Number field 20.20.10818327022517881605935827576487172209050638458251953125.1

• Label: 20.20.108...125.1
• Degree: $20$
• Signature: $[20, 0]$
• Discriminant: $1.082\times 10^{55}$
• Root discriminant: \(564.56\)
• Ramified primes: $5,11,41$
• Class number: $275$ (GRH)
• Class group: [5, 55] (GRH)
• Galois group: $C_{20}$ (as 20T1)

Normalized defining polynomial

x^20 - x^19 - 724*x^18 + 1933*x^17 + 207069*x^16 - 879414*x^15 - 29244939*x^14 + 168000394*x^13 + 2068450902*x^12 - 15163663108*x^11 - 65471435699*x^10 + 640279637052*x^9 + 723107842172*x^8 - 12962056979224*x^7 + 1904187630105*x^6 + 126523888290302*x^5 - 49209916894584*x^4 - 625770152130483*x^3 + 20489361496947*x^2 + 1354721938371869*x + 874798166506991

Invariants

Degree:		$20$
Signature:		$[20, 0]$
Discriminant:		\(10818327022517881605935827576487172209050638458251953125\) \(\medspace = 5^{15}\cdot 11^{18}\cdot 41^{16}\)
Root discriminant:		\(564.56\)
Galois root discriminant:		$5^{3/4}11^{9/10}41^{4/5}\approx 564.557271214292$
Ramified primes:		\(5\), \(11\), \(41\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_{20}$
This field is Galois and abelian over $\Q$.
Conductor:		\(2255=5\cdot 11\cdot 41\)
Dirichlet character group:		$\lbrace$$\chi_{2255}(1,·)$, $\chi_{2255}(57,·)$, $\chi_{2255}(182,·)$, $\chi_{2255}(201,·)$, $\chi_{2255}(283,·)$, $\chi_{2255}(346,·)$, $\chi_{2255}(502,·)$, $\chi_{2255}(508,·)$, $\chi_{2255}(633,·)$, $\chi_{2255}(953,·)$, $\chi_{2255}(994,·)$, $\chi_{2255}(1164,·)$, $\chi_{2255}(1354,·)$, $\chi_{2255}(1554,·)$, $\chi_{2255}(1682,·)$, $\chi_{2255}(1699,·)$, $\chi_{2255}(1896,·)$, $\chi_{2255}(2066,·)$, $\chi_{2255}(2087,·)$, $\chi_{2255}(2133,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{29321}a^{18}-\frac{5588}{29321}a^{17}+\frac{13627}{29321}a^{16}-\frac{3586}{29321}a^{15}+\frac{1233}{29321}a^{14}+\frac{11616}{29321}a^{13}-\frac{5226}{29321}a^{12}-\frac{7133}{29321}a^{11}-\frac{12058}{29321}a^{10}-\frac{6536}{29321}a^{9}+\frac{1988}{29321}a^{8}-\frac{247}{29321}a^{7}-\frac{5167}{29321}a^{6}-\frac{1924}{29321}a^{5}+\frac{7665}{29321}a^{4}-\frac{13380}{29321}a^{3}-\frac{1940}{29321}a^{2}-\frac{7454}{29321}a+\frac{13887}{29321}$, $\frac{1}{61\cdots 01}a^{19}+\frac{70\cdots 64}{61\cdots 01}a^{18}-\frac{20\cdots 82}{61\cdots 01}a^{17}-\frac{22\cdots 16}{61\cdots 01}a^{16}+\frac{19\cdots 27}{61\cdots 01}a^{15}+\frac{68\cdots 95}{61\cdots 01}a^{14}+\frac{18\cdots 00}{61\cdots 01}a^{13}+\frac{29\cdots 57}{61\cdots 01}a^{12}+\frac{68\cdots 02}{61\cdots 01}a^{11}-\frac{11\cdots 21}{61\cdots 01}a^{10}-\frac{40\cdots 06}{61\cdots 01}a^{9}+\frac{17\cdots 44}{61\cdots 01}a^{8}+\frac{25\cdots 50}{61\cdots 01}a^{7}+\frac{17\cdots 73}{61\cdots 01}a^{6}-\frac{12\cdots 03}{61\cdots 01}a^{5}+\frac{16\cdots 26}{61\cdots 01}a^{4}-\frac{14\cdots 70}{61\cdots 01}a^{3}-\frac{24\cdots 46}{61\cdots 01}a^{2}+\frac{46\cdots 52}{61\cdots 01}a-\frac{25\cdots 63}{61\cdots 01}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		$C_{5}\times C_{55}$, which has order $275$ (assuming GRH)
Narrow class group:		$C_{110}\times C_{10}$, which has order $1100$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{24\cdots 52}{50\cdots 99}a^{19}+\frac{15\cdots 19}{50\cdots 99}a^{18}-\frac{17\cdots 95}{50\cdots 99}a^{17}-\frac{83\cdots 65}{50\cdots 99}a^{16}+\frac{50\cdots 28}{50\cdots 99}a^{15}+\frac{15\cdots 04}{50\cdots 99}a^{14}-\frac{76\cdots 59}{50\cdots 99}a^{13}-\frac{12\cdots 53}{50\cdots 99}a^{12}+\frac{64\cdots 48}{50\cdots 99}a^{11}+\frac{27\cdots 78}{50\cdots 99}a^{10}-\frac{30\cdots 04}{50\cdots 99}a^{9}+\frac{64\cdots 54}{50\cdots 99}a^{8}+\frac{79\cdots 56}{50\cdots 99}a^{7}-\frac{34\cdots 58}{50\cdots 99}a^{6}-\frac{11\cdots 31}{50\cdots 99}a^{5}+\frac{34\cdots 95}{50\cdots 99}a^{4}+\frac{81\cdots 50}{50\cdots 99}a^{3}+\frac{13\cdots 77}{50\cdots 99}a^{2}-\frac{23\cdots 84}{50\cdots 99}a-\frac{18\cdots 55}{50\cdots 99}$, $\frac{13\cdots 30}{14\cdots 61}a^{19}+\frac{60\cdots 08}{14\cdots 61}a^{18}-\frac{97\cdots 52}{14\cdots 61}a^{17}-\frac{26\cdots 41}{14\cdots 61}a^{16}+\frac{27\cdots 78}{14\cdots 61}a^{15}+\frac{29\cdots 87}{14\cdots 61}a^{14}-\frac{40\cdots 98}{14\cdots 61}a^{13}+\frac{57\cdots 30}{54\cdots 69}a^{12}+\frac{32\cdots 78}{14\cdots 61}a^{11}-\frac{46\cdots 73}{14\cdots 61}a^{10}-\frac{13\cdots 24}{14\cdots 61}a^{9}+\frac{26\cdots 11}{14\cdots 61}a^{8}+\frac{31\cdots 10}{14\cdots 61}a^{7}-\frac{59\cdots 26}{14\cdots 61}a^{6}-\frac{38\cdots 40}{14\cdots 61}a^{5}+\frac{53\cdots 68}{14\cdots 61}a^{4}+\frac{24\cdots 23}{14\cdots 61}a^{3}-\frac{10\cdots 81}{14\cdots 61}a^{2}-\frac{65\cdots 49}{14\cdots 61}a-\frac{38\cdots 90}{14\cdots 61}$, $\frac{32\cdots 09}{14\cdots 61}a^{19}+\frac{51\cdots 13}{14\cdots 61}a^{18}-\frac{23\cdots 89}{14\cdots 61}a^{17}+\frac{28\cdots 99}{14\cdots 61}a^{16}+\frac{67\cdots 15}{14\cdots 61}a^{15}-\frac{11\cdots 75}{14\cdots 61}a^{14}-\frac{96\cdots 44}{14\cdots 61}a^{13}+\frac{29\cdots 78}{14\cdots 61}a^{12}+\frac{72\cdots 43}{14\cdots 61}a^{11}-\frac{29\cdots 01}{14\cdots 61}a^{10}-\frac{26\cdots 69}{14\cdots 61}a^{9}+\frac{12\cdots 96}{14\cdots 61}a^{8}+\frac{48\cdots 64}{14\cdots 61}a^{7}-\frac{24\cdots 57}{14\cdots 61}a^{6}-\frac{43\cdots 19}{14\cdots 61}a^{5}+\frac{19\cdots 16}{14\cdots 61}a^{4}+\frac{24\cdots 51}{14\cdots 61}a^{3}-\frac{57\cdots 60}{14\cdots 61}a^{2}-\frac{76\cdots 33}{14\cdots 61}a-\frac{16\cdots 10}{14\cdots 61}$, $\frac{17\cdots 35}{32\cdots 99}a^{19}+\frac{12\cdots 50}{32\cdots 99}a^{18}-\frac{12\cdots 85}{32\cdots 99}a^{17}-\frac{65\cdots 70}{32\cdots 99}a^{16}+\frac{34\cdots 70}{32\cdots 99}a^{15}+\frac{12\cdots 30}{32\cdots 99}a^{14}-\frac{51\cdots 70}{32\cdots 99}a^{13}-\frac{11\cdots 90}{32\cdots 99}a^{12}+\frac{42\cdots 25}{32\cdots 99}a^{11}+\frac{44\cdots 80}{32\cdots 99}a^{10}-\frac{19\cdots 95}{32\cdots 99}a^{9}-\frac{62\cdots 60}{32\cdots 99}a^{8}+\frac{50\cdots 90}{32\cdots 99}a^{7}+\frac{30\cdots 40}{32\cdots 99}a^{6}-\frac{70\cdots 31}{32\cdots 99}a^{5}-\frac{77\cdots 15}{32\cdots 99}a^{4}+\frac{48\cdots 85}{32\cdots 99}a^{3}+\frac{16\cdots 80}{32\cdots 99}a^{2}-\frac{13\cdots 95}{32\cdots 99}a-\frac{11\cdots 57}{32\cdots 99}$, $\frac{24\cdots 15}{32\cdots 99}a^{19}+\frac{66\cdots 35}{32\cdots 99}a^{18}-\frac{17\cdots 65}{32\cdots 99}a^{17}-\frac{18\cdots 55}{32\cdots 99}a^{16}+\frac{49\cdots 95}{32\cdots 99}a^{15}-\frac{25\cdots 80}{32\cdots 99}a^{14}-\frac{71\cdots 80}{32\cdots 99}a^{13}+\frac{13\cdots 65}{32\cdots 99}a^{12}+\frac{55\cdots 10}{32\cdots 99}a^{11}-\frac{15\cdots 24}{32\cdots 99}a^{10}-\frac{22\cdots 05}{32\cdots 99}a^{9}+\frac{72\cdots 30}{32\cdots 99}a^{8}+\frac{46\cdots 85}{32\cdots 99}a^{7}-\frac{14\cdots 35}{32\cdots 99}a^{6}-\frac{52\cdots 53}{32\cdots 99}a^{5}+\frac{12\cdots 90}{32\cdots 99}a^{4}+\frac{36\cdots 75}{32\cdots 99}a^{3}-\frac{28\cdots 30}{32\cdots 99}a^{2}-\frac{11\cdots 05}{32\cdots 99}a-\frac{69\cdots 55}{32\cdots 99}$, $\frac{37\cdots 51}{32\cdots 99}a^{19}-\frac{39\cdots 41}{32\cdots 99}a^{18}-\frac{29\cdots 33}{32\cdots 99}a^{17}+\frac{31\cdots 21}{32\cdots 99}a^{16}+\frac{87\cdots 24}{32\cdots 99}a^{15}-\frac{10\cdots 56}{32\cdots 99}a^{14}-\frac{12\cdots 79}{32\cdots 99}a^{13}+\frac{15\cdots 78}{32\cdots 99}a^{12}+\frac{85\cdots 73}{32\cdots 99}a^{11}-\frac{12\cdots 60}{32\cdots 99}a^{10}-\frac{22\cdots 82}{32\cdots 99}a^{9}+\frac{52\cdots 88}{32\cdots 99}a^{8}+\frac{66\cdots 79}{32\cdots 99}a^{7}-\frac{10\cdots 85}{32\cdots 99}a^{6}+\frac{41\cdots 27}{32\cdots 99}a^{5}+\frac{96\cdots 39}{32\cdots 99}a^{4}-\frac{20\cdots 42}{32\cdots 99}a^{3}-\frac{37\cdots 92}{32\cdots 99}a^{2}-\frac{10\cdots 79}{32\cdots 99}a+\frac{24\cdots 01}{32\cdots 99}$, $\frac{10\cdots 90}{22\cdots 29}a^{19}+\frac{35\cdots 98}{22\cdots 29}a^{18}-\frac{74\cdots 17}{22\cdots 29}a^{17}-\frac{11\cdots 16}{22\cdots 29}a^{16}+\frac{20\cdots 61}{22\cdots 29}a^{15}-\frac{42\cdots 42}{22\cdots 29}a^{14}-\frac{28\cdots 48}{22\cdots 29}a^{13}+\frac{49\cdots 13}{22\cdots 29}a^{12}+\frac{19\cdots 86}{22\cdots 29}a^{11}-\frac{58\cdots 72}{22\cdots 29}a^{10}-\frac{59\cdots 75}{22\cdots 29}a^{9}+\frac{23\cdots 74}{22\cdots 29}a^{8}+\frac{59\cdots 44}{22\cdots 29}a^{7}-\frac{32\cdots 29}{22\cdots 29}a^{6}+\frac{33\cdots 28}{22\cdots 29}a^{5}+\frac{77\cdots 06}{22\cdots 29}a^{4}-\frac{65\cdots 28}{22\cdots 29}a^{3}+\frac{27\cdots 08}{22\cdots 29}a^{2}+\frac{22\cdots 17}{22\cdots 29}a+\frac{11\cdots 53}{22\cdots 29}$, $\frac{14\cdots 05}{22\cdots 29}a^{19}+\frac{56\cdots 71}{22\cdots 29}a^{18}-\frac{10\cdots 04}{22\cdots 29}a^{17}-\frac{22\cdots 51}{22\cdots 29}a^{16}+\frac{29\cdots 69}{22\cdots 29}a^{15}+\frac{17\cdots 14}{22\cdots 29}a^{14}-\frac{42\cdots 91}{22\cdots 29}a^{13}+\frac{33\cdots 05}{22\cdots 29}a^{12}+\frac{33\cdots 09}{22\cdots 29}a^{11}-\frac{59\cdots 56}{22\cdots 29}a^{10}-\frac{13\cdots 61}{22\cdots 29}a^{9}+\frac{29\cdots 30}{22\cdots 29}a^{8}+\frac{30\cdots 87}{22\cdots 29}a^{7}-\frac{60\cdots 44}{22\cdots 29}a^{6}-\frac{35\cdots 69}{22\cdots 29}a^{5}+\frac{46\cdots 13}{22\cdots 29}a^{4}+\frac{23\cdots 02}{22\cdots 29}a^{3}-\frac{63\cdots 44}{22\cdots 29}a^{2}-\frac{70\cdots 36}{22\cdots 29}a-\frac{50\cdots 42}{22\cdots 29}$, $\frac{34\cdots 99}{22\cdots 29}a^{19}+\frac{43\cdots 48}{22\cdots 29}a^{18}-\frac{23\cdots 60}{22\cdots 29}a^{17}-\frac{27\cdots 50}{22\cdots 29}a^{16}+\frac{70\cdots 42}{22\cdots 29}a^{15}+\frac{65\cdots 28}{22\cdots 29}a^{14}-\frac{11\cdots 03}{22\cdots 29}a^{13}-\frac{78\cdots 06}{22\cdots 29}a^{12}+\frac{10\cdots 66}{22\cdots 29}a^{11}+\frac{48\cdots 04}{22\cdots 29}a^{10}-\frac{60\cdots 23}{22\cdots 29}a^{9}-\frac{14\cdots 32}{22\cdots 29}a^{8}+\frac{18\cdots 12}{22\cdots 29}a^{7}+\frac{19\cdots 89}{22\cdots 29}a^{6}-\frac{29\cdots 97}{22\cdots 29}a^{5}-\frac{98\cdots 00}{22\cdots 29}a^{4}+\frac{21\cdots 00}{22\cdots 29}a^{3}+\frac{76\cdots 54}{22\cdots 29}a^{2}-\frac{56\cdots 38}{22\cdots 29}a-\frac{47\cdots 73}{22\cdots 29}$, $\frac{20\cdots 86}{32\cdots 99}a^{19}+\frac{83\cdots 09}{32\cdots 99}a^{18}-\frac{14\cdots 18}{32\cdots 99}a^{17}-\frac{34\cdots 49}{32\cdots 99}a^{16}+\frac{43\cdots 94}{32\cdots 99}a^{15}+\frac{29\cdots 74}{32\cdots 99}a^{14}-\frac{63\cdots 49}{32\cdots 99}a^{13}+\frac{44\cdots 88}{32\cdots 99}a^{12}+\frac{51\cdots 98}{32\cdots 99}a^{11}-\frac{85\cdots 80}{32\cdots 99}a^{10}-\frac{22\cdots 77}{32\cdots 99}a^{9}+\frac{45\cdots 28}{32\cdots 99}a^{8}+\frac{51\cdots 69}{32\cdots 99}a^{7}-\frac{10\cdots 45}{32\cdots 99}a^{6}-\frac{66\cdots 04}{32\cdots 99}a^{5}+\frac{95\cdots 24}{32\cdots 99}a^{4}+\frac{46\cdots 43}{32\cdots 99}a^{3}-\frac{21\cdots 12}{32\cdots 99}a^{2}-\frac{14\cdots 74}{32\cdots 99}a-\frac{83\cdots 61}{32\cdots 99}$, $\frac{18\cdots 49}{22\cdots 29}a^{19}+\frac{60\cdots 83}{22\cdots 29}a^{18}-\frac{13\cdots 05}{22\cdots 29}a^{17}-\frac{20\cdots 35}{22\cdots 29}a^{16}+\frac{38\cdots 21}{22\cdots 29}a^{15}-\frac{33\cdots 12}{22\cdots 29}a^{14}-\frac{56\cdots 58}{22\cdots 29}a^{13}+\frac{81\cdots 89}{22\cdots 29}a^{12}+\frac{44\cdots 26}{22\cdots 29}a^{11}-\frac{10\cdots 28}{22\cdots 29}a^{10}-\frac{18\cdots 23}{22\cdots 29}a^{9}+\frac{50\cdots 98}{22\cdots 29}a^{8}+\frac{39\cdots 87}{22\cdots 29}a^{7}-\frac{10\cdots 21}{22\cdots 29}a^{6}-\frac{46\cdots 65}{22\cdots 29}a^{5}+\frac{86\cdots 80}{22\cdots 29}a^{4}+\frac{32\cdots 25}{22\cdots 29}a^{3}-\frac{18\cdots 96}{22\cdots 29}a^{2}-\frac{10\cdots 83}{22\cdots 29}a-\frac{65\cdots 02}{22\cdots 29}$, $\frac{16\cdots 03}{61\cdots 01}a^{19}+\frac{90\cdots 18}{61\cdots 01}a^{18}-\frac{11\cdots 68}{61\cdots 01}a^{17}-\frac{41\cdots 34}{61\cdots 01}a^{16}+\frac{32\cdots 05}{61\cdots 01}a^{15}+\frac{58\cdots 34}{61\cdots 01}a^{14}-\frac{46\cdots 63}{61\cdots 01}a^{13}-\frac{99\cdots 79}{61\cdots 01}a^{12}+\frac{35\cdots 70}{61\cdots 01}a^{11}-\frac{34\cdots 57}{61\cdots 01}a^{10}-\frac{13\cdots 02}{61\cdots 01}a^{9}+\frac{22\cdots 05}{61\cdots 01}a^{8}+\frac{28\cdots 39}{61\cdots 01}a^{7}-\frac{47\cdots 65}{61\cdots 01}a^{6}-\frac{31\cdots 09}{61\cdots 01}a^{5}+\frac{37\cdots 66}{61\cdots 01}a^{4}+\frac{18\cdots 37}{61\cdots 01}a^{3}-\frac{60\cdots 55}{61\cdots 01}a^{2}-\frac{48\cdots 08}{61\cdots 01}a-\frac{29\cdots 06}{61\cdots 01}$, $\frac{54\cdots 61}{61\cdots 01}a^{19}+\frac{14\cdots 47}{61\cdots 01}a^{18}-\frac{39\cdots 63}{61\cdots 01}a^{17}-\frac{36\cdots 51}{61\cdots 01}a^{16}+\frac{11\cdots 35}{61\cdots 01}a^{15}-\frac{76\cdots 68}{61\cdots 01}a^{14}-\frac{15\cdots 99}{61\cdots 01}a^{13}+\frac{33\cdots 73}{61\cdots 01}a^{12}+\frac{12\cdots 39}{61\cdots 01}a^{11}-\frac{38\cdots 91}{61\cdots 01}a^{10}-\frac{46\cdots 60}{61\cdots 01}a^{9}+\frac{17\cdots 28}{61\cdots 01}a^{8}+\frac{89\cdots 74}{61\cdots 01}a^{7}-\frac{34\cdots 94}{61\cdots 01}a^{6}-\frac{89\cdots 34}{61\cdots 01}a^{5}+\frac{28\cdots 58}{61\cdots 01}a^{4}+\frac{54\cdots 61}{61\cdots 01}a^{3}-\frac{77\cdots 87}{61\cdots 01}a^{2}-\frac{16\cdots 99}{61\cdots 01}a-\frac{60\cdots 85}{61\cdots 01}$, $\frac{37\cdots 30}{61\cdots 01}a^{19}+\frac{88\cdots 09}{61\cdots 01}a^{18}-\frac{26\cdots 71}{61\cdots 01}a^{17}-\frac{17\cdots 64}{61\cdots 01}a^{16}+\frac{76\cdots 06}{61\cdots 01}a^{15}-\frac{72\cdots 06}{61\cdots 01}a^{14}-\frac{11\cdots 16}{61\cdots 01}a^{13}+\frac{25\cdots 39}{61\cdots 01}a^{12}+\frac{84\cdots 65}{61\cdots 01}a^{11}-\frac{27\cdots 03}{61\cdots 01}a^{10}-\frac{32\cdots 32}{61\cdots 01}a^{9}+\frac{12\cdots 29}{61\cdots 01}a^{8}+\frac{64\cdots 64}{61\cdots 01}a^{7}-\frac{24\cdots 96}{61\cdots 01}a^{6}-\frac{67\cdots 83}{61\cdots 01}a^{5}+\frac{19\cdots 33}{61\cdots 01}a^{4}+\frac{44\cdots 06}{61\cdots 01}a^{3}-\frac{49\cdots 08}{61\cdots 01}a^{2}-\frac{14\cdots 78}{61\cdots 01}a-\frac{73\cdots 39}{61\cdots 01}$, $\frac{11\cdots 83}{61\cdots 01}a^{19}+\frac{33\cdots 48}{61\cdots 01}a^{18}-\frac{84\cdots 08}{61\cdots 01}a^{17}-\frac{94\cdots 20}{61\cdots 01}a^{16}+\frac{24\cdots 00}{61\cdots 01}a^{15}-\frac{12\cdots 25}{61\cdots 01}a^{14}-\frac{34\cdots 78}{61\cdots 01}a^{13}+\frac{65\cdots 10}{61\cdots 01}a^{12}+\frac{26\cdots 95}{61\cdots 01}a^{11}-\frac{76\cdots 94}{61\cdots 01}a^{10}-\frac{10\cdots 58}{61\cdots 01}a^{9}+\frac{34\cdots 77}{61\cdots 01}a^{8}+\frac{19\cdots 51}{61\cdots 01}a^{7}-\frac{66\cdots 19}{61\cdots 01}a^{6}-\frac{20\cdots 64}{61\cdots 01}a^{5}+\frac{51\cdots 92}{61\cdots 01}a^{4}+\frac{12\cdots 26}{61\cdots 01}a^{3}-\frac{11\cdots 76}{61\cdots 01}a^{2}-\frac{38\cdots 44}{61\cdots 01}a-\frac{20\cdots 23}{61\cdots 01}$, $\frac{51\cdots 63}{61\cdots 01}a^{19}+\frac{31\cdots 99}{61\cdots 01}a^{18}-\frac{36\cdots 74}{61\cdots 01}a^{17}-\frac{16\cdots 77}{61\cdots 01}a^{16}+\frac{95\cdots 39}{56\cdots 89}a^{15}+\frac{29\cdots 81}{61\cdots 01}a^{14}-\frac{15\cdots 63}{61\cdots 01}a^{13}-\frac{20\cdots 42}{61\cdots 01}a^{12}+\frac{13\cdots 55}{61\cdots 01}a^{11}+\frac{20\cdots 89}{61\cdots 01}a^{10}-\frac{60\cdots 74}{61\cdots 01}a^{9}+\frac{31\cdots 21}{61\cdots 01}a^{8}+\frac{15\cdots 91}{61\cdots 01}a^{7}-\frac{12\cdots 77}{61\cdots 01}a^{6}-\frac{20\cdots 95}{61\cdots 01}a^{5}+\frac{14\cdots 72}{61\cdots 01}a^{4}+\frac{13\cdots 51}{61\cdots 01}a^{3}-\frac{24\cdots 89}{61\cdots 01}a^{2}-\frac{36\cdots 13}{61\cdots 01}a-\frac{24\cdots 75}{61\cdots 01}$, $\frac{53\cdots 59}{61\cdots 01}a^{19}+\frac{15\cdots 69}{61\cdots 01}a^{18}-\frac{38\cdots 92}{61\cdots 01}a^{17}-\frac{42\cdots 83}{61\cdots 01}a^{16}+\frac{10\cdots 17}{61\cdots 01}a^{15}-\frac{52\cdots 62}{61\cdots 01}a^{14}-\frac{15\cdots 19}{61\cdots 01}a^{13}+\frac{29\cdots 05}{61\cdots 01}a^{12}+\frac{12\cdots 36}{61\cdots 01}a^{11}-\frac{34\cdots 32}{61\cdots 01}a^{10}-\frac{47\cdots 68}{61\cdots 01}a^{9}+\frac{15\cdots 23}{61\cdots 01}a^{8}+\frac{98\cdots 69}{61\cdots 01}a^{7}-\frac{31\cdots 72}{61\cdots 01}a^{6}-\frac{10\cdots 42}{61\cdots 01}a^{5}+\frac{26\cdots 32}{61\cdots 01}a^{4}+\frac{71\cdots 40}{61\cdots 01}a^{3}-\frac{64\cdots 32}{61\cdots 01}a^{2}-\frac{22\cdots 95}{61\cdots 01}a-\frac{11\cdots 63}{56\cdots 89}$, $\frac{35\cdots 90}{61\cdots 01}a^{19}+\frac{24\cdots 67}{61\cdots 01}a^{18}-\frac{25\cdots 56}{61\cdots 01}a^{17}-\frac{12\cdots 10}{61\cdots 01}a^{16}+\frac{76\cdots 20}{61\cdots 01}a^{15}+\frac{23\cdots 55}{61\cdots 01}a^{14}-\frac{12\cdots 40}{61\cdots 01}a^{13}-\frac{15\cdots 43}{61\cdots 01}a^{12}+\frac{11\cdots 11}{61\cdots 01}a^{11}-\frac{42\cdots 88}{61\cdots 01}a^{10}-\frac{55\cdots 58}{61\cdots 01}a^{9}+\frac{96\cdots 81}{61\cdots 01}a^{8}+\frac{14\cdots 18}{61\cdots 01}a^{7}-\frac{40\cdots 96}{61\cdots 01}a^{6}-\frac{17\cdots 25}{61\cdots 01}a^{5}+\frac{66\cdots 15}{61\cdots 01}a^{4}+\frac{60\cdots 57}{61\cdots 01}a^{3}-\frac{37\cdots 17}{61\cdots 01}a^{2}+\frac{39\cdots 79}{61\cdots 01}a+\frac{58\cdots 82}{61\cdots 01}$, $\frac{41\cdots 75}{61\cdots 01}a^{19}-\frac{43\cdots 85}{61\cdots 01}a^{18}-\frac{30\cdots 92}{61\cdots 01}a^{17}+\frac{36\cdots 45}{61\cdots 01}a^{16}+\frac{85\cdots 90}{61\cdots 01}a^{15}-\frac{11\cdots 69}{61\cdots 01}a^{14}-\frac{10\cdots 02}{61\cdots 01}a^{13}+\frac{18\cdots 42}{61\cdots 01}a^{12}+\frac{49\cdots 00}{61\cdots 01}a^{11}-\frac{14\cdots 76}{61\cdots 01}a^{10}+\frac{96\cdots 20}{61\cdots 01}a^{9}+\frac{58\cdots 86}{61\cdots 01}a^{8}-\frac{13\cdots 35}{61\cdots 01}a^{7}-\frac{10\cdots 62}{61\cdots 01}a^{6}+\frac{32\cdots 00}{61\cdots 01}a^{5}+\frac{97\cdots 39}{61\cdots 01}a^{4}-\frac{29\cdots 48}{61\cdots 01}a^{3}-\frac{47\cdots 30}{61\cdots 01}a^{2}+\frac{92\cdots 32}{61\cdots 01}a+\frac{12\cdots 14}{61\cdots 01}$ (assuming GRH)
Regulator:		\( 3313588880654819000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 3313588880654819000 \cdot 275}{2\cdot\sqrt{10818327022517881605935827576487172209050638458251953125}}\cr\approx \mathstrut & 0.145251686268944 \end{aligned}\] (assuming GRH)

Galois group

$C_{20}$ (as 20T1):

A cyclic group of order 20

The 20 conjugacy class representatives for $C_{20}$

Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 5.5.41371966801.1, 10.10.5348873865572019292503125.1

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	$20$	R	$20$	$20$	${\href{/padicField/19.5.0.1}{5} }^{4}$	$20$	${\href{/padicField/29.5.0.1}{5} }^{4}$	${\href{/padicField/31.5.0.1}{5} }^{4}$	$20$	R	$20$	$20$	$20$	${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(5\)	5.5.4.15a1.4	$x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 86$	$4$	$5$	$15$	20T1	not computed
\(11\)	11.1.10.9a1.3	$x^{10} + 33$	$10$	$1$	$9$	$C_{10}$	$$[\ ]_{10}$$
	11.1.10.9a1.3	$x^{10} + 33$	$10$	$1$	$9$	$C_{10}$	$$[\ ]_{10}$$
\(41\)	41.2.5.8a1.1	$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$	$5$	$2$	$8$	$C_{10}$	$$[\ ]_{5}^{2}$$
	41.2.5.8a1.1	$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$	$5$	$2$	$8$	$C_{10}$	$$[\ ]_{5}^{2}$$

Spectrum of ring of integers