Normalized defining polynomial
\( x^{20} - x^{19} - 51 x^{18} + 48 x^{17} + 998 x^{16} - 854 x^{15} - 9572 x^{14} + 7010 x^{13} + \cdots - 43 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(20, 0)$ |
| |
| Discriminant: |
\(284589332775604260722209388186521117\)
\(\medspace = 11^{18}\cdot 13^{15}\)
|
| |
| Root discriminant: | \(59.25\) |
| |
| Galois root discriminant: | $11^{9/10}13^{3/4}\approx 59.253080108324006$ | ||
| Ramified primes: |
\(11\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(143=11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(8,·)$, $\chi_{143}(73,·)$, $\chi_{143}(138,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(18,·)$, $\chi_{143}(83,·)$, $\chi_{143}(21,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(96,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(109,·)$, $\chi_{143}(112,·)$, $\chi_{143}(53,·)$, $\chi_{143}(57,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{419}a^{15}-\frac{204}{419}a^{14}+\frac{180}{419}a^{13}-\frac{150}{419}a^{12}+\frac{146}{419}a^{11}-\frac{141}{419}a^{10}+\frac{108}{419}a^{9}-\frac{162}{419}a^{8}+\frac{204}{419}a^{7}-\frac{75}{419}a^{6}-\frac{143}{419}a^{5}-\frac{86}{419}a^{4}+\frac{76}{419}a^{3}+\frac{49}{419}a^{2}+\frac{22}{419}a-\frac{91}{419}$, $\frac{1}{419}a^{16}+\frac{45}{419}a^{14}+\frac{117}{419}a^{13}+\frac{133}{419}a^{12}-\frac{106}{419}a^{11}-\frac{164}{419}a^{10}+\frac{82}{419}a^{9}-\frac{162}{419}a^{8}+\frac{60}{419}a^{7}+\frac{60}{419}a^{6}+\frac{72}{419}a^{5}+\frac{130}{419}a^{4}+\frac{50}{419}a^{3}-\frac{38}{419}a^{2}+\frac{207}{419}a-\frac{128}{419}$, $\frac{1}{419}a^{17}+\frac{79}{419}a^{14}-\frac{6}{419}a^{13}-\frac{60}{419}a^{12}-\frac{30}{419}a^{11}+\frac{142}{419}a^{10}+\frac{6}{419}a^{9}-\frac{192}{419}a^{8}+\frac{98}{419}a^{7}+\frac{95}{419}a^{6}-\frac{139}{419}a^{5}+\frac{149}{419}a^{4}-\frac{106}{419}a^{3}+\frac{97}{419}a^{2}+\frac{139}{419}a-\frac{95}{419}$, $\frac{1}{414391}a^{18}+\frac{24}{414391}a^{17}-\frac{381}{414391}a^{16}-\frac{149}{414391}a^{15}-\frac{34107}{414391}a^{14}+\frac{120746}{414391}a^{13}-\frac{188057}{414391}a^{12}+\frac{138086}{414391}a^{11}-\frac{382}{989}a^{10}+\frac{77747}{414391}a^{9}-\frac{2858}{9637}a^{8}+\frac{71345}{414391}a^{7}+\frac{29063}{414391}a^{6}-\frac{182794}{414391}a^{5}+\frac{179277}{414391}a^{4}+\frac{29472}{414391}a^{3}-\frac{180263}{414391}a^{2}-\frac{156900}{414391}a-\frac{3250}{9637}$, $\frac{1}{55\cdots 49}a^{19}+\frac{20\cdots 01}{55\cdots 49}a^{18}-\frac{13\cdots 92}{55\cdots 49}a^{17}-\frac{62\cdots 29}{55\cdots 49}a^{16}-\frac{24\cdots 70}{55\cdots 49}a^{15}+\frac{24\cdots 06}{55\cdots 49}a^{14}-\frac{26\cdots 70}{55\cdots 49}a^{13}-\frac{11\cdots 52}{23\cdots 63}a^{12}+\frac{21\cdots 66}{55\cdots 49}a^{11}-\frac{53\cdots 68}{23\cdots 63}a^{10}-\frac{21\cdots 27}{55\cdots 49}a^{9}-\frac{40\cdots 83}{55\cdots 49}a^{8}+\frac{63\cdots 91}{55\cdots 49}a^{7}-\frac{23\cdots 11}{55\cdots 49}a^{6}+\frac{20\cdots 13}{55\cdots 49}a^{5}+\frac{24\cdots 55}{55\cdots 49}a^{4}+\frac{27\cdots 63}{55\cdots 49}a^{3}-\frac{12\cdots 57}{55\cdots 49}a^{2}+\frac{14\cdots 37}{55\cdots 49}a+\frac{54\cdots 17}{12\cdots 43}$
| 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: | $C_{2}$, which has order $2$ (assuming GRH) |
|
Unit group
| Rank: | $19$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{97\cdots 54}{55\cdots 49}a^{19}-\frac{10\cdots 64}{55\cdots 49}a^{18}-\frac{49\cdots 49}{55\cdots 49}a^{17}+\frac{48\cdots 53}{55\cdots 49}a^{16}+\frac{96\cdots 37}{55\cdots 49}a^{15}-\frac{87\cdots 90}{55\cdots 49}a^{14}-\frac{92\cdots 16}{55\cdots 49}a^{13}+\frac{72\cdots 40}{55\cdots 49}a^{12}+\frac{47\cdots 64}{55\cdots 49}a^{11}-\frac{29\cdots 93}{55\cdots 49}a^{10}-\frac{13\cdots 49}{55\cdots 49}a^{9}+\frac{61\cdots 57}{55\cdots 49}a^{8}+\frac{92\cdots 02}{23\cdots 63}a^{7}-\frac{63\cdots 23}{55\cdots 49}a^{6}-\frac{18\cdots 05}{55\cdots 49}a^{5}+\frac{28\cdots 68}{55\cdots 49}a^{4}+\frac{71\cdots 21}{55\cdots 49}a^{3}-\frac{34\cdots 72}{55\cdots 49}a^{2}-\frac{90\cdots 38}{55\cdots 49}a-\frac{12\cdots 60}{12\cdots 43}$, $\frac{24\cdots 55}{55\cdots 49}a^{19}+\frac{70\cdots 85}{55\cdots 49}a^{18}-\frac{14\cdots 26}{55\cdots 49}a^{17}-\frac{34\cdots 48}{55\cdots 49}a^{16}+\frac{81\cdots 32}{12\cdots 43}a^{15}+\frac{64\cdots 89}{55\cdots 49}a^{14}-\frac{43\cdots 15}{55\cdots 49}a^{13}-\frac{57\cdots 09}{55\cdots 49}a^{12}+\frac{67\cdots 80}{12\cdots 43}a^{11}+\frac{25\cdots 10}{55\cdots 49}a^{10}-\frac{10\cdots 41}{55\cdots 49}a^{9}-\frac{63\cdots 36}{55\cdots 49}a^{8}+\frac{93\cdots 79}{23\cdots 63}a^{7}+\frac{85\cdots 48}{55\cdots 49}a^{6}-\frac{22\cdots 86}{55\cdots 49}a^{5}-\frac{57\cdots 67}{55\cdots 49}a^{4}+\frac{10\cdots 82}{55\cdots 49}a^{3}+\frac{14\cdots 80}{55\cdots 49}a^{2}-\frac{16\cdots 68}{55\cdots 49}a-\frac{17\cdots 19}{12\cdots 43}$, $\frac{99\cdots 09}{55\cdots 49}a^{19}-\frac{94\cdots 79}{55\cdots 49}a^{18}-\frac{51\cdots 75}{55\cdots 49}a^{17}+\frac{45\cdots 05}{55\cdots 49}a^{16}+\frac{10\cdots 13}{55\cdots 49}a^{15}-\frac{80\cdots 01}{55\cdots 49}a^{14}-\frac{96\cdots 31}{55\cdots 49}a^{13}+\frac{66\cdots 31}{55\cdots 49}a^{12}+\frac{49\cdots 04}{55\cdots 49}a^{11}-\frac{26\cdots 83}{55\cdots 49}a^{10}-\frac{33\cdots 30}{12\cdots 43}a^{9}+\frac{54\cdots 21}{55\cdots 49}a^{8}+\frac{10\cdots 81}{23\cdots 63}a^{7}-\frac{54\cdots 75}{55\cdots 49}a^{6}-\frac{20\cdots 91}{55\cdots 49}a^{5}+\frac{22\cdots 01}{55\cdots 49}a^{4}+\frac{81\cdots 03}{55\cdots 49}a^{3}-\frac{20\cdots 92}{55\cdots 49}a^{2}-\frac{24\cdots 42}{12\cdots 43}a-\frac{16\cdots 22}{12\cdots 43}$, $\frac{64\cdots 08}{55\cdots 49}a^{19}-\frac{88\cdots 59}{55\cdots 49}a^{18}-\frac{32\cdots 60}{55\cdots 49}a^{17}+\frac{42\cdots 84}{55\cdots 49}a^{16}+\frac{61\cdots 37}{55\cdots 49}a^{15}-\frac{76\cdots 14}{55\cdots 49}a^{14}-\frac{57\cdots 45}{55\cdots 49}a^{13}+\frac{64\cdots 81}{55\cdots 49}a^{12}+\frac{27\cdots 48}{55\cdots 49}a^{11}-\frac{26\cdots 84}{55\cdots 49}a^{10}-\frac{75\cdots 40}{55\cdots 49}a^{9}+\frac{55\cdots 29}{55\cdots 49}a^{8}+\frac{50\cdots 35}{23\cdots 63}a^{7}-\frac{57\cdots 60}{55\cdots 49}a^{6}-\frac{95\cdots 22}{55\cdots 49}a^{5}+\frac{26\cdots 45}{55\cdots 49}a^{4}+\frac{37\cdots 01}{55\cdots 49}a^{3}-\frac{29\cdots 80}{55\cdots 49}a^{2}-\frac{50\cdots 32}{55\cdots 49}a-\frac{81\cdots 29}{12\cdots 43}$, $\frac{27\cdots 77}{55\cdots 49}a^{19}-\frac{26\cdots 85}{55\cdots 49}a^{18}-\frac{13\cdots 95}{55\cdots 49}a^{17}+\frac{12\cdots 53}{55\cdots 49}a^{16}+\frac{27\cdots 71}{55\cdots 49}a^{15}-\frac{22\cdots 62}{55\cdots 49}a^{14}-\frac{26\cdots 28}{55\cdots 49}a^{13}+\frac{19\cdots 83}{55\cdots 49}a^{12}+\frac{13\cdots 00}{55\cdots 49}a^{11}-\frac{77\cdots 98}{55\cdots 49}a^{10}-\frac{38\cdots 16}{55\cdots 49}a^{9}+\frac{15\cdots 96}{55\cdots 49}a^{8}+\frac{27\cdots 07}{23\cdots 63}a^{7}-\frac{16\cdots 12}{55\cdots 49}a^{6}-\frac{54\cdots 60}{55\cdots 49}a^{5}+\frac{73\cdots 53}{55\cdots 49}a^{4}+\frac{22\cdots 37}{55\cdots 49}a^{3}-\frac{86\cdots 04}{55\cdots 49}a^{2}-\frac{28\cdots 62}{55\cdots 49}a-\frac{40\cdots 19}{12\cdots 43}$, $\frac{16\cdots 17}{55\cdots 49}a^{19}-\frac{18\cdots 38}{55\cdots 49}a^{18}-\frac{83\cdots 35}{55\cdots 49}a^{17}+\frac{88\cdots 89}{55\cdots 49}a^{16}+\frac{16\cdots 50}{55\cdots 49}a^{15}-\frac{15\cdots 15}{55\cdots 49}a^{14}-\frac{15\cdots 76}{55\cdots 49}a^{13}+\frac{13\cdots 12}{55\cdots 49}a^{12}+\frac{77\cdots 52}{55\cdots 49}a^{11}-\frac{53\cdots 67}{55\cdots 49}a^{10}-\frac{21\cdots 30}{55\cdots 49}a^{9}+\frac{10\cdots 50}{55\cdots 49}a^{8}+\frac{15\cdots 16}{23\cdots 63}a^{7}-\frac{11\cdots 35}{55\cdots 49}a^{6}-\frac{29\cdots 13}{55\cdots 49}a^{5}+\frac{48\cdots 46}{55\cdots 49}a^{4}+\frac{11\cdots 04}{55\cdots 49}a^{3}-\frac{49\cdots 72}{55\cdots 49}a^{2}-\frac{15\cdots 87}{55\cdots 49}a-\frac{24\cdots 51}{12\cdots 43}$, $\frac{18\cdots 27}{55\cdots 49}a^{19}-\frac{45\cdots 48}{12\cdots 43}a^{18}-\frac{95\cdots 58}{55\cdots 49}a^{17}+\frac{93\cdots 98}{55\cdots 49}a^{16}+\frac{18\cdots 64}{55\cdots 49}a^{15}-\frac{16\cdots 94}{55\cdots 49}a^{14}-\frac{17\cdots 48}{55\cdots 49}a^{13}+\frac{13\cdots 43}{55\cdots 49}a^{12}+\frac{91\cdots 86}{55\cdots 49}a^{11}-\frac{54\cdots 65}{55\cdots 49}a^{10}-\frac{26\cdots 04}{55\cdots 49}a^{9}+\frac{11\cdots 44}{55\cdots 49}a^{8}+\frac{18\cdots 11}{23\cdots 63}a^{7}-\frac{10\cdots 93}{55\cdots 49}a^{6}-\frac{37\cdots 09}{55\cdots 49}a^{5}+\frac{46\cdots 61}{55\cdots 49}a^{4}+\frac{15\cdots 44}{55\cdots 49}a^{3}-\frac{51\cdots 99}{55\cdots 49}a^{2}-\frac{20\cdots 74}{55\cdots 49}a-\frac{28\cdots 43}{12\cdots 43}$, $\frac{14\cdots 72}{55\cdots 49}a^{19}-\frac{18\cdots 28}{55\cdots 49}a^{18}-\frac{72\cdots 13}{55\cdots 49}a^{17}+\frac{90\cdots 67}{55\cdots 49}a^{16}+\frac{14\cdots 34}{55\cdots 49}a^{15}-\frac{16\cdots 62}{55\cdots 49}a^{14}-\frac{13\cdots 56}{55\cdots 49}a^{13}+\frac{13\cdots 21}{55\cdots 49}a^{12}+\frac{65\cdots 27}{55\cdots 49}a^{11}-\frac{57\cdots 23}{55\cdots 49}a^{10}-\frac{17\cdots 35}{55\cdots 49}a^{9}+\frac{12\cdots 61}{55\cdots 49}a^{8}+\frac{12\cdots 32}{23\cdots 63}a^{7}-\frac{13\cdots 43}{55\cdots 49}a^{6}-\frac{22\cdots 85}{55\cdots 49}a^{5}+\frac{62\cdots 23}{55\cdots 49}a^{4}+\frac{88\cdots 58}{55\cdots 49}a^{3}-\frac{81\cdots 24}{55\cdots 49}a^{2}-\frac{11\cdots 19}{55\cdots 49}a-\frac{18\cdots 87}{12\cdots 43}$, $\frac{49\cdots 86}{55\cdots 49}a^{19}-\frac{37\cdots 27}{55\cdots 49}a^{18}-\frac{25\cdots 71}{55\cdots 49}a^{17}+\frac{18\cdots 71}{55\cdots 49}a^{16}+\frac{51\cdots 26}{55\cdots 49}a^{15}-\frac{32\cdots 29}{55\cdots 49}a^{14}-\frac{50\cdots 59}{55\cdots 49}a^{13}+\frac{26\cdots 13}{55\cdots 49}a^{12}+\frac{26\cdots 97}{55\cdots 49}a^{11}-\frac{10\cdots 58}{55\cdots 49}a^{10}-\frac{79\cdots 51}{55\cdots 49}a^{9}+\frac{22\cdots 94}{55\cdots 49}a^{8}+\frac{13\cdots 42}{55\cdots 41}a^{7}-\frac{24\cdots 66}{55\cdots 49}a^{6}-\frac{12\cdots 61}{55\cdots 49}a^{5}+\frac{12\cdots 14}{55\cdots 49}a^{4}+\frac{51\cdots 24}{55\cdots 49}a^{3}-\frac{17\cdots 32}{55\cdots 49}a^{2}-\frac{68\cdots 69}{55\cdots 49}a-\frac{92\cdots 99}{12\cdots 43}$, $\frac{54\cdots 54}{55\cdots 49}a^{19}-\frac{63\cdots 17}{55\cdots 49}a^{18}-\frac{27\cdots 25}{55\cdots 49}a^{17}+\frac{30\cdots 62}{55\cdots 49}a^{16}+\frac{52\cdots 28}{55\cdots 49}a^{15}-\frac{53\cdots 05}{55\cdots 49}a^{14}-\frac{48\cdots 18}{55\cdots 49}a^{13}+\frac{43\cdots 17}{55\cdots 49}a^{12}+\frac{23\cdots 14}{55\cdots 49}a^{11}-\frac{17\cdots 47}{55\cdots 49}a^{10}-\frac{61\cdots 10}{55\cdots 49}a^{9}+\frac{33\cdots 14}{55\cdots 49}a^{8}+\frac{39\cdots 43}{23\cdots 63}a^{7}-\frac{31\cdots 54}{55\cdots 49}a^{6}-\frac{67\cdots 49}{55\cdots 49}a^{5}+\frac{11\cdots 78}{55\cdots 49}a^{4}+\frac{21\cdots 96}{55\cdots 49}a^{3}-\frac{74\cdots 15}{55\cdots 49}a^{2}-\frac{13\cdots 36}{55\cdots 49}a-\frac{17\cdots 03}{12\cdots 43}$, $\frac{16\cdots 00}{55\cdots 49}a^{19}-\frac{25\cdots 78}{55\cdots 49}a^{18}-\frac{83\cdots 23}{55\cdots 49}a^{17}+\frac{12\cdots 95}{55\cdots 49}a^{16}+\frac{15\cdots 56}{55\cdots 49}a^{15}-\frac{22\cdots 55}{55\cdots 49}a^{14}-\frac{61\cdots 14}{23\cdots 63}a^{13}+\frac{18\cdots 40}{55\cdots 49}a^{12}+\frac{65\cdots 61}{55\cdots 49}a^{11}-\frac{73\cdots 95}{55\cdots 49}a^{10}-\frac{16\cdots 66}{55\cdots 49}a^{9}+\frac{14\cdots 07}{55\cdots 49}a^{8}+\frac{23\cdots 26}{55\cdots 49}a^{7}-\frac{14\cdots 96}{55\cdots 49}a^{6}-\frac{16\cdots 25}{55\cdots 49}a^{5}+\frac{53\cdots 28}{55\cdots 49}a^{4}+\frac{55\cdots 59}{55\cdots 49}a^{3}-\frac{29\cdots 07}{55\cdots 49}a^{2}-\frac{52\cdots 44}{55\cdots 49}a-\frac{77\cdots 54}{12\cdots 43}$, $\frac{19\cdots 73}{55\cdots 49}a^{19}-\frac{21\cdots 88}{55\cdots 49}a^{18}-\frac{99\cdots 15}{55\cdots 49}a^{17}+\frac{10\cdots 43}{55\cdots 49}a^{16}+\frac{84\cdots 93}{23\cdots 63}a^{15}-\frac{18\cdots 69}{55\cdots 49}a^{14}-\frac{18\cdots 93}{55\cdots 49}a^{13}+\frac{15\cdots 40}{55\cdots 49}a^{12}+\frac{40\cdots 80}{23\cdots 63}a^{11}-\frac{64\cdots 75}{55\cdots 49}a^{10}-\frac{26\cdots 88}{55\cdots 49}a^{9}+\frac{13\cdots 42}{55\cdots 49}a^{8}+\frac{42\cdots 82}{55\cdots 49}a^{7}-\frac{59\cdots 87}{23\cdots 63}a^{6}-\frac{36\cdots 84}{55\cdots 49}a^{5}+\frac{61\cdots 85}{55\cdots 49}a^{4}+\frac{15\cdots 09}{55\cdots 49}a^{3}-\frac{70\cdots 18}{55\cdots 49}a^{2}-\frac{20\cdots 38}{55\cdots 49}a-\frac{28\cdots 24}{12\cdots 43}$, $\frac{13\cdots 77}{55\cdots 49}a^{19}-\frac{21\cdots 54}{55\cdots 49}a^{18}-\frac{65\cdots 86}{55\cdots 49}a^{17}+\frac{10\cdots 72}{55\cdots 49}a^{16}+\frac{12\cdots 48}{55\cdots 49}a^{15}-\frac{18\cdots 57}{55\cdots 49}a^{14}-\frac{11\cdots 05}{55\cdots 49}a^{13}+\frac{65\cdots 82}{23\cdots 63}a^{12}+\frac{52\cdots 84}{55\cdots 49}a^{11}-\frac{26\cdots 94}{23\cdots 63}a^{10}-\frac{13\cdots 25}{55\cdots 49}a^{9}+\frac{12\cdots 34}{55\cdots 49}a^{8}+\frac{19\cdots 67}{55\cdots 49}a^{7}-\frac{12\cdots 39}{55\cdots 49}a^{6}-\frac{16\cdots 02}{55\cdots 49}a^{5}+\frac{55\cdots 48}{55\cdots 49}a^{4}+\frac{62\cdots 57}{55\cdots 49}a^{3}-\frac{68\cdots 54}{55\cdots 49}a^{2}-\frac{82\cdots 93}{55\cdots 49}a-\frac{11\cdots 42}{12\cdots 43}$, $\frac{25\cdots 10}{55\cdots 49}a^{19}-\frac{32\cdots 04}{55\cdots 49}a^{18}-\frac{53\cdots 12}{23\cdots 63}a^{17}+\frac{15\cdots 15}{55\cdots 49}a^{16}+\frac{23\cdots 31}{55\cdots 49}a^{15}-\frac{26\cdots 49}{55\cdots 49}a^{14}-\frac{20\cdots 42}{55\cdots 49}a^{13}+\frac{20\cdots 15}{55\cdots 49}a^{12}+\frac{91\cdots 39}{55\cdots 49}a^{11}-\frac{71\cdots 53}{55\cdots 49}a^{10}-\frac{21\cdots 67}{55\cdots 49}a^{9}+\frac{11\cdots 61}{55\cdots 49}a^{8}+\frac{25\cdots 16}{55\cdots 49}a^{7}-\frac{59\cdots 73}{55\cdots 49}a^{6}-\frac{13\cdots 99}{55\cdots 49}a^{5}-\frac{14\cdots 75}{55\cdots 49}a^{4}+\frac{13\cdots 68}{55\cdots 49}a^{3}+\frac{10\cdots 64}{55\cdots 49}a^{2}+\frac{31\cdots 14}{55\cdots 49}a-\frac{61\cdots 80}{12\cdots 43}$, $\frac{77\cdots 57}{55\cdots 49}a^{19}-\frac{28\cdots 43}{55\cdots 49}a^{18}-\frac{34\cdots 81}{55\cdots 49}a^{17}+\frac{13\cdots 25}{55\cdots 49}a^{16}+\frac{52\cdots 29}{55\cdots 49}a^{15}-\frac{24\cdots 72}{55\cdots 49}a^{14}-\frac{30\cdots 93}{55\cdots 49}a^{13}+\frac{20\cdots 45}{55\cdots 49}a^{12}+\frac{16\cdots 56}{55\cdots 49}a^{11}-\frac{81\cdots 39}{55\cdots 49}a^{10}+\frac{16\cdots 08}{23\cdots 63}a^{9}+\frac{16\cdots 39}{55\cdots 49}a^{8}-\frac{12\cdots 91}{55\cdots 49}a^{7}-\frac{17\cdots 84}{55\cdots 49}a^{6}+\frac{15\cdots 32}{55\cdots 49}a^{5}+\frac{37\cdots 33}{23\cdots 63}a^{4}-\frac{72\cdots 57}{55\cdots 49}a^{3}-\frac{32\cdots 89}{12\cdots 43}a^{2}+\frac{87\cdots 48}{55\cdots 49}a+\frac{29\cdots 30}{12\cdots 43}$, $\frac{17\cdots 37}{55\cdots 49}a^{19}+\frac{16\cdots 39}{55\cdots 49}a^{18}-\frac{42\cdots 78}{23\cdots 63}a^{17}-\frac{79\cdots 86}{55\cdots 49}a^{16}+\frac{21\cdots 71}{55\cdots 49}a^{15}+\frac{14\cdots 60}{55\cdots 49}a^{14}-\frac{23\cdots 20}{55\cdots 49}a^{13}-\frac{12\cdots 05}{55\cdots 49}a^{12}+\frac{14\cdots 22}{55\cdots 49}a^{11}+\frac{51\cdots 85}{55\cdots 49}a^{10}-\frac{11\cdots 83}{12\cdots 43}a^{9}-\frac{10\cdots 49}{55\cdots 49}a^{8}+\frac{87\cdots 01}{55\cdots 49}a^{7}+\frac{11\cdots 05}{55\cdots 49}a^{6}-\frac{83\cdots 73}{55\cdots 49}a^{5}-\frac{51\cdots 76}{55\cdots 49}a^{4}+\frac{36\cdots 95}{55\cdots 49}a^{3}+\frac{69\cdots 49}{55\cdots 49}a^{2}-\frac{11\cdots 70}{12\cdots 43}a-\frac{75\cdots 85}{12\cdots 43}$, $\frac{73\cdots 88}{12\cdots 43}a^{19}-\frac{34\cdots 96}{55\cdots 49}a^{18}-\frac{16\cdots 17}{55\cdots 49}a^{17}+\frac{16\cdots 67}{55\cdots 49}a^{16}+\frac{31\cdots 03}{55\cdots 49}a^{15}-\frac{29\cdots 10}{55\cdots 49}a^{14}-\frac{29\cdots 82}{55\cdots 49}a^{13}+\frac{24\cdots 22}{55\cdots 49}a^{12}+\frac{15\cdots 96}{55\cdots 49}a^{11}-\frac{10\cdots 94}{55\cdots 49}a^{10}-\frac{42\cdots 10}{55\cdots 49}a^{9}+\frac{48\cdots 08}{12\cdots 43}a^{8}+\frac{68\cdots 67}{55\cdots 49}a^{7}-\frac{21\cdots 36}{55\cdots 49}a^{6}-\frac{59\cdots 47}{55\cdots 49}a^{5}+\frac{97\cdots 80}{55\cdots 49}a^{4}+\frac{24\cdots 07}{55\cdots 49}a^{3}-\frac{11\cdots 69}{55\cdots 49}a^{2}-\frac{31\cdots 46}{55\cdots 49}a-\frac{43\cdots 41}{12\cdots 43}$, $\frac{54\cdots 30}{55\cdots 49}a^{19}-\frac{80\cdots 11}{55\cdots 49}a^{18}-\frac{27\cdots 40}{55\cdots 49}a^{17}+\frac{38\cdots 89}{55\cdots 49}a^{16}+\frac{51\cdots 16}{55\cdots 49}a^{15}-\frac{69\cdots 69}{55\cdots 49}a^{14}-\frac{46\cdots 32}{55\cdots 49}a^{13}+\frac{57\cdots 59}{55\cdots 49}a^{12}+\frac{21\cdots 63}{55\cdots 49}a^{11}-\frac{22\cdots 52}{55\cdots 49}a^{10}-\frac{57\cdots 44}{55\cdots 49}a^{9}+\frac{46\cdots 90}{55\cdots 49}a^{8}+\frac{83\cdots 19}{55\cdots 49}a^{7}-\frac{46\cdots 63}{55\cdots 49}a^{6}-\frac{67\cdots 61}{55\cdots 49}a^{5}+\frac{19\cdots 48}{55\cdots 49}a^{4}+\frac{29\cdots 99}{55\cdots 49}a^{3}-\frac{25\cdots 57}{55\cdots 49}a^{2}-\frac{63\cdots 00}{55\cdots 49}a-\frac{94\cdots 06}{12\cdots 43}$, $\frac{18\cdots 61}{55\cdots 49}a^{19}-\frac{15\cdots 73}{55\cdots 49}a^{18}-\frac{96\cdots 91}{55\cdots 49}a^{17}+\frac{74\cdots 15}{55\cdots 49}a^{16}+\frac{18\cdots 89}{55\cdots 49}a^{15}-\frac{13\cdots 09}{55\cdots 49}a^{14}-\frac{18\cdots 80}{55\cdots 49}a^{13}+\frac{10\cdots 82}{55\cdots 49}a^{12}+\frac{92\cdots 72}{55\cdots 49}a^{11}-\frac{42\cdots 12}{55\cdots 49}a^{10}-\frac{26\cdots 41}{55\cdots 49}a^{9}+\frac{85\cdots 37}{55\cdots 49}a^{8}+\frac{40\cdots 36}{55\cdots 49}a^{7}-\frac{87\cdots 09}{55\cdots 49}a^{6}-\frac{32\cdots 41}{55\cdots 49}a^{5}+\frac{45\cdots 34}{55\cdots 49}a^{4}+\frac{11\cdots 75}{55\cdots 49}a^{3}-\frac{11\cdots 94}{55\cdots 49}a^{2}-\frac{82\cdots 28}{55\cdots 49}a-\frac{41\cdots 27}{55\cdots 41}$
|
| |
| Regulator: | \( 168224843913 \) (assuming GRH) |
| |
| Unit signature rank: | \( 19 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 168224843913 \cdot 1}{2\cdot\sqrt{284589332775604260722209388186521117}}\cr\approx \mathstrut & 0.165329648688129 \end{aligned}\] (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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{286 +66 \sqrt{13}})\), \(\Q(\zeta_{11})^+\), 10.10.79589952003133.1 |
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 | $20$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | $20$ | $20$ | R | R | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | $20$ | ${\href{/padicField/43.1.0.1}{1} }^{20}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.2.10.18a1.7 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35950 x + 1057$ | $10$ | $2$ | $18$ | 20T1 | $$[\ ]_{10}^{2}$$ |
|
\(13\)
| 13.5.4.15a1.2 | $x^{20} + 16 x^{16} + 44 x^{15} + 96 x^{12} + 528 x^{11} + 726 x^{10} + 256 x^{8} + 2112 x^{7} + 5808 x^{6} + 5324 x^{5} + 256 x^{4} + 2816 x^{3} + 11629 x^{2} + 21296 x + 14641$ | $4$ | $5$ | $15$ | 20T1 | $$[\ ]_{4}^{5}$$ |