Normalized defining polynomial
\( x^{20} - x^{19} - 60 x^{18} + 59 x^{17} + 1465 x^{16} - 1407 x^{15} - 18966 x^{14} + 17616 x^{13} + \cdots - 4099 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(20, 0)$ |
| |
| Discriminant: |
\(42913439156921905845805508304306640625\)
\(\medspace = 5^{10}\cdot 41^{19}\)
|
| |
| Root discriminant: | \(76.14\) |
| |
| Galois root discriminant: | $5^{1/2}41^{19/20}\approx 76.14294428654999$ | ||
| Ramified primes: |
\(5\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(205=5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(66,·)$, $\chi_{205}(9,·)$, $\chi_{205}(74,·)$, $\chi_{205}(141,·)$, $\chi_{205}(144,·)$, $\chi_{205}(81,·)$, $\chi_{205}(146,·)$, $\chi_{205}(84,·)$, $\chi_{205}(86,·)$, $\chi_{205}(31,·)$, $\chi_{205}(16,·)$, $\chi_{205}(39,·)$, $\chi_{205}(169,·)$, $\chi_{205}(49,·)$, $\chi_{205}(114,·)$, $\chi_{205}(51,·)$, $\chi_{205}(201,·)$, $\chi_{205}(184,·)$, $\chi_{205}(159,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{146}a^{17}+\frac{3}{146}a^{16}+\frac{2}{73}a^{15}+\frac{5}{73}a^{14}+\frac{14}{73}a^{13}+\frac{10}{73}a^{12}-\frac{3}{146}a^{11}+\frac{23}{73}a^{10}+\frac{26}{73}a^{9}-\frac{53}{146}a^{8}-\frac{29}{146}a^{7}-\frac{41}{146}a^{6}-\frac{21}{73}a^{5}-\frac{7}{146}a^{4}+\frac{13}{146}a^{3}-\frac{25}{146}a^{2}-\frac{1}{73}a-\frac{21}{73}$, $\frac{1}{146}a^{18}-\frac{5}{146}a^{16}-\frac{1}{73}a^{15}-\frac{1}{73}a^{14}-\frac{32}{73}a^{13}-\frac{63}{146}a^{12}+\frac{55}{146}a^{11}+\frac{30}{73}a^{10}-\frac{63}{146}a^{9}-\frac{8}{73}a^{8}+\frac{23}{73}a^{7}-\frac{65}{146}a^{6}-\frac{27}{146}a^{5}+\frac{17}{73}a^{4}-\frac{32}{73}a^{3}-\frac{1}{2}a^{2}-\frac{18}{73}a-\frac{10}{73}$, $\frac{1}{14\cdots 74}a^{19}+\frac{88\cdots 55}{14\cdots 74}a^{18}-\frac{24\cdots 75}{14\cdots 74}a^{17}+\frac{61\cdots 57}{14\cdots 74}a^{16}-\frac{16\cdots 14}{74\cdots 87}a^{15}+\frac{31\cdots 93}{74\cdots 87}a^{14}+\frac{58\cdots 03}{14\cdots 74}a^{13}+\frac{12\cdots 74}{74\cdots 87}a^{12}-\frac{53\cdots 93}{14\cdots 74}a^{11}-\frac{95\cdots 89}{14\cdots 74}a^{10}+\frac{73\cdots 29}{14\cdots 74}a^{9}+\frac{21\cdots 63}{74\cdots 87}a^{8}-\frac{31\cdots 59}{14\cdots 74}a^{7}-\frac{24\cdots 49}{74\cdots 87}a^{6}+\frac{22\cdots 09}{14\cdots 74}a^{5}+\frac{29\cdots 48}{74\cdots 87}a^{4}+\frac{39\cdots 81}{14\cdots 74}a^{3}-\frac{53\cdots 57}{14\cdots 74}a^{2}-\frac{11\cdots 03}{74\cdots 87}a-\frac{25\cdots 88}{74\cdots 87}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $19$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{31\cdots 83}{14\cdots 74}a^{19}-\frac{15\cdots 65}{14\cdots 74}a^{18}-\frac{92\cdots 14}{74\cdots 87}a^{17}+\frac{47\cdots 38}{74\cdots 87}a^{16}+\frac{42\cdots 07}{14\cdots 74}a^{15}-\frac{23\cdots 77}{14\cdots 74}a^{14}-\frac{51\cdots 75}{14\cdots 74}a^{13}+\frac{21\cdots 50}{10\cdots 19}a^{12}+\frac{17\cdots 72}{74\cdots 87}a^{11}-\frac{24\cdots 51}{14\cdots 74}a^{10}-\frac{66\cdots 93}{74\cdots 87}a^{9}+\frac{10\cdots 53}{14\cdots 74}a^{8}+\frac{13\cdots 35}{74\cdots 87}a^{7}-\frac{13\cdots 25}{74\cdots 87}a^{6}-\frac{13\cdots 68}{10\cdots 19}a^{5}+\frac{16\cdots 54}{74\cdots 87}a^{4}-\frac{24\cdots 81}{74\cdots 87}a^{3}-\frac{15\cdots 33}{14\cdots 74}a^{2}+\frac{92\cdots 63}{14\cdots 74}a-\frac{11\cdots 91}{14\cdots 74}$, $\frac{32\cdots 75}{14\cdots 74}a^{19}-\frac{34\cdots 69}{14\cdots 74}a^{18}-\frac{10\cdots 61}{74\cdots 87}a^{17}+\frac{18\cdots 29}{14\cdots 74}a^{16}+\frac{26\cdots 41}{74\cdots 87}a^{15}-\frac{40\cdots 45}{14\cdots 74}a^{14}-\frac{71\cdots 11}{14\cdots 74}a^{13}+\frac{21\cdots 22}{74\cdots 87}a^{12}+\frac{28\cdots 91}{74\cdots 87}a^{11}-\frac{11\cdots 78}{74\cdots 87}a^{10}-\frac{13\cdots 18}{74\cdots 87}a^{9}+\frac{55\cdots 95}{14\cdots 74}a^{8}+\frac{74\cdots 25}{14\cdots 74}a^{7}-\frac{96\cdots 95}{14\cdots 74}a^{6}-\frac{11\cdots 57}{14\cdots 74}a^{5}-\frac{77\cdots 73}{74\cdots 87}a^{4}+\frac{76\cdots 93}{14\cdots 74}a^{3}+\frac{60\cdots 19}{74\cdots 87}a^{2}-\frac{82\cdots 76}{74\cdots 87}a+\frac{13\cdots 73}{14\cdots 74}$, $\frac{57\cdots 33}{14\cdots 74}a^{19}-\frac{34\cdots 77}{14\cdots 74}a^{18}-\frac{14\cdots 99}{74\cdots 87}a^{17}+\frac{98\cdots 60}{74\cdots 87}a^{16}+\frac{57\cdots 39}{14\cdots 74}a^{15}-\frac{45\cdots 21}{14\cdots 74}a^{14}-\frac{46\cdots 95}{14\cdots 74}a^{13}+\frac{27\cdots 41}{74\cdots 87}a^{12}+\frac{41\cdots 57}{74\cdots 87}a^{11}-\frac{38\cdots 57}{14\cdots 74}a^{10}+\frac{57\cdots 24}{74\cdots 87}a^{9}+\frac{14\cdots 75}{14\cdots 74}a^{8}-\frac{36\cdots 71}{74\cdots 87}a^{7}-\frac{15\cdots 78}{74\cdots 87}a^{6}+\frac{70\cdots 50}{74\cdots 87}a^{5}+\frac{16\cdots 52}{74\cdots 87}a^{4}-\frac{13\cdots 54}{74\cdots 87}a^{3}-\frac{17\cdots 69}{14\cdots 74}a^{2}-\frac{63\cdots 35}{14\cdots 74}a+\frac{47\cdots 95}{14\cdots 74}$, $\frac{84\cdots 50}{74\cdots 87}a^{19}-\frac{27\cdots 65}{14\cdots 74}a^{18}-\frac{89\cdots 25}{14\cdots 74}a^{17}+\frac{15\cdots 45}{14\cdots 74}a^{16}+\frac{17\cdots 35}{14\cdots 74}a^{15}-\frac{17\cdots 30}{74\cdots 87}a^{14}-\frac{80\cdots 15}{74\cdots 87}a^{13}+\frac{42\cdots 75}{14\cdots 74}a^{12}+\frac{23\cdots 60}{74\cdots 87}a^{11}-\frac{27\cdots 25}{14\cdots 74}a^{10}+\frac{23\cdots 55}{14\cdots 74}a^{9}+\frac{88\cdots 15}{14\cdots 74}a^{8}-\frac{10\cdots 40}{74\cdots 87}a^{7}-\frac{11\cdots 95}{14\cdots 74}a^{6}+\frac{29\cdots 36}{74\cdots 87}a^{5}-\frac{39\cdots 95}{14\cdots 74}a^{4}-\frac{31\cdots 10}{74\cdots 87}a^{3}+\frac{13\cdots 55}{14\cdots 74}a^{2}+\frac{20\cdots 25}{14\cdots 74}a-\frac{11\cdots 85}{74\cdots 87}$, $\frac{28\cdots 77}{74\cdots 87}a^{19}+\frac{16\cdots 59}{14\cdots 74}a^{18}-\frac{16\cdots 64}{74\cdots 87}a^{17}-\frac{94\cdots 83}{14\cdots 74}a^{16}+\frac{38\cdots 28}{74\cdots 87}a^{15}+\frac{10\cdots 79}{74\cdots 87}a^{14}-\frac{47\cdots 57}{74\cdots 87}a^{13}-\frac{23\cdots 83}{14\cdots 74}a^{12}+\frac{65\cdots 63}{14\cdots 74}a^{11}+\frac{70\cdots 90}{74\cdots 87}a^{10}-\frac{26\cdots 03}{14\cdots 74}a^{9}-\frac{20\cdots 22}{74\cdots 87}a^{8}+\frac{28\cdots 51}{74\cdots 87}a^{7}+\frac{28\cdots 77}{14\cdots 74}a^{6}-\frac{62\cdots 49}{14\cdots 74}a^{5}+\frac{51\cdots 67}{74\cdots 87}a^{4}+\frac{15\cdots 54}{74\cdots 87}a^{3}-\frac{17\cdots 35}{14\cdots 74}a^{2}-\frac{42\cdots 77}{74\cdots 87}a+\frac{19\cdots 41}{74\cdots 87}$, $\frac{40\cdots 49}{14\cdots 74}a^{19}-\frac{45\cdots 28}{74\cdots 87}a^{18}-\frac{12\cdots 35}{74\cdots 87}a^{17}+\frac{26\cdots 57}{74\cdots 87}a^{16}+\frac{30\cdots 65}{74\cdots 87}a^{15}-\frac{12\cdots 07}{14\cdots 74}a^{14}-\frac{78\cdots 31}{14\cdots 74}a^{13}+\frac{15\cdots 73}{14\cdots 74}a^{12}+\frac{59\cdots 03}{14\cdots 74}a^{11}-\frac{53\cdots 54}{74\cdots 87}a^{10}-\frac{26\cdots 85}{14\cdots 74}a^{9}+\frac{41\cdots 05}{14\cdots 74}a^{8}+\frac{67\cdots 33}{14\cdots 74}a^{7}-\frac{44\cdots 68}{74\cdots 87}a^{6}-\frac{46\cdots 86}{74\cdots 87}a^{5}+\frac{43\cdots 11}{74\cdots 87}a^{4}+\frac{56\cdots 45}{14\cdots 74}a^{3}-\frac{32\cdots 71}{14\cdots 74}a^{2}-\frac{33\cdots 32}{74\cdots 87}a+\frac{13\cdots 69}{14\cdots 74}$, $\frac{22\cdots 56}{74\cdots 87}a^{19}-\frac{16\cdots 51}{14\cdots 74}a^{18}-\frac{25\cdots 75}{14\cdots 74}a^{17}+\frac{11\cdots 51}{14\cdots 74}a^{16}+\frac{55\cdots 93}{14\cdots 74}a^{15}-\frac{14\cdots 83}{74\cdots 87}a^{14}-\frac{31\cdots 97}{74\cdots 87}a^{13}+\frac{42\cdots 39}{14\cdots 74}a^{12}+\frac{18\cdots 23}{74\cdots 87}a^{11}-\frac{34\cdots 89}{14\cdots 74}a^{10}-\frac{10\cdots 59}{14\cdots 74}a^{9}+\frac{15\cdots 53}{14\cdots 74}a^{8}+\frac{43\cdots 01}{74\cdots 87}a^{7}-\frac{37\cdots 53}{14\cdots 74}a^{6}+\frac{11\cdots 68}{74\cdots 87}a^{5}+\frac{42\cdots 95}{14\cdots 74}a^{4}-\frac{21\cdots 95}{74\cdots 87}a^{3}-\frac{15\cdots 87}{14\cdots 74}a^{2}+\frac{19\cdots 31}{14\cdots 74}a-\frac{36\cdots 22}{74\cdots 87}$, $\frac{48\cdots 99}{14\cdots 74}a^{19}-\frac{81\cdots 99}{74\cdots 87}a^{18}-\frac{13\cdots 45}{74\cdots 87}a^{17}+\frac{47\cdots 24}{74\cdots 87}a^{16}+\frac{29\cdots 16}{74\cdots 87}a^{15}-\frac{21\cdots 71}{14\cdots 74}a^{14}-\frac{64\cdots 11}{14\cdots 74}a^{13}+\frac{26\cdots 05}{14\cdots 74}a^{12}+\frac{34\cdots 13}{14\cdots 74}a^{11}-\frac{87\cdots 32}{74\cdots 87}a^{10}-\frac{72\cdots 41}{14\cdots 74}a^{9}+\frac{63\cdots 57}{14\cdots 74}a^{8}-\frac{96\cdots 39}{14\cdots 74}a^{7}-\frac{57\cdots 16}{74\cdots 87}a^{6}+\frac{30\cdots 00}{74\cdots 87}a^{5}+\frac{39\cdots 14}{74\cdots 87}a^{4}-\frac{66\cdots 41}{14\cdots 74}a^{3}-\frac{70\cdots 97}{14\cdots 74}a^{2}+\frac{94\cdots 94}{74\cdots 87}a-\frac{44\cdots 33}{14\cdots 74}$, $\frac{12\cdots 64}{74\cdots 87}a^{19}+\frac{60\cdots 39}{14\cdots 74}a^{18}-\frac{74\cdots 62}{74\cdots 87}a^{17}-\frac{34\cdots 41}{14\cdots 74}a^{16}+\frac{18\cdots 83}{74\cdots 87}a^{15}+\frac{39\cdots 47}{74\cdots 87}a^{14}-\frac{25\cdots 18}{74\cdots 87}a^{13}-\frac{92\cdots 99}{14\cdots 74}a^{12}+\frac{41\cdots 63}{14\cdots 74}a^{11}+\frac{30\cdots 62}{74\cdots 87}a^{10}-\frac{20\cdots 91}{14\cdots 74}a^{9}-\frac{11\cdots 09}{74\cdots 87}a^{8}+\frac{31\cdots 44}{74\cdots 87}a^{7}+\frac{60\cdots 89}{20\cdots 38}a^{6}-\frac{11\cdots 37}{14\cdots 74}a^{5}-\frac{21\cdots 36}{74\cdots 87}a^{4}+\frac{45\cdots 80}{74\cdots 87}a^{3}+\frac{17\cdots 77}{14\cdots 74}a^{2}-\frac{10\cdots 29}{74\cdots 87}a-\frac{10\cdots 15}{74\cdots 87}$, $\frac{50\cdots 43}{11\cdots 58}a^{19}-\frac{39\cdots 67}{11\cdots 58}a^{18}-\frac{14\cdots 36}{57\cdots 29}a^{17}+\frac{22\cdots 89}{11\cdots 58}a^{16}+\frac{35\cdots 34}{57\cdots 29}a^{15}-\frac{50\cdots 57}{11\cdots 58}a^{14}-\frac{86\cdots 87}{11\cdots 58}a^{13}+\frac{29\cdots 22}{57\cdots 29}a^{12}+\frac{29\cdots 72}{57\cdots 29}a^{11}-\frac{18\cdots 95}{57\cdots 29}a^{10}-\frac{11\cdots 30}{57\cdots 29}a^{9}+\frac{12\cdots 97}{11\cdots 58}a^{8}+\frac{47\cdots 69}{11\cdots 58}a^{7}-\frac{18\cdots 75}{11\cdots 58}a^{6}-\frac{45\cdots 87}{11\cdots 58}a^{5}+\frac{41\cdots 68}{57\cdots 29}a^{4}+\frac{14\cdots 61}{11\cdots 58}a^{3}+\frac{60\cdots 54}{57\cdots 29}a^{2}-\frac{82\cdots 67}{57\cdots 29}a-\frac{75\cdots 93}{11\cdots 58}$, $\frac{14\cdots 11}{14\cdots 74}a^{19}-\frac{15\cdots 13}{74\cdots 87}a^{18}-\frac{34\cdots 42}{74\cdots 87}a^{17}+\frac{88\cdots 23}{74\cdots 87}a^{16}+\frac{50\cdots 58}{74\cdots 87}a^{15}-\frac{39\cdots 17}{14\cdots 74}a^{14}-\frac{17\cdots 53}{14\cdots 74}a^{13}+\frac{46\cdots 27}{14\cdots 74}a^{12}-\frac{10\cdots 01}{14\cdots 74}a^{11}-\frac{14\cdots 33}{74\cdots 87}a^{10}+\frac{11\cdots 91}{14\cdots 74}a^{9}+\frac{10\cdots 59}{14\cdots 74}a^{8}-\frac{52\cdots 47}{14\cdots 74}a^{7}-\frac{11\cdots 67}{74\cdots 87}a^{6}+\frac{58\cdots 80}{74\cdots 87}a^{5}+\frac{14\cdots 55}{74\cdots 87}a^{4}-\frac{10\cdots 13}{14\cdots 74}a^{3}-\frac{23\cdots 85}{14\cdots 74}a^{2}+\frac{11\cdots 43}{74\cdots 87}a-\frac{21\cdots 11}{14\cdots 74}$, $\frac{51\cdots 65}{14\cdots 74}a^{19}+\frac{14\cdots 80}{74\cdots 87}a^{18}-\frac{15\cdots 58}{74\cdots 87}a^{17}-\frac{18\cdots 10}{74\cdots 87}a^{16}+\frac{37\cdots 05}{74\cdots 87}a^{15}+\frac{14\cdots 65}{14\cdots 74}a^{14}-\frac{95\cdots 05}{14\cdots 74}a^{13}-\frac{27\cdots 95}{14\cdots 74}a^{12}+\frac{70\cdots 23}{14\cdots 74}a^{11}+\frac{14\cdots 18}{74\cdots 87}a^{10}-\frac{29\cdots 97}{14\cdots 74}a^{9}-\frac{20\cdots 87}{14\cdots 74}a^{8}+\frac{71\cdots 65}{14\cdots 74}a^{7}+\frac{35\cdots 56}{74\cdots 87}a^{6}-\frac{44\cdots 15}{74\cdots 87}a^{5}-\frac{54\cdots 65}{74\cdots 87}a^{4}+\frac{49\cdots 45}{14\cdots 74}a^{3}+\frac{41\cdots 01}{14\cdots 74}a^{2}-\frac{46\cdots 72}{74\cdots 87}a+\frac{19\cdots 23}{14\cdots 74}$, $\frac{62\cdots 87}{14\cdots 74}a^{19}-\frac{91\cdots 11}{14\cdots 74}a^{18}-\frac{37\cdots 25}{14\cdots 74}a^{17}+\frac{53\cdots 67}{14\cdots 74}a^{16}+\frac{92\cdots 03}{14\cdots 74}a^{15}-\frac{12\cdots 83}{14\cdots 74}a^{14}-\frac{12\cdots 03}{14\cdots 74}a^{13}+\frac{74\cdots 94}{74\cdots 87}a^{12}+\frac{92\cdots 45}{14\cdots 74}a^{11}-\frac{98\cdots 01}{14\cdots 74}a^{10}-\frac{21\cdots 30}{74\cdots 87}a^{9}+\frac{17\cdots 42}{74\cdots 87}a^{8}+\frac{11\cdots 01}{14\cdots 74}a^{7}-\frac{63\cdots 41}{14\cdots 74}a^{6}-\frac{83\cdots 32}{74\cdots 87}a^{5}+\frac{50\cdots 43}{20\cdots 38}a^{4}+\frac{11\cdots 33}{14\cdots 74}a^{3}+\frac{34\cdots 99}{74\cdots 87}a^{2}-\frac{14\cdots 05}{14\cdots 74}a+\frac{11\cdots 67}{14\cdots 74}$, $\frac{38\cdots 91}{14\cdots 74}a^{19}-\frac{23\cdots 59}{14\cdots 74}a^{18}-\frac{10\cdots 50}{74\cdots 87}a^{17}+\frac{26\cdots 35}{14\cdots 74}a^{16}+\frac{20\cdots 00}{74\cdots 87}a^{15}-\frac{10\cdots 69}{14\cdots 74}a^{14}-\frac{37\cdots 39}{14\cdots 74}a^{13}+\frac{98\cdots 27}{74\cdots 87}a^{12}+\frac{68\cdots 49}{74\cdots 87}a^{11}-\frac{98\cdots 69}{74\cdots 87}a^{10}+\frac{99\cdots 52}{74\cdots 87}a^{9}+\frac{10\cdots 31}{14\cdots 74}a^{8}-\frac{30\cdots 93}{14\cdots 74}a^{7}-\frac{31\cdots 13}{14\cdots 74}a^{6}+\frac{85\cdots 71}{14\cdots 74}a^{5}+\frac{24\cdots 66}{74\cdots 87}a^{4}-\frac{82\cdots 79}{14\cdots 74}a^{3}-\frac{15\cdots 93}{74\cdots 87}a^{2}+\frac{96\cdots 59}{74\cdots 87}a-\frac{10\cdots 25}{14\cdots 74}$, $\frac{11\cdots 33}{14\cdots 74}a^{19}+\frac{56\cdots 47}{74\cdots 87}a^{18}-\frac{32\cdots 17}{74\cdots 87}a^{17}-\frac{32\cdots 51}{74\cdots 87}a^{16}+\frac{13\cdots 67}{14\cdots 74}a^{15}+\frac{74\cdots 68}{74\cdots 87}a^{14}-\frac{14\cdots 29}{14\cdots 74}a^{13}-\frac{17\cdots 37}{14\cdots 74}a^{12}+\frac{81\cdots 65}{14\cdots 74}a^{11}+\frac{56\cdots 86}{74\cdots 87}a^{10}-\frac{12\cdots 10}{74\cdots 87}a^{9}-\frac{40\cdots 99}{14\cdots 74}a^{8}+\frac{40\cdots 09}{14\cdots 74}a^{7}+\frac{76\cdots 85}{14\cdots 74}a^{6}-\frac{29\cdots 03}{14\cdots 74}a^{5}-\frac{65\cdots 23}{14\cdots 74}a^{4}+\frac{37\cdots 09}{14\cdots 74}a^{3}+\frac{99\cdots 38}{74\cdots 87}a^{2}-\frac{26\cdots 87}{14\cdots 74}a-\frac{45\cdots 61}{74\cdots 87}$, $\frac{27\cdots 32}{74\cdots 87}a^{19}+\frac{49\cdots 87}{74\cdots 87}a^{18}-\frac{16\cdots 15}{74\cdots 87}a^{17}+\frac{43\cdots 57}{14\cdots 74}a^{16}+\frac{38\cdots 45}{74\cdots 87}a^{15}-\frac{38\cdots 05}{14\cdots 74}a^{14}-\frac{48\cdots 48}{74\cdots 87}a^{13}+\frac{41\cdots 68}{74\cdots 87}a^{12}+\frac{34\cdots 33}{74\cdots 87}a^{11}-\frac{74\cdots 97}{14\cdots 74}a^{10}-\frac{29\cdots 85}{14\cdots 74}a^{9}+\frac{12\cdots 65}{74\cdots 87}a^{8}+\frac{69\cdots 91}{14\cdots 74}a^{7}+\frac{61\cdots 74}{74\cdots 87}a^{6}-\frac{41\cdots 10}{74\cdots 87}a^{5}-\frac{21\cdots 41}{14\cdots 74}a^{4}+\frac{36\cdots 63}{14\cdots 74}a^{3}+\frac{14\cdots 96}{74\cdots 87}a^{2}+\frac{15\cdots 16}{74\cdots 87}a-\frac{41\cdots 23}{14\cdots 74}$, $\frac{64\cdots 77}{14\cdots 74}a^{19}-\frac{42\cdots 57}{74\cdots 87}a^{18}-\frac{23\cdots 04}{74\cdots 87}a^{17}+\frac{47\cdots 69}{14\cdots 74}a^{16}+\frac{73\cdots 73}{74\cdots 87}a^{15}-\frac{52\cdots 62}{74\cdots 87}a^{14}-\frac{24\cdots 03}{14\cdots 74}a^{13}+\frac{11\cdots 43}{14\cdots 74}a^{12}+\frac{24\cdots 25}{14\cdots 74}a^{11}-\frac{57\cdots 37}{14\cdots 74}a^{10}-\frac{72\cdots 11}{74\cdots 87}a^{9}+\frac{40\cdots 13}{14\cdots 74}a^{8}+\frac{25\cdots 53}{74\cdots 87}a^{7}+\frac{45\cdots 61}{74\cdots 87}a^{6}-\frac{48\cdots 57}{74\cdots 87}a^{5}-\frac{38\cdots 85}{14\cdots 74}a^{4}+\frac{39\cdots 81}{74\cdots 87}a^{3}+\frac{49\cdots 13}{14\cdots 74}a^{2}-\frac{38\cdots 17}{74\cdots 87}a-\frac{33\cdots 32}{74\cdots 87}$, $\frac{33\cdots 49}{14\cdots 74}a^{19}+\frac{21\cdots 71}{74\cdots 87}a^{18}-\frac{99\cdots 00}{74\cdots 87}a^{17}-\frac{26\cdots 89}{14\cdots 74}a^{16}+\frac{48\cdots 51}{14\cdots 74}a^{15}+\frac{63\cdots 51}{14\cdots 74}a^{14}-\frac{61\cdots 19}{14\cdots 74}a^{13}-\frac{80\cdots 03}{14\cdots 74}a^{12}+\frac{45\cdots 89}{14\cdots 74}a^{11}+\frac{58\cdots 99}{14\cdots 74}a^{10}-\frac{20\cdots 27}{14\cdots 74}a^{9}-\frac{24\cdots 17}{14\cdots 74}a^{8}+\frac{28\cdots 56}{74\cdots 87}a^{7}+\frac{56\cdots 79}{14\cdots 74}a^{6}-\frac{96\cdots 89}{14\cdots 74}a^{5}-\frac{31\cdots 64}{74\cdots 87}a^{4}+\frac{46\cdots 34}{74\cdots 87}a^{3}+\frac{12\cdots 34}{74\cdots 87}a^{2}-\frac{38\cdots 35}{14\cdots 74}a+\frac{15\cdots 81}{14\cdots 74}$, $\frac{96\cdots 77}{14\cdots 74}a^{19}-\frac{16\cdots 91}{14\cdots 74}a^{18}-\frac{56\cdots 91}{14\cdots 74}a^{17}+\frac{10\cdots 31}{14\cdots 74}a^{16}+\frac{13\cdots 03}{14\cdots 74}a^{15}-\frac{28\cdots 01}{14\cdots 74}a^{14}-\frac{15\cdots 99}{14\cdots 74}a^{13}+\frac{19\cdots 72}{74\cdots 87}a^{12}+\frac{10\cdots 69}{14\cdots 74}a^{11}-\frac{31\cdots 53}{14\cdots 74}a^{10}-\frac{20\cdots 78}{74\cdots 87}a^{9}+\frac{66\cdots 83}{74\cdots 87}a^{8}+\frac{79\cdots 67}{14\cdots 74}a^{7}-\frac{29\cdots 41}{14\cdots 74}a^{6}-\frac{33\cdots 47}{74\cdots 87}a^{5}+\frac{31\cdots 65}{14\cdots 74}a^{4}+\frac{58\cdots 55}{14\cdots 74}a^{3}-\frac{57\cdots 93}{74\cdots 87}a^{2}+\frac{82\cdots 13}{14\cdots 74}a-\frac{20\cdots 95}{14\cdots 74}$
|
| |
| Regulator: | \( 1163330087490 \) (assuming GRH) |
| |
| Unit signature rank: | \( 19 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 1163330087490 \cdot 2}{2\cdot\sqrt{42913439156921905845805508304306640625}}\cr\approx \mathstrut & 0.186211377693490 \end{aligned}\] (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
| \(\Q(\sqrt{41}) \), \(\Q(\sqrt{410 +50 \sqrt{41}})\), 5.5.2825761.1, 10.10.327381934393961.1 |
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 | ${\href{/padicField/2.5.0.1}{5} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | $20$ | $20$ | $20$ | $20$ | $20$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.5.2.5a1.1 | $x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 29 x + 9$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
| 5.5.2.5a1.1 | $x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 29 x + 9$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ | |
|
\(41\)
| 41.1.20.19a1.1 | $x^{20} + 41$ | $20$ | $1$ | $19$ | 20T1 | $$[\ ]_{20}$$ |