Normalized defining polynomial
\( x^{20} - x^{19} + 35 x^{18} - 36 x^{17} + 500 x^{16} - 536 x^{15} + 3874 x^{14} - 4410 x^{13} + \cdots + 1197901 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(10019151533337487082567413330078125\)
\(\medspace = 3^{10}\cdot 5^{15}\cdot 11^{18}\)
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| Root discriminant: | \(50.12\) |
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| Galois root discriminant: | $3^{1/2}5^{3/4}11^{9/10}\approx 50.12351825429183$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(165=3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(136,·)$, $\chi_{165}(98,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(83,·)$, $\chi_{165}(68,·)$, $\chi_{165}(91,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(64,·)$, $\chi_{165}(107,·)$, $\chi_{165}(49,·)$, $\chi_{165}(8,·)$, $\chi_{165}(124,·)$, $\chi_{165}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{89}a^{15}+\frac{42}{89}a^{14}-\frac{14}{89}a^{13}-\frac{44}{89}a^{12}+\frac{5}{89}a^{11}-\frac{19}{89}a^{10}+\frac{14}{89}a^{9}-\frac{34}{89}a^{8}-\frac{41}{89}a^{7}-\frac{36}{89}a^{6}-\frac{31}{89}a^{4}+\frac{7}{89}a^{3}-\frac{32}{89}a^{2}-\frac{43}{89}a-\frac{6}{89}$, $\frac{1}{184141}a^{16}-\frac{851}{184141}a^{15}+\frac{61270}{184141}a^{14}+\frac{70308}{184141}a^{13}+\frac{5388}{184141}a^{12}+\frac{53188}{184141}a^{11}+\frac{8793}{184141}a^{10}+\frac{27959}{184141}a^{9}+\frac{49990}{184141}a^{8}+\frac{33106}{184141}a^{7}-\frac{47507}{184141}a^{6}+\frac{41888}{184141}a^{5}+\frac{10068}{184141}a^{4}-\frac{5304}{184141}a^{3}-\frac{56195}{184141}a^{2}+\frac{27713}{184141}a+\frac{75935}{184141}$, $\frac{1}{184141}a^{17}-\frac{851}{184141}a^{15}-\frac{82897}{184141}a^{14}-\frac{791}{2069}a^{13}-\frac{2391}{184141}a^{12}-\frac{31043}{184141}a^{11}+\frac{87230}{184141}a^{10}-\frac{33161}{184141}a^{9}-\frac{7493}{184141}a^{8}+\frac{59714}{184141}a^{7}+\frac{43760}{184141}a^{6}-\frac{66598}{184141}a^{5}+\frac{7249}{184141}a^{4}+\frac{64661}{184141}a^{3}+\frac{72083}{184141}a^{2}-\frac{22076}{184141}a+\frac{65816}{184141}$, $\frac{1}{184141}a^{18}-\frac{188}{184141}a^{15}-\frac{33256}{184141}a^{14}-\frac{80247}{184141}a^{13}-\frac{14207}{184141}a^{12}+\frac{34980}{184141}a^{11}+\frac{36455}{184141}a^{10}-\frac{88575}{184141}a^{9}+\frac{66702}{184141}a^{8}-\frac{78678}{184141}a^{7}+\frac{61483}{184141}a^{6}-\frac{69417}{184141}a^{5}+\frac{6868}{184141}a^{4}-\frac{82238}{184141}a^{3}-\frac{8741}{184141}a^{2}+\frac{909}{184141}a-\frac{66600}{184141}$, $\frac{1}{44\cdots 91}a^{19}-\frac{49\cdots 61}{44\cdots 91}a^{18}+\frac{53\cdots 29}{44\cdots 91}a^{17}+\frac{55\cdots 86}{44\cdots 91}a^{16}-\frac{19\cdots 18}{44\cdots 91}a^{15}+\frac{15\cdots 05}{44\cdots 91}a^{14}-\frac{72\cdots 17}{44\cdots 91}a^{13}-\frac{17\cdots 88}{44\cdots 91}a^{12}+\frac{47\cdots 19}{44\cdots 91}a^{11}+\frac{11\cdots 98}{44\cdots 91}a^{10}-\frac{20\cdots 47}{44\cdots 91}a^{9}+\frac{14\cdots 50}{44\cdots 91}a^{8}+\frac{12\cdots 10}{44\cdots 91}a^{7}-\frac{45\cdots 33}{44\cdots 91}a^{6}-\frac{17\cdots 71}{44\cdots 91}a^{5}+\frac{31\cdots 74}{44\cdots 91}a^{4}-\frac{11\cdots 39}{44\cdots 91}a^{3}-\frac{13\cdots 54}{44\cdots 91}a^{2}-\frac{17\cdots 75}{44\cdots 91}a-\frac{11\cdots 59}{36\cdots 91}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{842}$, which has order $3368$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{842}$, which has order $3368$ (assuming GRH) |
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| Relative class number: | $3368$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{65\cdots 51}{21\cdots 39}a^{19}-\frac{15\cdots 59}{21\cdots 39}a^{18}+\frac{33\cdots 01}{21\cdots 39}a^{17}-\frac{47\cdots 07}{21\cdots 39}a^{16}+\frac{72\cdots 61}{21\cdots 39}a^{15}-\frac{58\cdots 84}{21\cdots 39}a^{14}+\frac{85\cdots 00}{21\cdots 39}a^{13}-\frac{38\cdots 19}{21\cdots 39}a^{12}+\frac{57\cdots 82}{21\cdots 39}a^{11}-\frac{15\cdots 18}{21\cdots 39}a^{10}+\frac{23\cdots 58}{21\cdots 39}a^{9}-\frac{45\cdots 76}{21\cdots 39}a^{8}+\frac{71\cdots 31}{21\cdots 39}a^{7}-\frac{11\cdots 00}{21\cdots 39}a^{6}+\frac{16\cdots 90}{21\cdots 39}a^{5}-\frac{27\cdots 91}{21\cdots 39}a^{4}+\frac{20\cdots 51}{21\cdots 39}a^{3}-\frac{39\cdots 52}{21\cdots 39}a^{2}-\frac{26\cdots 10}{21\cdots 39}a-\frac{93\cdots 05}{17\cdots 39}$, $\frac{25\cdots 25}{65\cdots 99}a^{19}+\frac{62\cdots 05}{65\cdots 99}a^{18}+\frac{12\cdots 80}{65\cdots 99}a^{17}+\frac{81\cdots 10}{65\cdots 99}a^{16}+\frac{25\cdots 00}{65\cdots 99}a^{15}-\frac{10\cdots 55}{65\cdots 99}a^{14}+\frac{24\cdots 85}{65\cdots 99}a^{13}-\frac{24\cdots 65}{65\cdots 99}a^{12}+\frac{12\cdots 50}{65\cdots 99}a^{11}-\frac{16\cdots 04}{65\cdots 99}a^{10}+\frac{41\cdots 55}{65\cdots 99}a^{9}-\frac{58\cdots 40}{65\cdots 99}a^{8}+\frac{10\cdots 95}{65\cdots 99}a^{7}-\frac{16\cdots 20}{65\cdots 99}a^{6}+\frac{28\cdots 06}{65\cdots 99}a^{5}-\frac{29\cdots 65}{65\cdots 99}a^{4}+\frac{60\cdots 10}{65\cdots 99}a^{3}+\frac{73\cdots 30}{65\cdots 99}a^{2}+\frac{62\cdots 50}{65\cdots 99}a+\frac{46\cdots 09}{54\cdots 99}$, $\frac{99\cdots 06}{65\cdots 99}a^{19}+\frac{26\cdots 60}{65\cdots 99}a^{18}+\frac{27\cdots 01}{65\cdots 99}a^{17}+\frac{78\cdots 30}{65\cdots 99}a^{16}+\frac{27\cdots 52}{65\cdots 99}a^{15}+\frac{89\cdots 46}{65\cdots 99}a^{14}+\frac{92\cdots 30}{65\cdots 99}a^{13}+\frac{49\cdots 41}{65\cdots 99}a^{12}-\frac{12\cdots 71}{65\cdots 99}a^{11}+\frac{16\cdots 72}{65\cdots 99}a^{10}-\frac{16\cdots 47}{65\cdots 99}a^{9}+\frac{47\cdots 16}{65\cdots 99}a^{8}-\frac{57\cdots 92}{65\cdots 99}a^{7}+\frac{15\cdots 91}{65\cdots 99}a^{6}-\frac{14\cdots 50}{65\cdots 99}a^{5}+\frac{44\cdots 39}{65\cdots 99}a^{4}-\frac{14\cdots 44}{65\cdots 99}a^{3}+\frac{76\cdots 08}{65\cdots 99}a^{2}+\frac{25\cdots 93}{65\cdots 99}a+\frac{35\cdots 19}{54\cdots 99}$, $\frac{11\cdots 11}{73\cdots 91}a^{19}-\frac{35\cdots 60}{73\cdots 91}a^{18}+\frac{38\cdots 01}{73\cdots 91}a^{17}-\frac{11\cdots 35}{73\cdots 91}a^{16}+\frac{54\cdots 22}{73\cdots 91}a^{15}-\frac{13\cdots 69}{73\cdots 91}a^{14}+\frac{46\cdots 10}{82\cdots 19}a^{13}-\frac{85\cdots 74}{73\cdots 91}a^{12}+\frac{19\cdots 14}{73\cdots 91}a^{11}-\frac{35\cdots 43}{73\cdots 91}a^{10}+\frac{67\cdots 98}{73\cdots 91}a^{9}-\frac{11\cdots 14}{73\cdots 91}a^{8}+\frac{18\cdots 38}{73\cdots 91}a^{7}-\frac{29\cdots 59}{73\cdots 91}a^{6}+\frac{42\cdots 41}{73\cdots 91}a^{5}-\frac{51\cdots 66}{73\cdots 91}a^{4}+\frac{71\cdots 56}{73\cdots 91}a^{3}-\frac{36\cdots 02}{73\cdots 91}a^{2}+\frac{42\cdots 58}{73\cdots 91}a+\frac{28\cdots 35}{61\cdots 91}$, $\frac{11\cdots 11}{73\cdots 91}a^{19}-\frac{35\cdots 60}{73\cdots 91}a^{18}+\frac{38\cdots 01}{73\cdots 91}a^{17}-\frac{11\cdots 35}{73\cdots 91}a^{16}+\frac{54\cdots 22}{73\cdots 91}a^{15}-\frac{13\cdots 69}{73\cdots 91}a^{14}+\frac{46\cdots 10}{82\cdots 19}a^{13}-\frac{85\cdots 74}{73\cdots 91}a^{12}+\frac{19\cdots 14}{73\cdots 91}a^{11}-\frac{35\cdots 43}{73\cdots 91}a^{10}+\frac{67\cdots 98}{73\cdots 91}a^{9}-\frac{11\cdots 14}{73\cdots 91}a^{8}+\frac{18\cdots 38}{73\cdots 91}a^{7}-\frac{29\cdots 59}{73\cdots 91}a^{6}+\frac{42\cdots 41}{73\cdots 91}a^{5}-\frac{51\cdots 66}{73\cdots 91}a^{4}+\frac{71\cdots 56}{73\cdots 91}a^{3}-\frac{36\cdots 02}{73\cdots 91}a^{2}+\frac{42\cdots 58}{73\cdots 91}a-\frac{32\cdots 56}{61\cdots 91}$, $\frac{19\cdots 03}{44\cdots 91}a^{19}-\frac{19\cdots 67}{44\cdots 91}a^{18}+\frac{68\cdots 40}{44\cdots 91}a^{17}-\frac{68\cdots 32}{44\cdots 91}a^{16}+\frac{96\cdots 19}{44\cdots 91}a^{15}-\frac{98\cdots 86}{44\cdots 91}a^{14}+\frac{72\cdots 43}{44\cdots 91}a^{13}-\frac{76\cdots 88}{44\cdots 91}a^{12}+\frac{33\cdots 12}{44\cdots 91}a^{11}-\frac{37\cdots 49}{44\cdots 91}a^{10}+\frac{10\cdots 42}{44\cdots 91}a^{9}-\frac{13\cdots 32}{44\cdots 91}a^{8}+\frac{26\cdots 98}{44\cdots 91}a^{7}-\frac{37\cdots 41}{44\cdots 91}a^{6}+\frac{64\cdots 44}{44\cdots 91}a^{5}-\frac{57\cdots 86}{44\cdots 91}a^{4}+\frac{13\cdots 13}{44\cdots 91}a^{3}+\frac{16\cdots 55}{44\cdots 91}a^{2}+\frac{12\cdots 36}{44\cdots 91}a+\frac{13\cdots 11}{36\cdots 91}$, $\frac{42\cdots 87}{44\cdots 91}a^{19}-\frac{13\cdots 69}{44\cdots 91}a^{18}+\frac{14\cdots 72}{44\cdots 91}a^{17}-\frac{45\cdots 52}{44\cdots 91}a^{16}+\frac{20\cdots 32}{44\cdots 91}a^{15}-\frac{59\cdots 48}{44\cdots 91}a^{14}+\frac{15\cdots 80}{44\cdots 91}a^{13}-\frac{41\cdots 43}{44\cdots 91}a^{12}+\frac{70\cdots 96}{44\cdots 91}a^{11}-\frac{17\cdots 27}{44\cdots 91}a^{10}+\frac{24\cdots 41}{44\cdots 91}a^{9}-\frac{55\cdots 85}{44\cdots 91}a^{8}+\frac{78\cdots 29}{44\cdots 91}a^{7}-\frac{14\cdots 89}{44\cdots 91}a^{6}+\frac{20\cdots 33}{44\cdots 91}a^{5}-\frac{29\cdots 27}{44\cdots 91}a^{4}+\frac{33\cdots 42}{44\cdots 91}a^{3}-\frac{30\cdots 64}{44\cdots 91}a^{2}+\frac{13\cdots 48}{44\cdots 91}a+\frac{28\cdots 12}{36\cdots 91}$, $\frac{41\cdots 81}{44\cdots 91}a^{19}-\frac{44\cdots 40}{44\cdots 91}a^{18}+\frac{14\cdots 24}{44\cdots 91}a^{17}-\frac{14\cdots 80}{44\cdots 91}a^{16}+\frac{19\cdots 79}{44\cdots 91}a^{15}-\frac{19\cdots 24}{44\cdots 91}a^{14}+\frac{14\cdots 10}{44\cdots 91}a^{13}-\frac{14\cdots 86}{44\cdots 91}a^{12}+\frac{63\cdots 17}{44\cdots 91}a^{11}-\frac{76\cdots 07}{44\cdots 91}a^{10}+\frac{18\cdots 68}{44\cdots 91}a^{9}-\frac{30\cdots 53}{44\cdots 91}a^{8}+\frac{40\cdots 87}{44\cdots 91}a^{7}-\frac{80\cdots 03}{44\cdots 91}a^{6}+\frac{10\cdots 66}{44\cdots 91}a^{5}-\frac{82\cdots 86}{44\cdots 91}a^{4}+\frac{28\cdots 67}{44\cdots 91}a^{3}+\frac{19\cdots 25}{44\cdots 91}a^{2}+\frac{37\cdots 43}{44\cdots 91}a+\frac{18\cdots 65}{36\cdots 91}$, $\frac{27\cdots 03}{44\cdots 91}a^{19}-\frac{16\cdots 02}{44\cdots 91}a^{18}+\frac{88\cdots 33}{44\cdots 91}a^{17}-\frac{47\cdots 41}{44\cdots 91}a^{16}+\frac{11\cdots 58}{44\cdots 91}a^{15}-\frac{50\cdots 27}{44\cdots 91}a^{14}+\frac{87\cdots 96}{44\cdots 91}a^{13}-\frac{26\cdots 02}{44\cdots 91}a^{12}+\frac{45\cdots 32}{44\cdots 91}a^{11}-\frac{81\cdots 83}{44\cdots 91}a^{10}+\frac{17\cdots 57}{44\cdots 91}a^{9}-\frac{23\cdots 48}{44\cdots 91}a^{8}+\frac{50\cdots 40}{44\cdots 91}a^{7}-\frac{53\cdots 45}{44\cdots 91}a^{6}+\frac{79\cdots 39}{44\cdots 91}a^{5}-\frac{11\cdots 63}{44\cdots 91}a^{4}+\frac{45\cdots 22}{44\cdots 91}a^{3}-\frac{24\cdots 01}{44\cdots 91}a^{2}-\frac{11\cdots 48}{44\cdots 91}a+\frac{14\cdots 04}{36\cdots 91}$
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| Regulator: | \( 140644.599182 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 3368}{2\cdot\sqrt{10019151533337487082567413330078125}}\cr\approx \mathstrut & 0.226907242423 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-330 +66 \sqrt{5}})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
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 | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.10.2.10a1.1 | $x^{20} + 4 x^{16} + 4 x^{15} + 4 x^{14} + 4 x^{12} + 10 x^{11} + 16 x^{10} + 8 x^{9} + 4 x^{8} + 4 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + x^{2} + 7 x + 4$ | $2$ | $10$ | $10$ | 20T1 | $$[\ ]_{2}^{10}$$ |
|
\(5\)
| 5.5.4.15a1.2 | $x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 869 x^{2} + 432 x + 81$ | $4$ | $5$ | $15$ | 20T1 | not computed |
|
\(11\)
| 11.1.10.9a1.10 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |
| 11.1.10.9a1.10 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |