Normalized defining polynomial
\( x^{20} - x^{19} - 74 x^{18} - 8 x^{17} + 2161 x^{16} + 2188 x^{15} - 30387 x^{14} - 53182 x^{13} + \cdots + 529 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(20, 0)$ |
| |
| Discriminant: |
\(138881946372451702408146629446494920973\)
\(\medspace = 3^{10}\cdot 11^{16}\cdot 13^{15}\)
|
| |
| Root discriminant: | \(80.75\) |
| |
| Galois root discriminant: | $3^{1/2}11^{4/5}13^{3/4}\approx 80.74809582166735$ | ||
| Ramified primes: |
\(3\), \(11\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(429=3\cdot 11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{429}(64,·)$, $\chi_{429}(1,·)$, $\chi_{429}(196,·)$, $\chi_{429}(5,·)$, $\chi_{429}(203,·)$, $\chi_{429}(320,·)$, $\chi_{429}(278,·)$, $\chi_{429}(25,·)$, $\chi_{429}(47,·)$, $\chi_{429}(86,·)$, $\chi_{429}(157,·)$, $\chi_{429}(356,·)$, $\chi_{429}(103,·)$, $\chi_{429}(298,·)$, $\chi_{429}(235,·)$, $\chi_{429}(125,·)$, $\chi_{429}(181,·)$, $\chi_{429}(313,·)$, $\chi_{429}(122,·)$, $\chi_{429}(317,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}+\frac{1}{23}a^{15}-\frac{7}{23}a^{14}-\frac{2}{23}a^{13}+\frac{5}{23}a^{12}-\frac{11}{23}a^{11}-\frac{10}{23}a^{10}+\frac{2}{23}a^{9}+\frac{8}{23}a^{8}+\frac{11}{23}a^{7}-\frac{4}{23}a^{5}+\frac{4}{23}a^{4}+\frac{5}{23}a^{3}+\frac{8}{23}a^{2}+\frac{5}{23}a$, $\frac{1}{23}a^{18}+\frac{1}{23}a^{16}-\frac{7}{23}a^{15}-\frac{2}{23}a^{14}+\frac{5}{23}a^{13}-\frac{11}{23}a^{12}-\frac{10}{23}a^{11}+\frac{2}{23}a^{10}+\frac{8}{23}a^{9}+\frac{11}{23}a^{8}-\frac{4}{23}a^{6}+\frac{4}{23}a^{5}+\frac{5}{23}a^{4}+\frac{8}{23}a^{3}+\frac{5}{23}a^{2}$, $\frac{1}{26\cdots 33}a^{19}-\frac{15\cdots 69}{26\cdots 33}a^{18}+\frac{12\cdots 61}{26\cdots 33}a^{17}+\frac{26\cdots 67}{26\cdots 33}a^{16}-\frac{46\cdots 74}{26\cdots 33}a^{15}-\frac{37\cdots 05}{26\cdots 33}a^{14}+\frac{26\cdots 66}{26\cdots 33}a^{13}-\frac{10\cdots 35}{26\cdots 33}a^{12}+\frac{47\cdots 73}{26\cdots 33}a^{11}+\frac{86\cdots 84}{26\cdots 33}a^{10}+\frac{11\cdots 12}{26\cdots 33}a^{9}+\frac{20\cdots 66}{26\cdots 33}a^{8}+\frac{88\cdots 87}{26\cdots 33}a^{7}+\frac{72\cdots 34}{26\cdots 33}a^{6}-\frac{11\cdots 29}{26\cdots 33}a^{5}-\frac{10\cdots 45}{26\cdots 33}a^{4}-\frac{95\cdots 75}{26\cdots 33}a^{3}+\frac{38\cdots 62}{26\cdots 33}a^{2}-\frac{72\cdots 16}{26\cdots 33}a-\frac{43\cdots 68}{11\cdots 71}$
| 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: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $19$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{17\cdots 10}{26\cdots 33}a^{19}-\frac{27\cdots 90}{26\cdots 33}a^{18}-\frac{12\cdots 01}{26\cdots 33}a^{17}+\frac{58\cdots 33}{26\cdots 33}a^{16}+\frac{37\cdots 81}{26\cdots 33}a^{15}+\frac{17\cdots 42}{26\cdots 33}a^{14}-\frac{53\cdots 42}{26\cdots 33}a^{13}-\frac{62\cdots 28}{26\cdots 33}a^{12}+\frac{38\cdots 88}{26\cdots 33}a^{11}+\frac{68\cdots 56}{26\cdots 33}a^{10}-\frac{12\cdots 53}{26\cdots 33}a^{9}-\frac{32\cdots 39}{26\cdots 33}a^{8}+\frac{81\cdots 48}{26\cdots 33}a^{7}+\frac{62\cdots 56}{26\cdots 33}a^{6}+\frac{30\cdots 27}{26\cdots 33}a^{5}-\frac{33\cdots 09}{26\cdots 33}a^{4}-\frac{33\cdots 64}{26\cdots 33}a^{3}-\frac{75\cdots 48}{26\cdots 33}a^{2}+\frac{41\cdots 85}{26\cdots 33}a+\frac{22\cdots 49}{11\cdots 71}$, $\frac{57\cdots 80}{26\cdots 33}a^{19}-\frac{17\cdots 99}{26\cdots 33}a^{18}-\frac{40\cdots 59}{26\cdots 33}a^{17}+\frac{77\cdots 46}{26\cdots 33}a^{16}+\frac{11\cdots 06}{26\cdots 33}a^{15}-\frac{11\cdots 75}{26\cdots 33}a^{14}-\frac{17\cdots 08}{26\cdots 33}a^{13}+\frac{41\cdots 86}{26\cdots 33}a^{12}+\frac{14\cdots 68}{26\cdots 33}a^{11}+\frac{37\cdots 14}{26\cdots 33}a^{10}-\frac{62\cdots 83}{26\cdots 33}a^{9}-\frac{36\cdots 24}{26\cdots 33}a^{8}+\frac{61\cdots 26}{11\cdots 71}a^{7}+\frac{10\cdots 33}{26\cdots 33}a^{6}-\frac{14\cdots 38}{26\cdots 33}a^{5}-\frac{12\cdots 73}{26\cdots 33}a^{4}+\frac{52\cdots 78}{26\cdots 33}a^{3}+\frac{45\cdots 22}{26\cdots 33}a^{2}+\frac{15\cdots 90}{26\cdots 33}a-\frac{20\cdots 93}{11\cdots 71}$, $\frac{57\cdots 80}{26\cdots 33}a^{19}-\frac{17\cdots 99}{26\cdots 33}a^{18}-\frac{40\cdots 59}{26\cdots 33}a^{17}+\frac{77\cdots 46}{26\cdots 33}a^{16}+\frac{11\cdots 06}{26\cdots 33}a^{15}-\frac{11\cdots 75}{26\cdots 33}a^{14}-\frac{17\cdots 08}{26\cdots 33}a^{13}+\frac{41\cdots 86}{26\cdots 33}a^{12}+\frac{14\cdots 68}{26\cdots 33}a^{11}+\frac{37\cdots 14}{26\cdots 33}a^{10}-\frac{62\cdots 83}{26\cdots 33}a^{9}-\frac{36\cdots 24}{26\cdots 33}a^{8}+\frac{61\cdots 26}{11\cdots 71}a^{7}+\frac{10\cdots 33}{26\cdots 33}a^{6}-\frac{14\cdots 38}{26\cdots 33}a^{5}-\frac{12\cdots 73}{26\cdots 33}a^{4}+\frac{52\cdots 78}{26\cdots 33}a^{3}+\frac{45\cdots 22}{26\cdots 33}a^{2}+\frac{15\cdots 90}{26\cdots 33}a-\frac{32\cdots 64}{11\cdots 71}$, $\frac{36\cdots 94}{26\cdots 33}a^{19}-\frac{93\cdots 80}{26\cdots 33}a^{18}-\frac{25\cdots 66}{26\cdots 33}a^{17}+\frac{38\cdots 28}{26\cdots 33}a^{16}+\frac{73\cdots 98}{26\cdots 33}a^{15}-\frac{40\cdots 78}{26\cdots 33}a^{14}-\frac{10\cdots 06}{26\cdots 33}a^{13}-\frac{19\cdots 58}{26\cdots 33}a^{12}+\frac{84\cdots 74}{26\cdots 33}a^{11}+\frac{57\cdots 82}{26\cdots 33}a^{10}-\frac{34\cdots 54}{26\cdots 33}a^{9}-\frac{15\cdots 99}{11\cdots 71}a^{8}+\frac{64\cdots 08}{26\cdots 33}a^{7}+\frac{82\cdots 64}{26\cdots 33}a^{6}-\frac{39\cdots 72}{26\cdots 33}a^{5}-\frac{66\cdots 87}{26\cdots 33}a^{4}-\frac{11\cdots 44}{26\cdots 33}a^{3}+\frac{94\cdots 03}{26\cdots 33}a^{2}+\frac{23\cdots 53}{26\cdots 33}a+\frac{14\cdots 94}{11\cdots 71}$, $\frac{25\cdots 18}{19\cdots 07}a^{19}-\frac{48\cdots 23}{19\cdots 07}a^{18}-\frac{18\cdots 31}{19\cdots 07}a^{17}+\frac{14\cdots 27}{19\cdots 07}a^{16}+\frac{54\cdots 00}{19\cdots 07}a^{15}+\frac{90\cdots 57}{19\cdots 07}a^{14}-\frac{78\cdots 96}{19\cdots 07}a^{13}-\frac{68\cdots 00}{19\cdots 07}a^{12}+\frac{57\cdots 70}{19\cdots 07}a^{11}+\frac{81\cdots 96}{19\cdots 07}a^{10}-\frac{88\cdots 41}{84\cdots 09}a^{9}-\frac{40\cdots 55}{19\cdots 07}a^{8}+\frac{24\cdots 08}{19\cdots 07}a^{7}+\frac{83\cdots 79}{19\cdots 07}a^{6}+\frac{18\cdots 66}{19\cdots 07}a^{5}-\frac{52\cdots 95}{19\cdots 07}a^{4}-\frac{32\cdots 10}{19\cdots 07}a^{3}-\frac{28\cdots 77}{19\cdots 07}a^{2}+\frac{76\cdots 43}{19\cdots 07}a+\frac{13\cdots 80}{84\cdots 09}$, $\frac{72\cdots 33}{26\cdots 33}a^{19}-\frac{20\cdots 91}{26\cdots 33}a^{18}-\frac{50\cdots 92}{26\cdots 33}a^{17}+\frac{88\cdots 44}{26\cdots 33}a^{16}+\frac{14\cdots 03}{26\cdots 33}a^{15}-\frac{11\cdots 26}{26\cdots 33}a^{14}-\frac{22\cdots 06}{26\cdots 33}a^{13}+\frac{17\cdots 77}{26\cdots 33}a^{12}+\frac{18\cdots 79}{26\cdots 33}a^{11}+\frac{75\cdots 77}{26\cdots 33}a^{10}-\frac{80\cdots 14}{26\cdots 33}a^{9}-\frac{59\cdots 67}{26\cdots 33}a^{8}+\frac{17\cdots 04}{26\cdots 33}a^{7}+\frac{17\cdots 06}{26\cdots 33}a^{6}-\frac{17\cdots 88}{26\cdots 33}a^{5}-\frac{19\cdots 24}{26\cdots 33}a^{4}+\frac{20\cdots 24}{11\cdots 71}a^{3}+\frac{76\cdots 72}{26\cdots 33}a^{2}+\frac{12\cdots 78}{26\cdots 33}a+\frac{15\cdots 15}{11\cdots 71}$, $\frac{19\cdots 68}{26\cdots 33}a^{19}-\frac{45\cdots 76}{26\cdots 33}a^{18}-\frac{14\cdots 08}{26\cdots 33}a^{17}+\frac{16\cdots 06}{26\cdots 33}a^{16}+\frac{41\cdots 39}{26\cdots 33}a^{15}-\frac{10\cdots 45}{26\cdots 33}a^{14}-\frac{60\cdots 15}{26\cdots 33}a^{13}-\frac{12\cdots 76}{11\cdots 71}a^{12}+\frac{45\cdots 52}{26\cdots 33}a^{11}+\frac{44\cdots 09}{26\cdots 33}a^{10}-\frac{17\cdots 72}{26\cdots 33}a^{9}-\frac{23\cdots 58}{26\cdots 33}a^{8}+\frac{27\cdots 33}{26\cdots 33}a^{7}+\frac{51\cdots 25}{26\cdots 33}a^{6}-\frac{42\cdots 03}{26\cdots 33}a^{5}-\frac{34\cdots 16}{26\cdots 33}a^{4}-\frac{12\cdots 36}{26\cdots 33}a^{3}+\frac{24\cdots 81}{26\cdots 33}a^{2}+\frac{40\cdots 07}{26\cdots 33}a+\frac{43\cdots 75}{11\cdots 71}$, $\frac{11\cdots 55}{26\cdots 33}a^{19}-\frac{15\cdots 07}{26\cdots 33}a^{18}-\frac{84\cdots 47}{26\cdots 33}a^{17}+\frac{24\cdots 12}{26\cdots 33}a^{16}+\frac{25\cdots 81}{26\cdots 33}a^{15}+\frac{14\cdots 41}{26\cdots 33}a^{14}-\frac{36\cdots 97}{26\cdots 33}a^{13}-\frac{45\cdots 33}{26\cdots 33}a^{12}+\frac{26\cdots 53}{26\cdots 33}a^{11}+\frac{47\cdots 28}{26\cdots 33}a^{10}-\frac{91\cdots 61}{26\cdots 33}a^{9}-\frac{22\cdots 87}{26\cdots 33}a^{8}+\frac{41\cdots 49}{11\cdots 71}a^{7}+\frac{44\cdots 99}{26\cdots 33}a^{6}+\frac{12\cdots 88}{26\cdots 33}a^{5}-\frac{26\cdots 19}{26\cdots 33}a^{4}-\frac{17\cdots 14}{26\cdots 33}a^{3}-\frac{22\cdots 72}{26\cdots 33}a^{2}-\frac{21\cdots 33}{26\cdots 33}a-\frac{15\cdots 46}{11\cdots 71}$, $\frac{13\cdots 16}{26\cdots 33}a^{19}-\frac{87\cdots 34}{11\cdots 71}a^{18}-\frac{95\cdots 66}{26\cdots 33}a^{17}+\frac{38\cdots 22}{26\cdots 33}a^{16}+\frac{27\cdots 38}{26\cdots 33}a^{15}+\frac{14\cdots 69}{26\cdots 33}a^{14}-\frac{39\cdots 71}{26\cdots 33}a^{13}-\frac{49\cdots 14}{26\cdots 33}a^{12}+\frac{28\cdots 91}{26\cdots 33}a^{11}+\frac{52\cdots 53}{26\cdots 33}a^{10}-\frac{87\cdots 79}{26\cdots 33}a^{9}-\frac{24\cdots 57}{26\cdots 33}a^{8}+\frac{43\cdots 38}{26\cdots 33}a^{7}+\frac{47\cdots 76}{26\cdots 33}a^{6}+\frac{26\cdots 39}{26\cdots 33}a^{5}-\frac{23\cdots 45}{26\cdots 33}a^{4}-\frac{26\cdots 70}{26\cdots 33}a^{3}-\frac{70\cdots 52}{26\cdots 33}a^{2}-\frac{93\cdots 12}{26\cdots 33}a+\frac{29\cdots 74}{11\cdots 71}$, $\frac{17\cdots 20}{26\cdots 33}a^{19}-\frac{31\cdots 20}{26\cdots 33}a^{18}-\frac{12\cdots 88}{26\cdots 33}a^{17}+\frac{85\cdots 19}{26\cdots 33}a^{16}+\frac{36\cdots 32}{26\cdots 33}a^{15}+\frac{85\cdots 88}{26\cdots 33}a^{14}-\frac{52\cdots 72}{26\cdots 33}a^{13}-\frac{49\cdots 86}{26\cdots 33}a^{12}+\frac{38\cdots 64}{26\cdots 33}a^{11}+\frac{57\cdots 38}{26\cdots 33}a^{10}-\frac{13\cdots 06}{26\cdots 33}a^{9}-\frac{27\cdots 97}{26\cdots 33}a^{8}+\frac{14\cdots 36}{26\cdots 33}a^{7}+\frac{57\cdots 74}{26\cdots 33}a^{6}+\frac{15\cdots 86}{26\cdots 33}a^{5}-\frac{34\cdots 17}{26\cdots 33}a^{4}-\frac{23\cdots 34}{26\cdots 33}a^{3}-\frac{32\cdots 41}{26\cdots 33}a^{2}+\frac{37\cdots 69}{26\cdots 33}a+\frac{31\cdots 88}{11\cdots 71}$, $\frac{64\cdots 54}{26\cdots 33}a^{19}-\frac{16\cdots 80}{26\cdots 33}a^{18}-\frac{45\cdots 31}{26\cdots 33}a^{17}+\frac{64\cdots 98}{26\cdots 33}a^{16}+\frac{13\cdots 43}{26\cdots 33}a^{15}-\frac{59\cdots 28}{26\cdots 33}a^{14}-\frac{19\cdots 26}{26\cdots 33}a^{13}-\frac{54\cdots 18}{26\cdots 33}a^{12}+\frac{14\cdots 84}{26\cdots 33}a^{11}+\frac{12\cdots 62}{26\cdots 33}a^{10}-\frac{59\cdots 89}{26\cdots 33}a^{9}-\frac{71\cdots 62}{26\cdots 33}a^{8}+\frac{10\cdots 68}{26\cdots 33}a^{7}+\frac{17\cdots 84}{26\cdots 33}a^{6}-\frac{58\cdots 99}{26\cdots 33}a^{5}-\frac{15\cdots 72}{26\cdots 33}a^{4}-\frac{14\cdots 24}{26\cdots 33}a^{3}+\frac{39\cdots 68}{26\cdots 33}a^{2}+\frac{65\cdots 63}{26\cdots 33}a+\frac{90\cdots 96}{11\cdots 71}$, $\frac{14\cdots 59}{26\cdots 33}a^{19}-\frac{29\cdots 49}{26\cdots 33}a^{18}-\frac{10\cdots 39}{26\cdots 33}a^{17}+\frac{94\cdots 39}{26\cdots 33}a^{16}+\frac{30\cdots 91}{26\cdots 33}a^{15}+\frac{90\cdots 54}{26\cdots 33}a^{14}-\frac{44\cdots 67}{26\cdots 33}a^{13}-\frac{32\cdots 59}{26\cdots 33}a^{12}+\frac{33\cdots 94}{26\cdots 33}a^{11}+\frac{41\cdots 39}{26\cdots 33}a^{10}-\frac{12\cdots 77}{26\cdots 33}a^{9}-\frac{21\cdots 38}{26\cdots 33}a^{8}+\frac{16\cdots 25}{26\cdots 33}a^{7}+\frac{44\cdots 50}{26\cdots 33}a^{6}+\frac{45\cdots 81}{26\cdots 33}a^{5}-\frac{12\cdots 33}{11\cdots 71}a^{4}-\frac{14\cdots 49}{26\cdots 33}a^{3}-\frac{13\cdots 41}{26\cdots 33}a^{2}+\frac{24\cdots 65}{26\cdots 33}a+\frac{30\cdots 63}{11\cdots 71}$, $\frac{10\cdots 49}{26\cdots 33}a^{19}-\frac{35\cdots 71}{26\cdots 33}a^{18}-\frac{68\cdots 42}{26\cdots 33}a^{17}+\frac{17\cdots 24}{26\cdots 33}a^{16}+\frac{19\cdots 04}{26\cdots 33}a^{15}-\frac{29\cdots 60}{26\cdots 33}a^{14}-\frac{29\cdots 04}{26\cdots 33}a^{13}+\frac{21\cdots 49}{26\cdots 33}a^{12}+\frac{25\cdots 29}{26\cdots 33}a^{11}-\frac{38\cdots 30}{26\cdots 33}a^{10}-\frac{11\cdots 28}{26\cdots 33}a^{9}-\frac{26\cdots 92}{26\cdots 33}a^{8}+\frac{29\cdots 82}{26\cdots 33}a^{7}+\frac{13\cdots 57}{26\cdots 33}a^{6}-\frac{36\cdots 95}{26\cdots 33}a^{5}-\frac{23\cdots 14}{26\cdots 33}a^{4}+\frac{15\cdots 70}{26\cdots 33}a^{3}+\frac{12\cdots 14}{26\cdots 33}a^{2}+\frac{10\cdots 81}{26\cdots 33}a-\frac{23\cdots 03}{11\cdots 71}$, $\frac{14\cdots 39}{26\cdots 33}a^{19}-\frac{27\cdots 75}{26\cdots 33}a^{18}-\frac{10\cdots 87}{26\cdots 33}a^{17}+\frac{85\cdots 63}{26\cdots 33}a^{16}+\frac{29\cdots 81}{26\cdots 33}a^{15}+\frac{26\cdots 44}{26\cdots 33}a^{14}-\frac{43\cdots 47}{26\cdots 33}a^{13}-\frac{34\cdots 63}{26\cdots 33}a^{12}+\frac{32\cdots 38}{26\cdots 33}a^{11}+\frac{42\cdots 86}{26\cdots 33}a^{10}-\frac{11\cdots 41}{26\cdots 33}a^{9}-\frac{21\cdots 61}{26\cdots 33}a^{8}+\frac{14\cdots 57}{26\cdots 33}a^{7}+\frac{44\cdots 96}{26\cdots 33}a^{6}+\frac{68\cdots 75}{26\cdots 33}a^{5}-\frac{28\cdots 19}{26\cdots 33}a^{4}-\frac{15\cdots 01}{26\cdots 33}a^{3}-\frac{98\cdots 67}{26\cdots 33}a^{2}+\frac{19\cdots 17}{26\cdots 33}a+\frac{13\cdots 18}{11\cdots 71}$, $\frac{14\cdots 65}{26\cdots 33}a^{19}-\frac{28\cdots 58}{26\cdots 33}a^{18}-\frac{10\cdots 53}{26\cdots 33}a^{17}+\frac{87\cdots 37}{26\cdots 33}a^{16}+\frac{30\cdots 12}{26\cdots 33}a^{15}+\frac{26\cdots 21}{26\cdots 33}a^{14}-\frac{45\cdots 52}{26\cdots 33}a^{13}-\frac{34\cdots 96}{26\cdots 33}a^{12}+\frac{34\cdots 84}{26\cdots 33}a^{11}+\frac{43\cdots 18}{26\cdots 33}a^{10}-\frac{12\cdots 86}{26\cdots 33}a^{9}-\frac{21\cdots 77}{26\cdots 33}a^{8}+\frac{18\cdots 85}{26\cdots 33}a^{7}+\frac{46\cdots 33}{26\cdots 33}a^{6}+\frac{11\cdots 56}{26\cdots 33}a^{5}-\frac{31\cdots 79}{26\cdots 33}a^{4}-\frac{11\cdots 94}{26\cdots 33}a^{3}+\frac{13\cdots 25}{26\cdots 33}a^{2}+\frac{23\cdots 49}{26\cdots 33}a-\frac{87\cdots 72}{11\cdots 71}$, $\frac{33\cdots 39}{26\cdots 33}a^{19}-\frac{54\cdots 04}{26\cdots 33}a^{18}-\frac{24\cdots 36}{26\cdots 33}a^{17}+\frac{12\cdots 75}{26\cdots 33}a^{16}+\frac{71\cdots 68}{26\cdots 33}a^{15}+\frac{26\cdots 62}{26\cdots 33}a^{14}-\frac{10\cdots 02}{26\cdots 33}a^{13}-\frac{11\cdots 10}{26\cdots 33}a^{12}+\frac{78\cdots 03}{26\cdots 33}a^{11}+\frac{12\cdots 25}{26\cdots 33}a^{10}-\frac{28\cdots 42}{26\cdots 33}a^{9}-\frac{62\cdots 82}{26\cdots 33}a^{8}+\frac{36\cdots 60}{26\cdots 33}a^{7}+\frac{13\cdots 10}{26\cdots 33}a^{6}+\frac{20\cdots 74}{26\cdots 33}a^{5}-\frac{10\cdots 84}{26\cdots 33}a^{4}-\frac{48\cdots 51}{26\cdots 33}a^{3}+\frac{19\cdots 73}{26\cdots 33}a^{2}+\frac{86\cdots 11}{26\cdots 33}a-\frac{25\cdots 37}{11\cdots 71}$, $\frac{31\cdots 68}{26\cdots 33}a^{19}-\frac{49\cdots 26}{26\cdots 33}a^{18}-\frac{99\cdots 45}{11\cdots 71}a^{17}+\frac{10\cdots 03}{26\cdots 33}a^{16}+\frac{66\cdots 23}{26\cdots 33}a^{15}+\frac{31\cdots 29}{26\cdots 33}a^{14}-\frac{94\cdots 24}{26\cdots 33}a^{13}-\frac{11\cdots 47}{26\cdots 33}a^{12}+\frac{67\cdots 02}{26\cdots 33}a^{11}+\frac{12\cdots 05}{26\cdots 33}a^{10}-\frac{21\cdots 78}{26\cdots 33}a^{9}-\frac{56\cdots 29}{26\cdots 33}a^{8}+\frac{11\cdots 36}{26\cdots 33}a^{7}+\frac{47\cdots 95}{11\cdots 71}a^{6}+\frac{58\cdots 77}{26\cdots 33}a^{5}-\frac{52\cdots 26}{26\cdots 33}a^{4}-\frac{61\cdots 24}{26\cdots 33}a^{3}-\frac{17\cdots 05}{26\cdots 33}a^{2}-\frac{81\cdots 78}{26\cdots 33}a+\frac{14\cdots 20}{11\cdots 71}$, $\frac{31\cdots 21}{26\cdots 33}a^{19}-\frac{74\cdots 33}{26\cdots 33}a^{18}-\frac{22\cdots 25}{26\cdots 33}a^{17}+\frac{27\cdots 55}{26\cdots 33}a^{16}+\frac{64\cdots 93}{26\cdots 33}a^{15}-\frac{15\cdots 69}{26\cdots 33}a^{14}-\frac{93\cdots 22}{26\cdots 33}a^{13}-\frac{46\cdots 43}{26\cdots 33}a^{12}+\frac{69\cdots 86}{26\cdots 33}a^{11}+\frac{73\cdots 85}{26\cdots 33}a^{10}-\frac{25\cdots 85}{26\cdots 33}a^{9}-\frac{38\cdots 77}{26\cdots 33}a^{8}+\frac{35\cdots 30}{26\cdots 33}a^{7}+\frac{83\cdots 55}{26\cdots 33}a^{6}+\frac{66\cdots 33}{26\cdots 33}a^{5}-\frac{53\cdots 24}{26\cdots 33}a^{4}-\frac{27\cdots 33}{26\cdots 33}a^{3}-\frac{18\cdots 54}{26\cdots 33}a^{2}+\frac{20\cdots 08}{26\cdots 33}a+\frac{21\cdots 98}{11\cdots 71}$, $\frac{37\cdots 17}{26\cdots 33}a^{19}-\frac{90\cdots 90}{26\cdots 33}a^{18}-\frac{26\cdots 24}{26\cdots 33}a^{17}+\frac{34\cdots 86}{26\cdots 33}a^{16}+\frac{77\cdots 49}{26\cdots 33}a^{15}-\frac{27\cdots 45}{26\cdots 33}a^{14}-\frac{11\cdots 34}{26\cdots 33}a^{13}-\frac{41\cdots 45}{26\cdots 33}a^{12}+\frac{86\cdots 62}{26\cdots 33}a^{11}+\frac{33\cdots 82}{11\cdots 71}a^{10}-\frac{33\cdots 58}{26\cdots 33}a^{9}-\frac{18\cdots 47}{11\cdots 71}a^{8}+\frac{54\cdots 68}{26\cdots 33}a^{7}+\frac{98\cdots 39}{26\cdots 33}a^{6}-\frac{14\cdots 77}{26\cdots 33}a^{5}-\frac{74\cdots 68}{26\cdots 33}a^{4}-\frac{20\cdots 57}{26\cdots 33}a^{3}+\frac{77\cdots 82}{26\cdots 33}a^{2}+\frac{21\cdots 20}{26\cdots 33}a+\frac{82\cdots 67}{11\cdots 71}$
|
| |
| Regulator: | \( 5124246016990 \) (assuming GRH) |
| |
| Unit signature rank: | \( 18 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 5124246016990 \cdot 1}{2\cdot\sqrt{138881946372451702408146629446494920973}}\cr\approx \mathstrut & 0.227969629110476 \end{aligned}\] (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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{78 +18 \sqrt{13}})\), \(\Q(\zeta_{11})^+\), 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 | $20$ | R | $20$ | $20$ | R | R | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/23.1.0.1}{1} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | $20$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | $20$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.5.2.5a1.2 | $x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
| 3.5.2.5a1.2 | $x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ | |
|
\(11\)
| 11.4.5.16a1.4 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1600 x^{15} + 6440 x^{14} + 19600 x^{13} + 48360 x^{12} + 122000 x^{11} + 252208 x^{10} + 442800 x^{9} + 706720 x^{8} + 904000 x^{7} + 817760 x^{6} + 498400 x^{5} + 201200 x^{4} + 52800 x^{3} + 8640 x^{2} + 800 x + 43$ | $5$ | $4$ | $16$ | 20T1 | $$[\ ]_{5}^{4}$$ |
|
\(13\)
| 13.5.4.15a1.4 | $x^{20} + 16 x^{16} + 44 x^{15} + 96 x^{12} + 528 x^{11} + 726 x^{10} + 256 x^{8} + 2112 x^{7} + 5808 x^{6} + 5324 x^{5} + 256 x^{4} + 2816 x^{3} + 11616 x^{2} + 21296 x + 14654$ | $4$ | $5$ | $15$ | 20T1 | $$[\ ]_{4}^{5}$$ |