Normalized defining polynomial
\( x^{20} - 4 x^{19} - 35 x^{18} + 146 x^{17} + 455 x^{16} - 2044 x^{15} - 2696 x^{14} + 13960 x^{13} + \cdots + 1 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(20, 0)$ |
| |
| Discriminant: |
\(1470391355634309152000000000000000\)
\(\medspace = 2^{20}\cdot 5^{15}\cdot 11^{16}\)
|
| |
| Root discriminant: | \(45.54\) |
| |
| Galois root discriminant: | $2\cdot 5^{3/4}11^{4/5}\approx 45.53775823431064$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(67,·)$, $\chi_{220}(69,·)$, $\chi_{220}(9,·)$, $\chi_{220}(203,·)$, $\chi_{220}(141,·)$, $\chi_{220}(207,·)$, $\chi_{220}(81,·)$, $\chi_{220}(3,·)$, $\chi_{220}(23,·)$, $\chi_{220}(89,·)$, $\chi_{220}(27,·)$, $\chi_{220}(163,·)$, $\chi_{220}(103,·)$, $\chi_{220}(169,·)$, $\chi_{220}(47,·)$, $\chi_{220}(49,·)$, $\chi_{220}(147,·)$, $\chi_{220}(181,·)$, $\chi_{220}(201,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{884372941}a^{18}-\frac{433934556}{884372941}a^{17}+\frac{282192243}{884372941}a^{16}-\frac{394641417}{884372941}a^{15}+\frac{139771924}{884372941}a^{14}-\frac{322974840}{884372941}a^{13}+\frac{30938985}{884372941}a^{12}-\frac{151298292}{884372941}a^{11}-\frac{112545307}{884372941}a^{10}-\frac{215392746}{884372941}a^{9}+\frac{388285222}{884372941}a^{8}+\frac{64050219}{884372941}a^{7}+\frac{71148378}{884372941}a^{6}+\frac{174835562}{884372941}a^{5}+\frac{438006565}{884372941}a^{4}-\frac{263417180}{884372941}a^{3}+\frac{55954187}{884372941}a^{2}-\frac{704594}{884372941}a-\frac{82193344}{884372941}$, $\frac{1}{23\cdots 39}a^{19}+\frac{132546192707036}{23\cdots 39}a^{18}-\frac{23\cdots 25}{23\cdots 39}a^{17}+\frac{79\cdots 12}{23\cdots 39}a^{16}-\frac{21\cdots 62}{23\cdots 39}a^{15}-\frac{27\cdots 64}{23\cdots 39}a^{14}-\frac{88\cdots 02}{23\cdots 39}a^{13}-\frac{27\cdots 16}{23\cdots 39}a^{12}-\frac{90\cdots 53}{23\cdots 39}a^{11}-\frac{10\cdots 56}{23\cdots 39}a^{10}-\frac{72\cdots 42}{23\cdots 39}a^{9}-\frac{45\cdots 87}{23\cdots 39}a^{8}+\frac{52\cdots 81}{23\cdots 39}a^{7}-\frac{83\cdots 56}{23\cdots 39}a^{6}+\frac{41\cdots 37}{23\cdots 39}a^{5}-\frac{11\cdots 31}{23\cdots 39}a^{4}-\frac{47\cdots 53}{23\cdots 39}a^{3}+\frac{90\cdots 98}{23\cdots 39}a^{2}+\frac{21\cdots 43}{23\cdots 39}a-\frac{41\cdots 78}{23\cdots 39}$
| 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: |
$\frac{174929876}{884372941}a^{19}-\frac{673899290}{884372941}a^{18}-\frac{6218535724}{884372941}a^{17}+\frac{24606686920}{884372941}a^{16}+\frac{83108702336}{884372941}a^{15}-\frac{344738896726}{884372941}a^{14}-\frac{521103623880}{884372941}a^{13}+\frac{2357487850459}{884372941}a^{12}+\frac{1524396168412}{884372941}a^{11}-\frac{8266906522891}{884372941}a^{10}-\frac{1610301276772}{884372941}a^{9}+\frac{14405100955089}{884372941}a^{8}-\frac{544506565172}{884372941}a^{7}-\frac{11348805468261}{884372941}a^{6}+\frac{1341886555748}{884372941}a^{5}+\frac{3648990445211}{884372941}a^{4}-\frac{498517107844}{884372941}a^{3}-\frac{354778587694}{884372941}a^{2}+\frac{70667211302}{884372941}a-\frac{1635833215}{884372941}$, $\frac{12\cdots 64}{23\cdots 39}a^{19}-\frac{48\cdots 20}{23\cdots 39}a^{18}-\frac{44\cdots 36}{23\cdots 39}a^{17}+\frac{17\cdots 30}{23\cdots 39}a^{16}+\frac{59\cdots 24}{23\cdots 39}a^{15}-\frac{24\cdots 24}{23\cdots 39}a^{14}-\frac{37\cdots 00}{23\cdots 39}a^{13}+\frac{17\cdots 76}{23\cdots 39}a^{12}+\frac{11\cdots 16}{23\cdots 39}a^{11}-\frac{60\cdots 68}{23\cdots 39}a^{10}-\frac{12\cdots 28}{23\cdots 39}a^{9}+\frac{10\cdots 11}{23\cdots 39}a^{8}-\frac{23\cdots 68}{23\cdots 39}a^{7}-\frac{87\cdots 34}{23\cdots 39}a^{6}+\frac{92\cdots 76}{23\cdots 39}a^{5}+\frac{30\cdots 29}{23\cdots 39}a^{4}-\frac{39\cdots 36}{23\cdots 39}a^{3}-\frac{35\cdots 32}{23\cdots 39}a^{2}+\frac{57\cdots 32}{23\cdots 39}a+\frac{36\cdots 24}{23\cdots 39}$, $\frac{47\cdots 04}{23\cdots 39}a^{19}-\frac{18\cdots 40}{23\cdots 39}a^{18}-\frac{16\cdots 46}{23\cdots 39}a^{17}+\frac{67\cdots 40}{23\cdots 39}a^{16}+\frac{22\cdots 48}{23\cdots 39}a^{15}-\frac{94\cdots 54}{23\cdots 39}a^{14}-\frac{14\cdots 20}{23\cdots 39}a^{13}+\frac{64\cdots 56}{23\cdots 39}a^{12}+\frac{41\cdots 16}{23\cdots 39}a^{11}-\frac{22\cdots 04}{23\cdots 39}a^{10}-\frac{42\cdots 58}{23\cdots 39}a^{9}+\frac{39\cdots 16}{23\cdots 39}a^{8}-\frac{19\cdots 88}{23\cdots 39}a^{7}-\frac{31\cdots 64}{23\cdots 39}a^{6}+\frac{44\cdots 70}{23\cdots 39}a^{5}+\frac{10\cdots 39}{23\cdots 39}a^{4}-\frac{18\cdots 86}{23\cdots 39}a^{3}-\frac{10\cdots 97}{23\cdots 39}a^{2}+\frac{25\cdots 62}{23\cdots 39}a-\frac{12\cdots 68}{23\cdots 39}$, $\frac{31\cdots 24}{23\cdots 39}a^{19}-\frac{12\cdots 90}{23\cdots 39}a^{18}-\frac{11\cdots 26}{23\cdots 39}a^{17}+\frac{43\cdots 20}{23\cdots 39}a^{16}+\frac{15\cdots 64}{23\cdots 39}a^{15}-\frac{61\cdots 74}{23\cdots 39}a^{14}-\frac{97\cdots 00}{23\cdots 39}a^{13}+\frac{42\cdots 41}{23\cdots 39}a^{12}+\frac{29\cdots 16}{23\cdots 39}a^{11}-\frac{14\cdots 89}{23\cdots 39}a^{10}-\frac{36\cdots 78}{23\cdots 39}a^{9}+\frac{25\cdots 71}{23\cdots 39}a^{8}+\frac{11\cdots 32}{23\cdots 39}a^{7}-\frac{20\cdots 19}{23\cdots 39}a^{6}+\frac{17\cdots 62}{23\cdots 39}a^{5}+\frac{67\cdots 89}{23\cdots 39}a^{4}-\frac{81\cdots 26}{23\cdots 39}a^{3}-\frac{66\cdots 27}{23\cdots 39}a^{2}+\frac{13\cdots 92}{23\cdots 39}a-\frac{61\cdots 84}{23\cdots 39}$, $\frac{18\cdots 56}{23\cdots 39}a^{19}-\frac{69\cdots 20}{23\cdots 39}a^{18}-\frac{67\cdots 74}{23\cdots 39}a^{17}+\frac{25\cdots 55}{23\cdots 39}a^{16}+\frac{93\cdots 56}{23\cdots 39}a^{15}-\frac{35\cdots 66}{23\cdots 39}a^{14}-\frac{62\cdots 60}{23\cdots 39}a^{13}+\frac{24\cdots 64}{23\cdots 39}a^{12}+\frac{20\cdots 24}{23\cdots 39}a^{11}-\frac{86\cdots 86}{23\cdots 39}a^{10}-\frac{31\cdots 22}{23\cdots 39}a^{9}+\frac{15\cdots 94}{23\cdots 39}a^{8}+\frac{17\cdots 28}{23\cdots 39}a^{7}-\frac{12\cdots 16}{23\cdots 39}a^{6}-\frac{24\cdots 46}{23\cdots 39}a^{5}+\frac{43\cdots 96}{23\cdots 39}a^{4}-\frac{83\cdots 54}{23\cdots 39}a^{3}-\frac{47\cdots 73}{23\cdots 39}a^{2}+\frac{45\cdots 78}{23\cdots 39}a+\frac{26\cdots 19}{23\cdots 39}$, $\frac{25\cdots 60}{23\cdots 39}a^{19}-\frac{94\cdots 95}{23\cdots 39}a^{18}-\frac{90\cdots 70}{23\cdots 39}a^{17}+\frac{34\cdots 73}{23\cdots 39}a^{16}+\frac{12\cdots 80}{23\cdots 39}a^{15}-\frac{48\cdots 25}{23\cdots 39}a^{14}-\frac{79\cdots 60}{23\cdots 39}a^{13}+\frac{33\cdots 60}{23\cdots 39}a^{12}+\frac{25\cdots 00}{23\cdots 39}a^{11}-\frac{11\cdots 60}{23\cdots 39}a^{10}-\frac{34\cdots 00}{23\cdots 39}a^{9}+\frac{21\cdots 20}{23\cdots 39}a^{8}+\frac{99\cdots 24}{23\cdots 39}a^{7}-\frac{17\cdots 06}{23\cdots 39}a^{6}+\frac{87\cdots 86}{23\cdots 39}a^{5}+\frac{60\cdots 15}{23\cdots 39}a^{4}-\frac{59\cdots 70}{23\cdots 39}a^{3}-\frac{67\cdots 25}{23\cdots 39}a^{2}+\frac{12\cdots 70}{23\cdots 39}a-\frac{26\cdots 55}{23\cdots 39}$, $\frac{63\cdots 42}{23\cdots 39}a^{19}-\frac{23\cdots 17}{23\cdots 39}a^{18}-\frac{22\cdots 28}{23\cdots 39}a^{17}+\frac{86\cdots 53}{23\cdots 39}a^{16}+\frac{30\cdots 04}{23\cdots 39}a^{15}-\frac{12\cdots 52}{23\cdots 39}a^{14}-\frac{20\cdots 98}{23\cdots 39}a^{13}+\frac{83\cdots 71}{23\cdots 39}a^{12}+\frac{63\cdots 80}{23\cdots 39}a^{11}-\frac{29\cdots 26}{23\cdots 39}a^{10}-\frac{85\cdots 22}{23\cdots 39}a^{9}+\frac{51\cdots 27}{23\cdots 39}a^{8}+\frac{24\cdots 68}{23\cdots 39}a^{7}-\frac{41\cdots 58}{23\cdots 39}a^{6}+\frac{18\cdots 60}{23\cdots 39}a^{5}+\frac{13\cdots 80}{23\cdots 39}a^{4}-\frac{10\cdots 32}{23\cdots 39}a^{3}-\frac{13\cdots 89}{23\cdots 39}a^{2}+\frac{21\cdots 62}{23\cdots 39}a-\frac{41\cdots 53}{23\cdots 39}$, $\frac{15\cdots 22}{23\cdots 39}a^{19}-\frac{61\cdots 92}{23\cdots 39}a^{18}-\frac{54\cdots 38}{23\cdots 39}a^{17}+\frac{22\cdots 26}{23\cdots 39}a^{16}+\frac{72\cdots 44}{23\cdots 39}a^{15}-\frac{31\cdots 67}{23\cdots 39}a^{14}-\frac{43\cdots 22}{23\cdots 39}a^{13}+\frac{21\cdots 50}{23\cdots 39}a^{12}+\frac{11\cdots 32}{23\cdots 39}a^{11}-\frac{74\cdots 12}{23\cdots 39}a^{10}-\frac{84\cdots 42}{23\cdots 39}a^{9}+\frac{12\cdots 47}{23\cdots 39}a^{8}-\frac{15\cdots 72}{23\cdots 39}a^{7}-\frac{98\cdots 24}{23\cdots 39}a^{6}+\frac{20\cdots 70}{23\cdots 39}a^{5}+\frac{29\cdots 44}{23\cdots 39}a^{4}-\frac{72\cdots 14}{23\cdots 39}a^{3}-\frac{23\cdots 24}{23\cdots 39}a^{2}+\frac{88\cdots 96}{23\cdots 39}a-\frac{91\cdots 05}{23\cdots 39}$, $\frac{23\cdots 76}{23\cdots 39}a^{19}-\frac{88\cdots 92}{23\cdots 39}a^{18}-\frac{85\cdots 80}{23\cdots 39}a^{17}+\frac{32\cdots 26}{23\cdots 39}a^{16}+\frac{11\cdots 04}{23\cdots 39}a^{15}-\frac{45\cdots 01}{23\cdots 39}a^{14}-\frac{75\cdots 92}{23\cdots 39}a^{13}+\frac{31\cdots 27}{23\cdots 39}a^{12}+\frac{24\cdots 08}{23\cdots 39}a^{11}-\frac{10\cdots 82}{23\cdots 39}a^{10}-\frac{33\cdots 40}{23\cdots 39}a^{9}+\frac{19\cdots 33}{23\cdots 39}a^{8}+\frac{10\cdots 72}{23\cdots 39}a^{7}-\frac{15\cdots 45}{23\cdots 39}a^{6}+\frac{75\cdots 30}{23\cdots 39}a^{5}+\frac{48\cdots 68}{23\cdots 39}a^{4}-\frac{52\cdots 40}{23\cdots 39}a^{3}-\frac{40\cdots 32}{23\cdots 39}a^{2}+\frac{11\cdots 90}{23\cdots 39}a-\frac{97\cdots 75}{23\cdots 39}$, $\frac{15\cdots 84}{23\cdots 39}a^{19}-\frac{63\cdots 70}{23\cdots 39}a^{18}-\frac{54\cdots 86}{23\cdots 39}a^{17}+\frac{23\cdots 95}{23\cdots 39}a^{16}+\frac{70\cdots 64}{23\cdots 39}a^{15}-\frac{32\cdots 64}{23\cdots 39}a^{14}-\frac{40\cdots 60}{23\cdots 39}a^{13}+\frac{21\cdots 21}{23\cdots 39}a^{12}+\frac{92\cdots 08}{23\cdots 39}a^{11}-\frac{75\cdots 35}{23\cdots 39}a^{10}+\frac{87\cdots 62}{23\cdots 39}a^{9}+\frac{12\cdots 26}{23\cdots 39}a^{8}-\frac{30\cdots 48}{23\cdots 39}a^{7}-\frac{90\cdots 69}{23\cdots 39}a^{6}+\frac{29\cdots 62}{23\cdots 39}a^{5}+\frac{24\cdots 44}{23\cdots 39}a^{4}-\frac{86\cdots 86}{23\cdots 39}a^{3}-\frac{12\cdots 21}{23\cdots 39}a^{2}+\frac{89\cdots 87}{23\cdots 39}a-\frac{98\cdots 67}{23\cdots 39}$, $\frac{38\cdots 28}{23\cdots 39}a^{19}-\frac{14\cdots 03}{23\cdots 39}a^{18}-\frac{13\cdots 72}{23\cdots 39}a^{17}+\frac{52\cdots 32}{23\cdots 39}a^{16}+\frac{18\cdots 42}{23\cdots 39}a^{15}-\frac{73\cdots 98}{23\cdots 39}a^{14}-\frac{12\cdots 17}{23\cdots 39}a^{13}+\frac{50\cdots 28}{23\cdots 39}a^{12}+\frac{38\cdots 27}{23\cdots 39}a^{11}-\frac{17\cdots 65}{23\cdots 39}a^{10}-\frac{52\cdots 23}{23\cdots 39}a^{9}+\frac{30\cdots 13}{23\cdots 39}a^{8}+\frac{19\cdots 57}{23\cdots 39}a^{7}-\frac{23\cdots 67}{23\cdots 39}a^{6}+\frac{41\cdots 29}{23\cdots 39}a^{5}+\frac{75\cdots 85}{23\cdots 39}a^{4}-\frac{25\cdots 02}{23\cdots 39}a^{3}-\frac{67\cdots 03}{23\cdots 39}a^{2}+\frac{70\cdots 51}{23\cdots 39}a+\frac{33\cdots 83}{23\cdots 39}$, $\frac{32\cdots 04}{23\cdots 39}a^{19}-\frac{27\cdots 50}{23\cdots 39}a^{18}-\frac{45\cdots 16}{23\cdots 39}a^{17}+\frac{92\cdots 35}{23\cdots 39}a^{16}-\frac{10\cdots 16}{23\cdots 39}a^{15}-\frac{11\cdots 84}{23\cdots 39}a^{14}+\frac{25\cdots 60}{23\cdots 39}a^{13}+\frac{61\cdots 01}{23\cdots 39}a^{12}-\frac{21\cdots 24}{23\cdots 39}a^{11}-\frac{10\cdots 35}{23\cdots 39}a^{10}+\frac{80\cdots 72}{23\cdots 39}a^{9}-\frac{18\cdots 34}{23\cdots 39}a^{8}-\frac{14\cdots 28}{23\cdots 39}a^{7}+\frac{77\cdots 31}{23\cdots 39}a^{6}+\frac{10\cdots 32}{23\cdots 39}a^{5}-\frac{66\cdots 36}{23\cdots 39}a^{4}-\frac{34\cdots 36}{23\cdots 39}a^{3}+\frac{16\cdots 78}{23\cdots 39}a^{2}+\frac{38\cdots 21}{23\cdots 39}a-\frac{92\cdots 27}{23\cdots 39}$, $a$, $\frac{15\cdots 40}{23\cdots 39}a^{19}-\frac{53\cdots 90}{23\cdots 39}a^{18}-\frac{55\cdots 90}{23\cdots 39}a^{17}+\frac{19\cdots 65}{23\cdots 39}a^{16}+\frac{79\cdots 96}{23\cdots 39}a^{15}-\frac{27\cdots 10}{23\cdots 39}a^{14}-\frac{55\cdots 40}{23\cdots 39}a^{13}+\frac{19\cdots 25}{23\cdots 39}a^{12}+\frac{20\cdots 40}{23\cdots 39}a^{11}-\frac{68\cdots 39}{23\cdots 39}a^{10}-\frac{37\cdots 70}{23\cdots 39}a^{9}+\frac{12\cdots 60}{23\cdots 39}a^{8}+\frac{36\cdots 80}{23\cdots 39}a^{7}-\frac{10\cdots 05}{23\cdots 39}a^{6}-\frac{19\cdots 78}{23\cdots 39}a^{5}+\frac{38\cdots 75}{23\cdots 39}a^{4}+\frac{50\cdots 70}{23\cdots 39}a^{3}-\frac{46\cdots 35}{23\cdots 39}a^{2}-\frac{19\cdots 99}{23\cdots 39}a+\frac{69\cdots 88}{23\cdots 39}$, $\frac{19\cdots 60}{23\cdots 39}a^{19}-\frac{71\cdots 70}{23\cdots 39}a^{18}-\frac{68\cdots 90}{23\cdots 39}a^{17}+\frac{26\cdots 90}{23\cdots 39}a^{16}+\frac{92\cdots 40}{23\cdots 39}a^{15}-\frac{36\cdots 50}{23\cdots 39}a^{14}-\frac{59\cdots 00}{23\cdots 39}a^{13}+\frac{25\cdots 65}{23\cdots 39}a^{12}+\frac{18\cdots 00}{23\cdots 39}a^{11}-\frac{87\cdots 21}{23\cdots 39}a^{10}-\frac{23\cdots 50}{23\cdots 39}a^{9}+\frac{15\cdots 60}{23\cdots 39}a^{8}+\frac{35\cdots 00}{23\cdots 39}a^{7}-\frac{11\cdots 85}{23\cdots 39}a^{6}+\frac{84\cdots 86}{23\cdots 39}a^{5}+\frac{37\cdots 60}{23\cdots 39}a^{4}-\frac{42\cdots 90}{23\cdots 39}a^{3}-\frac{31\cdots 95}{23\cdots 39}a^{2}+\frac{83\cdots 99}{23\cdots 39}a-\frac{65\cdots 08}{23\cdots 39}$, $\frac{96\cdots 14}{23\cdots 39}a^{19}-\frac{36\cdots 00}{23\cdots 39}a^{18}-\frac{34\cdots 36}{23\cdots 39}a^{17}+\frac{13\cdots 71}{23\cdots 39}a^{16}+\frac{46\cdots 24}{23\cdots 39}a^{15}-\frac{18\cdots 54}{23\cdots 39}a^{14}-\frac{29\cdots 95}{23\cdots 39}a^{13}+\frac{12\cdots 86}{23\cdots 39}a^{12}+\frac{91\cdots 23}{23\cdots 39}a^{11}-\frac{45\cdots 84}{23\cdots 39}a^{10}-\frac{11\cdots 93}{23\cdots 39}a^{9}+\frac{79\cdots 36}{23\cdots 39}a^{8}+\frac{75\cdots 57}{23\cdots 39}a^{7}-\frac{63\cdots 42}{23\cdots 39}a^{6}+\frac{48\cdots 98}{23\cdots 39}a^{5}+\frac{20\cdots 94}{23\cdots 39}a^{4}-\frac{21\cdots 06}{23\cdots 39}a^{3}-\frac{20\cdots 36}{23\cdots 39}a^{2}+\frac{36\cdots 13}{23\cdots 39}a-\frac{12\cdots 48}{23\cdots 39}$, $\frac{11\cdots 20}{23\cdots 39}a^{19}-\frac{47\cdots 08}{23\cdots 39}a^{18}-\frac{38\cdots 89}{23\cdots 39}a^{17}+\frac{17\cdots 76}{23\cdots 39}a^{16}+\frac{47\cdots 96}{23\cdots 39}a^{15}-\frac{24\cdots 59}{23\cdots 39}a^{14}-\frac{25\cdots 16}{23\cdots 39}a^{13}+\frac{16\cdots 62}{23\cdots 39}a^{12}+\frac{43\cdots 48}{23\cdots 39}a^{11}-\frac{57\cdots 52}{23\cdots 39}a^{10}+\frac{82\cdots 99}{23\cdots 39}a^{9}+\frac{98\cdots 57}{23\cdots 39}a^{8}-\frac{34\cdots 75}{23\cdots 39}a^{7}-\frac{76\cdots 67}{23\cdots 39}a^{6}+\frac{31\cdots 81}{23\cdots 39}a^{5}+\frac{24\cdots 71}{23\cdots 39}a^{4}-\frac{91\cdots 43}{23\cdots 39}a^{3}-\frac{28\cdots 46}{23\cdots 39}a^{2}+\frac{75\cdots 43}{23\cdots 39}a-\frac{28\cdots 46}{23\cdots 39}$, $\frac{16\cdots 02}{23\cdots 39}a^{19}-\frac{66\cdots 74}{23\cdots 39}a^{18}-\frac{59\cdots 94}{23\cdots 39}a^{17}+\frac{24\cdots 86}{23\cdots 39}a^{16}+\frac{77\cdots 21}{23\cdots 39}a^{15}-\frac{33\cdots 36}{23\cdots 39}a^{14}-\frac{46\cdots 15}{23\cdots 39}a^{13}+\frac{22\cdots 28}{23\cdots 39}a^{12}+\frac{12\cdots 38}{23\cdots 39}a^{11}-\frac{79\cdots 38}{23\cdots 39}a^{10}-\frac{77\cdots 11}{23\cdots 39}a^{9}+\frac{13\cdots 76}{23\cdots 39}a^{8}-\frac{16\cdots 77}{23\cdots 39}a^{7}-\frac{10\cdots 87}{23\cdots 39}a^{6}+\frac{19\cdots 59}{23\cdots 39}a^{5}+\frac{29\cdots 71}{23\cdots 39}a^{4}-\frac{53\cdots 59}{23\cdots 39}a^{3}-\frac{25\cdots 07}{23\cdots 39}a^{2}+\frac{51\cdots 26}{23\cdots 39}a-\frac{12\cdots 92}{23\cdots 39}$, $\frac{92\cdots 61}{23\cdots 39}a^{19}-\frac{29\cdots 44}{23\cdots 39}a^{18}-\frac{35\cdots 19}{23\cdots 39}a^{17}+\frac{10\cdots 59}{23\cdots 39}a^{16}+\frac{51\cdots 28}{23\cdots 39}a^{15}-\frac{15\cdots 11}{23\cdots 39}a^{14}-\frac{38\cdots 49}{23\cdots 39}a^{13}+\frac{10\cdots 16}{23\cdots 39}a^{12}+\frac{15\cdots 21}{23\cdots 39}a^{11}-\frac{36\cdots 29}{23\cdots 39}a^{10}-\frac{33\cdots 19}{23\cdots 39}a^{9}+\frac{63\cdots 22}{23\cdots 39}a^{8}+\frac{40\cdots 58}{23\cdots 39}a^{7}-\frac{49\cdots 59}{23\cdots 39}a^{6}-\frac{25\cdots 34}{23\cdots 39}a^{5}+\frac{13\cdots 45}{23\cdots 39}a^{4}+\frac{63\cdots 39}{23\cdots 39}a^{3}-\frac{57\cdots 93}{23\cdots 39}a^{2}-\frac{16\cdots 68}{23\cdots 39}a-\frac{44\cdots 96}{23\cdots 39}$
|
| |
| Regulator: | \( 8953413920.01 \) (assuming GRH) |
| |
| Unit signature rank: | \( 19 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 8953413920.01 \cdot 1}{2\cdot\sqrt{1470391355634309152000000000000000}}\cr\approx \mathstrut & 0.122417106397 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | R | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.2.20a1.2 | $x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 5 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 7 x^{6} + 10 x^{5} + 3 x^{4} + 6 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $10$ | $20$ | 20T1 | not computed |
|
\(5\)
| 5.5.4.15a1.4 | $x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 86$ | $4$ | $5$ | $15$ | 20T1 | not computed |
|
\(11\)
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |