Number field 20.0.8341936223273428359616333847680741.1

• Label: 20.0.834...741.1
• Degree: $20$
• Signature: $[0, 10]$
• Discriminant: $8.342\times 10^{33}$
• Root discriminant: \(49.67\)
• Ramified prime: $61$
• Class number: $41$ (GRH)
• Class group: [41] (GRH)
• Galois group: $C_{20}$ (as 20T1)

Normalized defining polynomial

x^20 - x^19 + 2*x^18 + 36*x^17 - 29*x^16 + 51*x^15 + 413*x^14 - 267*x^13 + 414*x^12 + 1778*x^11 - 745*x^10 + 691*x^9 + 3278*x^8 - 541*x^7 - 1330*x^6 + 1820*x^5 - 261*x^4 - 439*x^3 + 2983*x^2 - 522*x + 611

Invariants

Degree:		$20$
Signature:		$[0, 10]$
Discriminant:		\(8341936223273428359616333847680741\) \(\medspace = 61^{19}\)
Root discriminant:		\(49.67\)
Galois root discriminant:		$61^{19/20}\approx 49.6664756335031$
Ramified primes:		\(61\)
Discriminant root field:		\(\Q(\sqrt{61}) \)
$\card{ \Gal(K/\Q) }$:		$20$
This field is Galois and abelian over $\Q$.
Conductor:		\(61\)
Dirichlet character group:		$\lbrace$$\chi_{61}(1,·)$, $\chi_{61}(3,·)$, $\chi_{61}(8,·)$, $\chi_{61}(9,·)$, $\chi_{61}(11,·)$, $\chi_{61}(20,·)$, $\chi_{61}(23,·)$, $\chi_{61}(24,·)$, $\chi_{61}(27,·)$, $\chi_{61}(28,·)$, $\chi_{61}(33,·)$, $\chi_{61}(34,·)$, $\chi_{61}(37,·)$, $\chi_{61}(38,·)$, $\chi_{61}(41,·)$, $\chi_{61}(50,·)$, $\chi_{61}(52,·)$, $\chi_{61}(53,·)$, $\chi_{61}(58,·)$, $\chi_{61}(60,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13}a^{11}+\frac{3}{13}a^{10}-\frac{1}{13}a^{8}-\frac{3}{13}a^{7}+\frac{1}{13}a^{5}+\frac{3}{13}a^{4}-\frac{1}{13}a^{2}-\frac{3}{13}a$, $\frac{1}{13}a^{12}+\frac{4}{13}a^{10}-\frac{1}{13}a^{9}-\frac{4}{13}a^{7}+\frac{1}{13}a^{6}+\frac{4}{13}a^{4}-\frac{1}{13}a^{3}-\frac{4}{13}a$, $\frac{1}{13}a^{13}-\frac{1}{13}a$, $\frac{1}{13}a^{14}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}-\frac{1}{13}a^{3}$, $\frac{1}{611}a^{16}+\frac{17}{611}a^{15}+\frac{1}{47}a^{14}-\frac{1}{611}a^{13}+\frac{4}{611}a^{12}+\frac{4}{611}a^{11}-\frac{271}{611}a^{10}+\frac{113}{611}a^{9}+\frac{87}{611}a^{8}-\frac{223}{611}a^{7}+\frac{277}{611}a^{6}+\frac{69}{611}a^{5}-\frac{168}{611}a^{4}-\frac{255}{611}a^{3}+\frac{295}{611}a^{2}+\frac{38}{611}a$, $\frac{1}{611}a^{17}+\frac{6}{611}a^{15}+\frac{1}{47}a^{14}+\frac{21}{611}a^{13}-\frac{17}{611}a^{12}-\frac{10}{611}a^{11}-\frac{215}{611}a^{10}-\frac{48}{611}a^{9}-\frac{198}{611}a^{8}-\frac{162}{611}a^{7}+\frac{295}{611}a^{6}+\frac{210}{611}a^{5}+\frac{110}{611}a^{4}+\frac{24}{611}a^{3}-\frac{42}{611}a^{2}+\frac{12}{611}a$, $\frac{1}{7943}a^{18}+\frac{6}{7943}a^{17}-\frac{4}{7943}a^{16}-\frac{215}{7943}a^{15}-\frac{172}{7943}a^{14}+\frac{119}{7943}a^{13}-\frac{105}{7943}a^{12}+\frac{202}{7943}a^{11}+\frac{3722}{7943}a^{10}-\frac{1052}{7943}a^{9}-\frac{2737}{7943}a^{8}+\frac{1036}{7943}a^{7}-\frac{132}{7943}a^{6}-\frac{3691}{7943}a^{5}-\frac{2007}{7943}a^{4}-\frac{2189}{7943}a^{3}+\frac{1322}{7943}a^{2}-\frac{3880}{7943}a+\frac{3}{13}$, $\frac{1}{14\!\cdots\!23}a^{19}+\frac{71\!\cdots\!02}{14\!\cdots\!23}a^{18}-\frac{98\!\cdots\!90}{31\!\cdots\!09}a^{17}-\frac{80\!\cdots\!77}{14\!\cdots\!23}a^{16}-\frac{26\!\cdots\!63}{14\!\cdots\!23}a^{15}+\frac{29\!\cdots\!32}{14\!\cdots\!23}a^{14}-\frac{22\!\cdots\!02}{14\!\cdots\!23}a^{13}-\frac{53\!\cdots\!38}{14\!\cdots\!23}a^{12}+\frac{50\!\cdots\!84}{14\!\cdots\!23}a^{11}-\frac{52\!\cdots\!52}{11\!\cdots\!71}a^{10}-\frac{60\!\cdots\!32}{14\!\cdots\!23}a^{9}+\frac{60\!\cdots\!44}{14\!\cdots\!23}a^{8}-\frac{32\!\cdots\!63}{14\!\cdots\!23}a^{7}-\frac{28\!\cdots\!36}{14\!\cdots\!23}a^{6}+\frac{42\!\cdots\!73}{14\!\cdots\!23}a^{5}-\frac{85\!\cdots\!54}{14\!\cdots\!23}a^{4}-\frac{72\!\cdots\!64}{14\!\cdots\!23}a^{3}+\frac{63\!\cdots\!01}{14\!\cdots\!23}a^{2}+\frac{54\!\cdots\!72}{14\!\cdots\!23}a-\frac{59\!\cdots\!10}{23\!\cdots\!93}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$13$

Class group and class number

$C_{41}$, which has order $41$ (assuming GRH)

Unit group

Rank:		$9$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{60\!\cdots\!32}{14\!\cdots\!23}a^{19}+\frac{10\!\cdots\!60}{14\!\cdots\!23}a^{18}+\frac{95\!\cdots\!05}{14\!\cdots\!23}a^{17}+\frac{24\!\cdots\!37}{14\!\cdots\!23}a^{16}+\frac{45\!\cdots\!89}{14\!\cdots\!23}a^{15}+\frac{56\!\cdots\!47}{14\!\cdots\!23}a^{14}+\frac{30\!\cdots\!63}{14\!\cdots\!23}a^{13}+\frac{60\!\cdots\!40}{14\!\cdots\!23}a^{12}-\frac{21\!\cdots\!46}{14\!\cdots\!23}a^{11}+\frac{11\!\cdots\!05}{11\!\cdots\!71}a^{10}+\frac{31\!\cdots\!05}{14\!\cdots\!23}a^{9}-\frac{37\!\cdots\!24}{14\!\cdots\!23}a^{8}+\frac{26\!\cdots\!24}{14\!\cdots\!23}a^{7}+\frac{66\!\cdots\!21}{14\!\cdots\!23}a^{6}-\frac{15\!\cdots\!75}{14\!\cdots\!23}a^{5}-\frac{10\!\cdots\!57}{14\!\cdots\!23}a^{4}+\frac{22\!\cdots\!49}{14\!\cdots\!23}a^{3}-\frac{11\!\cdots\!04}{14\!\cdots\!23}a^{2}+\frac{50\!\cdots\!73}{14\!\cdots\!23}a+\frac{49\!\cdots\!23}{23\!\cdots\!93}$, $\frac{49\!\cdots\!62}{14\!\cdots\!23}a^{19}-\frac{45\!\cdots\!57}{14\!\cdots\!23}a^{18}+\frac{14\!\cdots\!47}{14\!\cdots\!23}a^{17}+\frac{17\!\cdots\!52}{14\!\cdots\!23}a^{16}-\frac{11\!\cdots\!79}{14\!\cdots\!23}a^{15}+\frac{42\!\cdots\!48}{14\!\cdots\!23}a^{14}+\frac{19\!\cdots\!68}{14\!\cdots\!23}a^{13}-\frac{91\!\cdots\!57}{14\!\cdots\!23}a^{12}+\frac{39\!\cdots\!26}{14\!\cdots\!23}a^{11}+\frac{60\!\cdots\!43}{11\!\cdots\!71}a^{10}-\frac{11\!\cdots\!52}{14\!\cdots\!23}a^{9}+\frac{11\!\cdots\!49}{14\!\cdots\!23}a^{8}+\frac{13\!\cdots\!48}{14\!\cdots\!23}a^{7}+\frac{12\!\cdots\!39}{14\!\cdots\!23}a^{6}+\frac{63\!\cdots\!71}{14\!\cdots\!23}a^{5}+\frac{62\!\cdots\!58}{14\!\cdots\!23}a^{4}-\frac{70\!\cdots\!27}{14\!\cdots\!23}a^{3}+\frac{57\!\cdots\!08}{14\!\cdots\!23}a^{2}-\frac{19\!\cdots\!60}{14\!\cdots\!23}a+\frac{45\!\cdots\!31}{23\!\cdots\!93}$, $\frac{35\!\cdots\!50}{14\!\cdots\!23}a^{19}+\frac{13\!\cdots\!74}{14\!\cdots\!23}a^{18}-\frac{15\!\cdots\!35}{14\!\cdots\!23}a^{17}+\frac{15\!\cdots\!57}{14\!\cdots\!23}a^{16}+\frac{52\!\cdots\!97}{14\!\cdots\!23}a^{15}-\frac{51\!\cdots\!31}{14\!\cdots\!23}a^{14}+\frac{21\!\cdots\!03}{14\!\cdots\!23}a^{13}+\frac{61\!\cdots\!92}{14\!\cdots\!23}a^{12}-\frac{55\!\cdots\!32}{14\!\cdots\!23}a^{11}+\frac{85\!\cdots\!13}{11\!\cdots\!71}a^{10}+\frac{27\!\cdots\!53}{14\!\cdots\!23}a^{9}-\frac{21\!\cdots\!73}{14\!\cdots\!23}a^{8}+\frac{80\!\cdots\!68}{14\!\cdots\!23}a^{7}+\frac{54\!\cdots\!06}{14\!\cdots\!23}a^{6}-\frac{26\!\cdots\!48}{14\!\cdots\!23}a^{5}-\frac{50\!\cdots\!02}{14\!\cdots\!23}a^{4}+\frac{41\!\cdots\!79}{14\!\cdots\!23}a^{3}+\frac{24\!\cdots\!36}{14\!\cdots\!23}a^{2}+\frac{23\!\cdots\!63}{14\!\cdots\!23}a+\frac{17\!\cdots\!97}{23\!\cdots\!93}$, $\frac{79\!\cdots\!19}{14\!\cdots\!23}a^{19}+\frac{57\!\cdots\!47}{14\!\cdots\!23}a^{18}-\frac{53\!\cdots\!40}{14\!\cdots\!23}a^{17}+\frac{22\!\cdots\!60}{14\!\cdots\!23}a^{16}+\frac{19\!\cdots\!75}{14\!\cdots\!23}a^{15}-\frac{12\!\cdots\!12}{14\!\cdots\!23}a^{14}+\frac{60\!\cdots\!78}{14\!\cdots\!23}a^{13}+\frac{19\!\cdots\!83}{14\!\cdots\!23}a^{12}-\frac{55\!\cdots\!97}{14\!\cdots\!23}a^{11}+\frac{42\!\cdots\!26}{11\!\cdots\!71}a^{10}+\frac{57\!\cdots\!57}{14\!\cdots\!23}a^{9}+\frac{36\!\cdots\!12}{14\!\cdots\!23}a^{8}+\frac{15\!\cdots\!26}{14\!\cdots\!23}a^{7}+\frac{53\!\cdots\!86}{14\!\cdots\!23}a^{6}+\frac{12\!\cdots\!12}{14\!\cdots\!23}a^{5}+\frac{14\!\cdots\!26}{31\!\cdots\!09}a^{4}+\frac{25\!\cdots\!51}{14\!\cdots\!23}a^{3}-\frac{21\!\cdots\!10}{14\!\cdots\!23}a^{2}+\frac{31\!\cdots\!34}{14\!\cdots\!23}a-\frac{28\!\cdots\!14}{23\!\cdots\!93}$, $\frac{18\!\cdots\!85}{56\!\cdots\!39}a^{19}+\frac{28\!\cdots\!84}{56\!\cdots\!39}a^{18}-\frac{733348794514119}{56\!\cdots\!39}a^{17}+\frac{75\!\cdots\!75}{56\!\cdots\!39}a^{16}+\frac{11\!\cdots\!67}{56\!\cdots\!39}a^{15}-\frac{39\!\cdots\!54}{56\!\cdots\!39}a^{14}+\frac{95\!\cdots\!34}{56\!\cdots\!39}a^{13}+\frac{13\!\cdots\!26}{56\!\cdots\!39}a^{12}-\frac{60\!\cdots\!87}{56\!\cdots\!39}a^{11}+\frac{32\!\cdots\!68}{43\!\cdots\!03}a^{10}+\frac{63\!\cdots\!57}{56\!\cdots\!39}a^{9}-\frac{37\!\cdots\!54}{56\!\cdots\!39}a^{8}+\frac{25\!\cdots\!67}{56\!\cdots\!39}a^{7}+\frac{12\!\cdots\!88}{56\!\cdots\!39}a^{6}-\frac{65\!\cdots\!31}{56\!\cdots\!39}a^{5}-\frac{16\!\cdots\!37}{56\!\cdots\!39}a^{4}+\frac{10\!\cdots\!53}{56\!\cdots\!39}a^{3}+\frac{13\!\cdots\!75}{56\!\cdots\!39}a^{2}-\frac{52\!\cdots\!24}{56\!\cdots\!39}a-\frac{62\!\cdots\!68}{92\!\cdots\!49}$, $\frac{16\!\cdots\!67}{14\!\cdots\!23}a^{19}+\frac{29\!\cdots\!73}{14\!\cdots\!23}a^{18}-\frac{95\!\cdots\!99}{14\!\cdots\!23}a^{17}+\frac{63\!\cdots\!66}{14\!\cdots\!23}a^{16}+\frac{11\!\cdots\!17}{14\!\cdots\!23}a^{15}-\frac{36\!\cdots\!91}{14\!\cdots\!23}a^{14}+\frac{63\!\cdots\!85}{14\!\cdots\!23}a^{13}+\frac{15\!\cdots\!20}{14\!\cdots\!23}a^{12}-\frac{44\!\cdots\!45}{14\!\cdots\!23}a^{11}+\frac{68\!\cdots\!45}{11\!\cdots\!71}a^{10}+\frac{83\!\cdots\!83}{14\!\cdots\!23}a^{9}-\frac{20\!\cdots\!85}{14\!\cdots\!23}a^{8}-\frac{11\!\cdots\!98}{14\!\cdots\!23}a^{7}+\frac{22\!\cdots\!14}{14\!\cdots\!23}a^{6}-\frac{27\!\cdots\!62}{14\!\cdots\!23}a^{5}-\frac{44\!\cdots\!20}{14\!\cdots\!23}a^{4}+\frac{25\!\cdots\!56}{14\!\cdots\!23}a^{3}+\frac{39\!\cdots\!27}{14\!\cdots\!23}a^{2}-\frac{19\!\cdots\!17}{14\!\cdots\!23}a+\frac{13\!\cdots\!45}{23\!\cdots\!93}$, $\frac{13\!\cdots\!80}{14\!\cdots\!23}a^{19}+\frac{90\!\cdots\!50}{14\!\cdots\!23}a^{18}+\frac{33\!\cdots\!17}{31\!\cdots\!09}a^{17}+\frac{51\!\cdots\!99}{14\!\cdots\!23}a^{16}+\frac{43\!\cdots\!48}{14\!\cdots\!23}a^{15}-\frac{88\!\cdots\!85}{14\!\cdots\!23}a^{14}+\frac{61\!\cdots\!58}{14\!\cdots\!23}a^{13}+\frac{60\!\cdots\!01}{14\!\cdots\!23}a^{12}-\frac{19\!\cdots\!08}{14\!\cdots\!23}a^{11}+\frac{20\!\cdots\!51}{11\!\cdots\!71}a^{10}+\frac{33\!\cdots\!14}{14\!\cdots\!23}a^{9}-\frac{13\!\cdots\!01}{14\!\cdots\!23}a^{8}+\frac{32\!\cdots\!44}{14\!\cdots\!23}a^{7}+\frac{79\!\cdots\!52}{14\!\cdots\!23}a^{6}-\frac{29\!\cdots\!82}{14\!\cdots\!23}a^{5}-\frac{57\!\cdots\!54}{14\!\cdots\!23}a^{4}+\frac{43\!\cdots\!55}{14\!\cdots\!23}a^{3}+\frac{39\!\cdots\!07}{14\!\cdots\!23}a^{2}+\frac{29\!\cdots\!70}{14\!\cdots\!23}a+\frac{16\!\cdots\!96}{23\!\cdots\!93}$, $\frac{43\!\cdots\!11}{14\!\cdots\!23}a^{19}+\frac{15\!\cdots\!37}{14\!\cdots\!23}a^{18}-\frac{98\!\cdots\!90}{14\!\cdots\!23}a^{17}+\frac{19\!\cdots\!21}{14\!\cdots\!23}a^{16}+\frac{60\!\cdots\!87}{14\!\cdots\!23}a^{15}-\frac{31\!\cdots\!81}{14\!\cdots\!23}a^{14}+\frac{26\!\cdots\!49}{14\!\cdots\!23}a^{13}+\frac{71\!\cdots\!80}{14\!\cdots\!23}a^{12}-\frac{32\!\cdots\!79}{14\!\cdots\!23}a^{11}+\frac{11\!\cdots\!77}{11\!\cdots\!71}a^{10}+\frac{32\!\cdots\!87}{14\!\cdots\!23}a^{9}-\frac{11\!\cdots\!99}{14\!\cdots\!23}a^{8}+\frac{20\!\cdots\!85}{14\!\cdots\!23}a^{7}+\frac{62\!\cdots\!25}{14\!\cdots\!23}a^{6}-\frac{16\!\cdots\!56}{14\!\cdots\!23}a^{5}-\frac{29\!\cdots\!44}{14\!\cdots\!23}a^{4}+\frac{34\!\cdots\!19}{14\!\cdots\!23}a^{3}-\frac{13\!\cdots\!40}{14\!\cdots\!23}a^{2}+\frac{49\!\cdots\!17}{14\!\cdots\!23}a+\frac{76\!\cdots\!21}{23\!\cdots\!93}$, $\frac{43\!\cdots\!05}{14\!\cdots\!23}a^{19}+\frac{46\!\cdots\!23}{14\!\cdots\!23}a^{18}-\frac{35\!\cdots\!80}{14\!\cdots\!23}a^{17}+\frac{25\!\cdots\!19}{14\!\cdots\!23}a^{16}+\frac{16\!\cdots\!86}{14\!\cdots\!23}a^{15}-\frac{10\!\cdots\!38}{14\!\cdots\!23}a^{14}+\frac{41\!\cdots\!97}{14\!\cdots\!23}a^{13}+\frac{18\!\cdots\!58}{14\!\cdots\!23}a^{12}-\frac{91\!\cdots\!77}{14\!\cdots\!23}a^{11}+\frac{19\!\cdots\!26}{11\!\cdots\!71}a^{10}+\frac{77\!\cdots\!07}{14\!\cdots\!23}a^{9}-\frac{25\!\cdots\!88}{14\!\cdots\!23}a^{8}+\frac{38\!\cdots\!81}{14\!\cdots\!23}a^{7}+\frac{12\!\cdots\!31}{14\!\cdots\!23}a^{6}-\frac{28\!\cdots\!82}{14\!\cdots\!23}a^{5}-\frac{57\!\cdots\!45}{14\!\cdots\!23}a^{4}+\frac{79\!\cdots\!26}{14\!\cdots\!23}a^{3}-\frac{21\!\cdots\!83}{14\!\cdots\!23}a^{2}+\frac{14\!\cdots\!59}{14\!\cdots\!23}a+\frac{18\!\cdots\!31}{23\!\cdots\!93}$ (assuming GRH)
Regulator:		\( 36549838.4715 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 36549838.4715 \cdot 41}{2\cdot\sqrt{8341936223273428359616333847680741}}\cr\approx \mathstrut & 0.786691710589 \end{aligned}\] (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

\(\Q(\sqrt{61}) \), 4.0.226981.1, 5.5.13845841.1, 10.10.11694146092834141.1

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	$20$	${\href{/padicField/3.10.0.1}{10} }^{2}$	${\href{/padicField/5.10.0.1}{10} }^{2}$	$20$	${\href{/padicField/11.4.0.1}{4} }^{5}$	${\href{/padicField/13.1.0.1}{1} }^{20}$	$20$	${\href{/padicField/19.10.0.1}{10} }^{2}$	$20$	${\href{/padicField/29.4.0.1}{4} }^{5}$	$20$	$20$	${\href{/padicField/41.10.0.1}{10} }^{2}$	$20$	${\href{/padicField/47.1.0.1}{1} }^{20}$	$20$	$20$

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
\(61\)	61.20.19.6	$x^{20} + 61$	$20$	$1$	$19$	20T1	$[\ ]_{20}$