^{14}-\frac{33\cdots 32}{11\cdots 25}a^{13}-\frac{83\cdots 49}{11\cdots 25}a^{12}+\frac{18\cdots 23}{11\cdots 25}a^{11}+\frac{44\cdots 91}{11\cdots 25}a^{10}-\frac{62\cdots 58}{11\cdots 25}a^{9}-\frac{25\cdots 09}{22\cdots 45}a^{8}+\frac{11\cdots 83}{11\cdots 25}a^{7}+\frac{41\cdots 78}{22\cdots 45}a^{6}-\frac{12\cdots 87}{11\cdots 25}a^{5}-\frac{16\cdots 73}{11\cdots 25}a^{4}+\frac{11\cdots 24}{22\cdots 45}a^{3}+\frac{11\cdots 17}{22\cdots 45}a^{2}-\frac{12\cdots 47}{22\cdots 45}a-\frac{87\cdots 13}{22\cdots 45}$, $\frac{17\cdots 59}{11\cdots 25}a^{19}-\frac{59\cdots 06}{11\cdots 25}a^{18}-\frac{19\cdots 14}{22\cdots 45}a^{17}+\frac{48\cdots 24}{11\cdots 25}a^{16}+\frac{42\cdots 93}{22\cdots 45}a^{15}-\frac{12\cdots 24}{11\cdots 25}a^{14}-\frac{22\cdots 13}{11\cdots 25}a^{13}+\frac{14\cdots 44}{11\cdots 25}a^{12}+\frac{12\cdots 57}{11\cdots 25}a^{11}-\frac{93\cdots 71}{11\cdots 25}a^{10}-\frac{39\cdots 22}{11\cdots 25}a^{9}+\frac{64\cdots 62}{22\cdots 45}a^{8}+\frac{65\cdots 57}{11\cdots 25}a^{7}-\frac{11\cdots 44}{22\cdots 45}a^{6}-\frac{51\cdots 38}{11\cdots 25}a^{5}+\frac{48\cdots 33}{11\cdots 25}a^{4}+\frac{28\cdots 71}{22\cdots 45}a^{3}-\frac{30\cdots 47}{22\cdots 45}a^{2}-\frac{14\cdots 93}{22\cdots 45}a+\frac{20\cdots 73}{22\cdots 45}$, $\frac{55\cdots 26}{11\cdots 25}a^{19}-\frac{96\cdots 59}{11\cdots 25}a^{18}-\frac{64\cdots 07}{22\cdots 45}a^{17}+\frac{58\cdots 66}{11\cdots 25}a^{16}+\frac{14\cdots 43}{22\cdots 45}a^{15}-\frac{12\cdots 86}{11\cdots 25}a^{14}-\frac{75\cdots 62}{11\cdots 25}a^{13}+\frac{13\cdots 06}{11\cdots 25}a^{12}+\frac{43\cdots 68}{11\cdots 25}a^{11}-\frac{76\cdots 54}{11\cdots 25}a^{10}-\frac{13\cdots 13}{11\cdots 25}a^{9}+\frac{46\cdots 11}{22\cdots 45}a^{8}+\frac{19\cdots 73}{11\cdots 25}a^{7}-\frac{75\cdots 49}{22\cdots 45}a^{6}-\frac{12\cdots 67}{11\cdots 25}a^{5}+\frac{28\cdots 02}{11\cdots 25}a^{4}+\frac{10\cdots 64}{22\cdots 45}a^{3}-\frac{14\cdots 73}{22\cdots 45}a^{2}+\frac{40\cdots 23}{22\cdots 45}a+\frac{98\cdots 82}{22\cdots 45}$, $\frac{93\cdots 59}{45\cdots 49}a^{19}+\frac{42\cdots 33}{22\cdots 45}a^{18}-\frac{52\cdots 94}{45\cdots 49}a^{17}+\frac{38\cdots 03}{22\cdots 45}a^{16}+\frac{11\cdots 04}{45\cdots 49}a^{15}-\frac{14\cdots 77}{22\cdots 45}a^{14}-\frac{57\cdots 31}{22\cdots 45}a^{13}+\frac{20\cdots 43}{22\cdots 45}a^{12}+\frac{31\cdots 14}{22\cdots 45}a^{11}-\frac{15\cdots 27}{22\cdots 45}a^{10}-\frac{95\cdots 56}{22\cdots 45}a^{9}+\frac{11\cdots 25}{45\cdots 49}a^{8}+\frac{15\cdots 34}{22\cdots 45}a^{7}-\frac{21\cdots 46}{45\cdots 49}a^{6}-\frac{11\cdots 86}{22\cdots 45}a^{5}+\frac{18\cdots 47}{45\cdots 49}a^{4}+\frac{66\cdots 38}{45\cdots 49}a^{3}-\frac{55\cdots 38}{45\cdots 49}a^{2}-\frac{43\cdots 67}{45\cdots 49}a+\frac{35\cdots 72}{45\cdots 49}$, $\frac{29\cdots 62}{22\cdots 45}a^{19}+\frac{37\cdots 42}{22\cdots 45}a^{18}-\frac{16\cdots 92}{22\cdots 45}a^{17}+\frac{55\cdots 32}{22\cdots 45}a^{16}+\frac{33\cdots 78}{22\cdots 45}a^{15}-\frac{47\cdots 57}{22\cdots 45}a^{14}-\frac{32\cdots 11}{22\cdots 45}a^{13}+\frac{85\cdots 81}{22\cdots 45}a^{12}+\frac{16\cdots 69}{22\cdots 45}a^{11}-\frac{65\cdots 19}{22\cdots 45}a^{10}-\frac{42\cdots 68}{22\cdots 45}a^{9}+\frac{22\cdots 64}{22\cdots 45}a^{8}+\frac{55\cdots 91}{22\cdots 45}a^{7}-\frac{34\cdots 66}{22\cdots 45}a^{6}-\frac{32\cdots 09}{22\cdots 45}a^{5}+\frac{23\cdots 68}{22\cdots 45}a^{4}+\frac{13\cdots 70}{45\cdots 49}a^{3}-\frac{10\cdots 16}{45\cdots 49}a^{2}-\frac{65\cdots 46}{45\cdots 49}a+\frac{67\cdots 06}{45\cdots 49}$
|
| |
| Regulator: | \( 3696529687160 \) (assuming GRH) |
| |
| Unit signature rank: | \( 14 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 3696529687160 \cdot 1}{2\cdot\sqrt{21333423461884919389012763702392578125}}\cr\approx \mathstrut & 0.419598414981616 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{310 -62 \sqrt{5}})\), 5.5.923521.1, 10.10.2665284492003125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | $20$ | R | $20$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | $20$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(31\)
| 31.1.10.9a1.1 | $x^{10} + 31$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |
| 31.1.10.9a1.1 | $x^{10} + 31$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |