Normalized defining polynomial
\( x^{20} - x^{19} + 2 x^{18} + 36 x^{17} - 29 x^{16} + 51 x^{15} + 413 x^{14} - 267 x^{13} + 414 x^{12} + \cdots + 611 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(8341936223273428359616333847680741\)
\(\medspace = 61^{19}\)
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| Root discriminant: | \(49.67\) |
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| Galois root discriminant: | $61^{19/20}\approx 49.6664756335031$ | ||
| Ramified primes: |
\(61\)
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| Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
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| 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
| Ideal class group: | $C_{41}$, which has order $41$ (assuming GRH) |
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| Narrow class group: | $C_{41}$, which has order $41$ (assuming GRH) |
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| Relative class number: | $41$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{60\cdots 32}{14\cdots 23}a^{19}+\frac{10\cdots 60}{14\cdots 23}a^{18}+\frac{95\cdots 05}{14\cdots 23}a^{17}+\frac{24\cdots 37}{14\cdots 23}a^{16}+\frac{45\cdots 89}{14\cdots 23}a^{15}+\frac{56\cdots 47}{14\cdots 23}a^{14}+\frac{30\cdots 63}{14\cdots 23}a^{13}+\frac{60\cdots 40}{14\cdots 23}a^{12}-\frac{21\cdots 46}{14\cdots 23}a^{11}+\frac{11\cdots 05}{11\cdots 71}a^{10}+\frac{31\cdots 05}{14\cdots 23}a^{9}-\frac{37\cdots 24}{14\cdots 23}a^{8}+\frac{26\cdots 24}{14\cdots 23}a^{7}+\frac{66\cdots 21}{14\cdots 23}a^{6}-\frac{15\cdots 75}{14\cdots 23}a^{5}-\frac{10\cdots 57}{14\cdots 23}a^{4}+\frac{22\cdots 49}{14\cdots 23}a^{3}-\frac{11\cdots 04}{14\cdots 23}a^{2}+\frac{50\cdots 73}{14\cdots 23}a+\frac{49\cdots 23}{23\cdots 93}$, $\frac{49\cdots 62}{14\cdots 23}a^{19}-\frac{45\cdots 57}{14\cdots 23}a^{18}+\frac{14\cdots 47}{14\cdots 23}a^{17}+\frac{17\cdots 52}{14\cdots 23}a^{16}-\frac{11\cdots 79}{14\cdots 23}a^{15}+\frac{42\cdots 48}{14\cdots 23}a^{14}+\frac{19\cdots 68}{14\cdots 23}a^{13}-\frac{91\cdots 57}{14\cdots 23}a^{12}+\frac{39\cdots 26}{14\cdots 23}a^{11}+\frac{60\cdots 43}{11\cdots 71}a^{10}-\frac{11\cdots 52}{14\cdots 23}a^{9}+\frac{11\cdots 49}{14\cdots 23}a^{8}+\frac{13\cdots 48}{14\cdots 23}a^{7}+\frac{12\cdots 39}{14\cdots 23}a^{6}+\frac{63\cdots 71}{14\cdots 23}a^{5}+\frac{62\cdots 58}{14\cdots 23}a^{4}-\frac{70\cdots 27}{14\cdots 23}a^{3}+\frac{57\cdots 08}{14\cdots 23}a^{2}-\frac{19\cdots 60}{14\cdots 23}a+\frac{45\cdots 31}{23\cdots 93}$, $\frac{35\cdots 50}{14\cdots 23}a^{19}+\frac{13\cdots 74}{14\cdots 23}a^{18}-\frac{15\cdots 35}{14\cdots 23}a^{17}+\frac{15\cdots 57}{14\cdots 23}a^{16}+\frac{52\cdots 97}{14\cdots 23}a^{15}-\frac{51\cdots 31}{14\cdots 23}a^{14}+\frac{21\cdots 03}{14\cdots 23}a^{13}+\frac{61\cdots 92}{14\cdots 23}a^{12}-\frac{55\cdots 32}{14\cdots 23}a^{11}+\frac{85\cdots 13}{11\cdots 71}a^{10}+\frac{27\cdots 53}{14\cdots 23}a^{9}-\frac{21\cdots 73}{14\cdots 23}a^{8}+\frac{80\cdots 68}{14\cdots 23}a^{7}+\frac{54\cdots 06}{14\cdots 23}a^{6}-\frac{26\cdots 48}{14\cdots 23}a^{5}-\frac{50\cdots 02}{14\cdots 23}a^{4}+\frac{41\cdots 79}{14\cdots 23}a^{3}+\frac{24\cdots 36}{14\cdots 23}a^{2}+\frac{23\cdots 63}{14\cdots 23}a+\frac{17\cdots 97}{23\cdots 93}$, $\frac{79\cdots 19}{14\cdots 23}a^{19}+\frac{57\cdots 47}{14\cdots 23}a^{18}-\frac{53\cdots 40}{14\cdots 23}a^{17}+\frac{22\cdots 60}{14\cdots 23}a^{16}+\frac{19\cdots 75}{14\cdots 23}a^{15}-\frac{12\cdots 12}{14\cdots 23}a^{14}+\frac{60\cdots 78}{14\cdots 23}a^{13}+\frac{19\cdots 83}{14\cdots 23}a^{12}-\frac{55\cdots 97}{14\cdots 23}a^{11}+\frac{42\cdots 26}{11\cdots 71}a^{10}+\frac{57\cdots 57}{14\cdots 23}a^{9}+\frac{36\cdots 12}{14\cdots 23}a^{8}+\frac{15\cdots 26}{14\cdots 23}a^{7}+\frac{53\cdots 86}{14\cdots 23}a^{6}+\frac{12\cdots 12}{14\cdots 23}a^{5}+\frac{14\cdots 26}{31\cdots 09}a^{4}+\frac{25\cdots 51}{14\cdots 23}a^{3}-\frac{21\cdots 10}{14\cdots 23}a^{2}+\frac{31\cdots 34}{14\cdots 23}a-\frac{28\cdots 14}{23\cdots 93}$, $\frac{18\cdots 85}{56\cdots 39}a^{19}+\frac{28\cdots 84}{56\cdots 39}a^{18}-\frac{733348794514119}{56\cdots 39}a^{17}+\frac{75\cdots 75}{56\cdots 39}a^{16}+\frac{11\cdots 67}{56\cdots 39}a^{15}-\frac{39\cdots 54}{56\cdots 39}a^{14}+\frac{95\cdots 34}{56\cdots 39}a^{13}+\frac{13\cdots 26}{56\cdots 39}a^{12}-\frac{60\cdots 87}{56\cdots 39}a^{11}+\frac{32\cdots 68}{43\cdots 03}a^{10}+\frac{63\cdots 57}{56\cdots 39}a^{9}-\frac{37\cdots 54}{56\cdots 39}a^{8}+\frac{25\cdots 67}{56\cdots 39}a^{7}+\frac{12\cdots 88}{56\cdots 39}a^{6}-\frac{65\cdots 31}{56\cdots 39}a^{5}-\frac{16\cdots 37}{56\cdots 39}a^{4}+\frac{10\cdots 53}{56\cdots 39}a^{3}+\frac{13\cdots 75}{56\cdots 39}a^{2}-\frac{52\cdots 24}{56\cdots 39}a-\frac{62\cdots 68}{92\cdots 49}$, $\frac{16\cdots 67}{14\cdots 23}a^{19}+\frac{29\cdots 73}{14\cdots 23}a^{18}-\frac{95\cdots 99}{14\cdots 23}a^{17}+\frac{63\cdots 66}{14\cdots 23}a^{16}+\frac{11\cdots 17}{14\cdots 23}a^{15}-\frac{36\cdots 91}{14\cdots 23}a^{14}+\frac{63\cdots 85}{14\cdots 23}a^{13}+\frac{15\cdots 20}{14\cdots 23}a^{12}-\frac{44\cdots 45}{14\cdots 23}a^{11}+\frac{68\cdots 45}{11\cdots 71}a^{10}+\frac{83\cdots 83}{14\cdots 23}a^{9}-\frac{20\cdots 85}{14\cdots 23}a^{8}-\frac{11\cdots 98}{14\cdots 23}a^{7}+\frac{22\cdots 14}{14\cdots 23}a^{6}-\frac{27\cdots 62}{14\cdots 23}a^{5}-\frac{44\cdots 20}{14\cdots 23}a^{4}+\frac{25\cdots 56}{14\cdots 23}a^{3}+\frac{39\cdots 27}{14\cdots 23}a^{2}-\frac{19\cdots 17}{14\cdots 23}a+\frac{13\cdots 45}{23\cdots 93}$, $\frac{13\cdots 80}{14\cdots 23}a^{19}+\frac{90\cdots 50}{14\cdots 23}a^{18}+\frac{33\cdots 17}{31\cdots 09}a^{17}+\frac{51\cdots 99}{14\cdots 23}a^{16}+\frac{43\cdots 48}{14\cdots 23}a^{15}-\frac{88\cdots 85}{14\cdots 23}a^{14}+\frac{61\cdots 58}{14\cdots 23}a^{13}+\frac{60\cdots 01}{14\cdots 23}a^{12}-\frac{19\cdots 08}{14\cdots 23}a^{11}+\frac{20\cdots 51}{11\cdots 71}a^{10}+\frac{33\cdots 14}{14\cdots 23}a^{9}-\frac{13\cdots 01}{14\cdots 23}a^{8}+\frac{32\cdots 44}{14\cdots 23}a^{7}+\frac{79\cdots 52}{14\cdots 23}a^{6}-\frac{29\cdots 82}{14\cdots 23}a^{5}-\frac{57\cdots 54}{14\cdots 23}a^{4}+\frac{43\cdots 55}{14\cdots 23}a^{3}+\frac{39\cdots 07}{14\cdots 23}a^{2}+\frac{29\cdots 70}{14\cdots 23}a+\frac{16\cdots 96}{23\cdots 93}$, $\frac{43\cdots 11}{14\cdots 23}a^{19}+\frac{15\cdots 37}{14\cdots 23}a^{18}-\frac{98\cdots 90}{14\cdots 23}a^{17}+\frac{19\cdots 21}{14\cdots 23}a^{16}+\frac{60\cdots 87}{14\cdots 23}a^{15}-\frac{31\cdots 81}{14\cdots 23}a^{14}+\frac{26\cdots 49}{14\cdots 23}a^{13}+\frac{71\cdots 80}{14\cdots 23}a^{12}-\frac{32\cdots 79}{14\cdots 23}a^{11}+\frac{11\cdots 77}{11\cdots 71}a^{10}+\frac{32\cdots 87}{14\cdots 23}a^{9}-\frac{11\cdots 99}{14\cdots 23}a^{8}+\frac{20\cdots 85}{14\cdots 23}a^{7}+\frac{62\cdots 25}{14\cdots 23}a^{6}-\frac{16\cdots 56}{14\cdots 23}a^{5}-\frac{29\cdots 44}{14\cdots 23}a^{4}+\frac{34\cdots 19}{14\cdots 23}a^{3}-\frac{13\cdots 40}{14\cdots 23}a^{2}+\frac{49\cdots 17}{14\cdots 23}a+\frac{76\cdots 21}{23\cdots 93}$, $\frac{43\cdots 05}{14\cdots 23}a^{19}+\frac{46\cdots 23}{14\cdots 23}a^{18}-\frac{35\cdots 80}{14\cdots 23}a^{17}+\frac{25\cdots 19}{14\cdots 23}a^{16}+\frac{16\cdots 86}{14\cdots 23}a^{15}-\frac{10\cdots 38}{14\cdots 23}a^{14}+\frac{41\cdots 97}{14\cdots 23}a^{13}+\frac{18\cdots 58}{14\cdots 23}a^{12}-\frac{91\cdots 77}{14\cdots 23}a^{11}+\frac{19\cdots 26}{11\cdots 71}a^{10}+\frac{77\cdots 07}{14\cdots 23}a^{9}-\frac{25\cdots 88}{14\cdots 23}a^{8}+\frac{38\cdots 81}{14\cdots 23}a^{7}+\frac{12\cdots 31}{14\cdots 23}a^{6}-\frac{28\cdots 82}{14\cdots 23}a^{5}-\frac{57\cdots 45}{14\cdots 23}a^{4}+\frac{79\cdots 26}{14\cdots 23}a^{3}-\frac{21\cdots 83}{14\cdots 23}a^{2}+\frac{14\cdots 59}{14\cdots 23}a+\frac{18\cdots 31}{23\cdots 93}$
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| Regulator: | \( 36549838.4715 \) (assuming GRH) |
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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
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-122 +10 \sqrt{61}})\), 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 |
|---|---|---|---|---|---|---|---|
|
\(61\)
| 61.1.20.19a1.1 | $x^{20} + 61$ | $20$ | $1$ | $19$ | 20T1 | $$[\ ]_{20}$$ |