Normalized defining polynomial
\( x^{20} - 4 x^{19} + 65 x^{18} - 214 x^{17} + 2035 x^{16} - 5724 x^{15} + 39824 x^{14} + \cdots + 2065660201 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(86825139158850321116448000000000000000\)
\(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{15}\cdot 11^{16}\)
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| Root discriminant: | \(78.87\) |
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| Galois root discriminant: | $2\cdot 3^{1/2}5^{3/4}11^{4/5}\approx 78.87371092461403$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(11\)
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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: | \(660=2^{2}\cdot 3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(323,·)$, $\chi_{660}(647,·)$, $\chi_{660}(587,·)$, $\chi_{660}(529,·)$, $\chi_{660}(467,·)$, $\chi_{660}(203,·)$, $\chi_{660}(23,·)$, $\chi_{660}(229,·)$, $\chi_{660}(289,·)$, $\chi_{660}(421,·)$, $\chi_{660}(169,·)$, $\chi_{660}(301,·)$, $\chi_{660}(47,·)$, $\chi_{660}(287,·)$, $\chi_{660}(49,·)$, $\chi_{660}(181,·)$, $\chi_{660}(361,·)$, $\chi_{660}(443,·)$, $\chi_{660}(383,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{27\cdots 83}a^{18}-\frac{18\cdots 84}{27\cdots 83}a^{17}+\frac{49\cdots 02}{91\cdots 61}a^{16}+\frac{22\cdots 77}{27\cdots 83}a^{15}+\frac{12\cdots 99}{91\cdots 61}a^{14}+\frac{27\cdots 64}{27\cdots 83}a^{13}+\frac{14\cdots 93}{27\cdots 83}a^{12}+\frac{76\cdots 51}{27\cdots 83}a^{11}+\frac{33\cdots 65}{27\cdots 83}a^{10}-\frac{17\cdots 28}{91\cdots 61}a^{9}-\frac{73\cdots 73}{27\cdots 83}a^{8}+\frac{36\cdots 21}{91\cdots 61}a^{7}-\frac{10\cdots 60}{27\cdots 83}a^{6}-\frac{11\cdots 23}{27\cdots 83}a^{5}-\frac{84\cdots 87}{27\cdots 83}a^{4}-\frac{11\cdots 17}{27\cdots 83}a^{3}-\frac{10\cdots 35}{27\cdots 83}a^{2}+\frac{93\cdots 15}{27\cdots 83}a-\frac{11\cdots 37}{27\cdots 83}$, $\frac{1}{23\cdots 43}a^{19}-\frac{11\cdots 75}{77\cdots 81}a^{18}+\frac{30\cdots 87}{23\cdots 43}a^{17}+\frac{23\cdots 95}{23\cdots 43}a^{16}+\frac{14\cdots 86}{23\cdots 43}a^{15}+\frac{73\cdots 55}{23\cdots 43}a^{14}-\frac{32\cdots 00}{77\cdots 81}a^{13}+\frac{17\cdots 27}{23\cdots 43}a^{12}-\frac{10\cdots 54}{77\cdots 81}a^{11}+\frac{49\cdots 87}{23\cdots 43}a^{10}-\frac{59\cdots 37}{77\cdots 81}a^{9}+\frac{24\cdots 61}{23\cdots 43}a^{8}-\frac{63\cdots 91}{23\cdots 43}a^{7}+\frac{24\cdots 33}{23\cdots 43}a^{6}-\frac{34\cdots 28}{77\cdots 81}a^{5}-\frac{28\cdots 07}{77\cdots 81}a^{4}+\frac{12\cdots 70}{77\cdots 81}a^{3}-\frac{19\cdots 27}{23\cdots 43}a^{2}-\frac{88\cdots 22}{23\cdots 43}a+\frac{27\cdots 56}{23\cdots 43}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{10}\times C_{31810}$, which has order $318100$ (assuming GRH) |
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| Narrow class group: | $C_{10}\times C_{31810}$, which has order $318100$ (assuming GRH) |
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| Relative class number: | $318100$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{13568180049644}{27\cdots 83}a^{19}+\frac{134897714527690}{27\cdots 83}a^{18}+\frac{104276651336804}{27\cdots 83}a^{17}+\frac{71\cdots 20}{27\cdots 83}a^{16}-\frac{21\cdots 12}{91\cdots 61}a^{15}+\frac{18\cdots 86}{27\cdots 83}a^{14}-\frac{24\cdots 80}{27\cdots 83}a^{13}+\frac{30\cdots 61}{27\cdots 83}a^{12}-\frac{39\cdots 92}{27\cdots 83}a^{11}+\frac{34\cdots 51}{27\cdots 83}a^{10}-\frac{40\cdots 08}{27\cdots 83}a^{9}+\frac{90\cdots 97}{91\cdots 61}a^{8}-\frac{27\cdots 28}{27\cdots 83}a^{7}+\frac{53\cdots 27}{91\cdots 61}a^{6}-\frac{44\cdots 16}{91\cdots 61}a^{5}+\frac{66\cdots 29}{27\cdots 83}a^{4}-\frac{38\cdots 96}{27\cdots 83}a^{3}+\frac{17\cdots 74}{27\cdots 83}a^{2}-\frac{18\cdots 94}{91\cdots 61}a+\frac{40\cdots 64}{27\cdots 83}$, $\frac{11\cdots 56}{77\cdots 81}a^{19}-\frac{71\cdots 10}{77\cdots 81}a^{18}+\frac{32\cdots 76}{77\cdots 81}a^{17}+\frac{68\cdots 10}{77\cdots 81}a^{16}+\frac{10\cdots 32}{77\cdots 81}a^{15}+\frac{38\cdots 24}{77\cdots 81}a^{14}-\frac{13\cdots 20}{77\cdots 81}a^{13}+\frac{98\cdots 69}{77\cdots 81}a^{12}-\frac{37\cdots 16}{77\cdots 81}a^{11}+\frac{14\cdots 23}{77\cdots 81}a^{10}-\frac{52\cdots 52}{77\cdots 81}a^{9}+\frac{14\cdots 64}{77\cdots 81}a^{8}-\frac{45\cdots 92}{77\cdots 81}a^{7}+\frac{10\cdots 39}{77\cdots 81}a^{6}-\frac{27\cdots 04}{77\cdots 81}a^{5}+\frac{49\cdots 61}{77\cdots 81}a^{4}-\frac{10\cdots 64}{77\cdots 81}a^{3}+\frac{15\cdots 22}{77\cdots 81}a^{2}-\frac{20\cdots 82}{77\cdots 81}a+\frac{22\cdots 80}{77\cdots 81}$, $\frac{49\cdots 20}{77\cdots 81}a^{19}+\frac{13\cdots 90}{77\cdots 81}a^{18}-\frac{35\cdots 50}{77\cdots 81}a^{17}+\frac{87\cdots 50}{77\cdots 81}a^{16}-\frac{24\cdots 00}{77\cdots 81}a^{15}+\frac{27\cdots 90}{77\cdots 81}a^{14}-\frac{69\cdots 60}{77\cdots 81}a^{13}+\frac{52\cdots 05}{77\cdots 81}a^{12}-\frac{11\cdots 40}{77\cdots 81}a^{11}+\frac{66\cdots 59}{77\cdots 81}a^{10}-\frac{12\cdots 30}{77\cdots 81}a^{9}+\frac{60\cdots 40}{77\cdots 81}a^{8}-\frac{97\cdots 40}{77\cdots 81}a^{7}+\frac{39\cdots 35}{77\cdots 81}a^{6}-\frac{51\cdots 74}{77\cdots 81}a^{5}+\frac{18\cdots 80}{77\cdots 81}a^{4}-\frac{18\cdots 90}{77\cdots 81}a^{3}+\frac{54\cdots 85}{77\cdots 81}a^{2}-\frac{31\cdots 40}{77\cdots 81}a+\frac{73\cdots 31}{77\cdots 81}$, $\frac{63\cdots 36}{77\cdots 81}a^{19}-\frac{13\cdots 00}{77\cdots 81}a^{18}+\frac{68\cdots 26}{77\cdots 81}a^{17}-\frac{80\cdots 40}{77\cdots 81}a^{16}+\frac{25\cdots 32}{77\cdots 81}a^{15}-\frac{23\cdots 66}{77\cdots 81}a^{14}+\frac{55\cdots 40}{77\cdots 81}a^{13}-\frac{42\cdots 36}{77\cdots 81}a^{12}+\frac{78\cdots 24}{77\cdots 81}a^{11}-\frac{52\cdots 36}{77\cdots 81}a^{10}+\frac{75\cdots 78}{77\cdots 81}a^{9}-\frac{45\cdots 76}{77\cdots 81}a^{8}+\frac{51\cdots 48}{77\cdots 81}a^{7}-\frac{29\cdots 96}{77\cdots 81}a^{6}+\frac{24\cdots 70}{77\cdots 81}a^{5}-\frac{13\cdots 19}{77\cdots 81}a^{4}+\frac{75\cdots 26}{77\cdots 81}a^{3}-\frac{39\cdots 63}{77\cdots 81}a^{2}+\frac{11\cdots 58}{77\cdots 81}a-\frac{58\cdots 32}{77\cdots 81}$, $\frac{23\cdots 60}{77\cdots 81}a^{19}+\frac{14\cdots 30}{77\cdots 81}a^{18}-\frac{42\cdots 20}{77\cdots 81}a^{17}+\frac{92\cdots 15}{77\cdots 81}a^{16}-\frac{25\cdots 20}{77\cdots 81}a^{15}+\frac{28\cdots 80}{77\cdots 81}a^{14}-\frac{67\cdots 00}{77\cdots 81}a^{13}+\frac{52\cdots 45}{77\cdots 81}a^{12}-\frac{10\cdots 80}{77\cdots 81}a^{11}+\frac{66\cdots 05}{77\cdots 81}a^{10}-\frac{11\cdots 00}{77\cdots 81}a^{9}+\frac{59\cdots 05}{77\cdots 81}a^{8}-\frac{87\cdots 40}{77\cdots 81}a^{7}+\frac{38\cdots 05}{77\cdots 81}a^{6}-\frac{44\cdots 24}{77\cdots 81}a^{5}+\frac{17\cdots 05}{77\cdots 81}a^{4}-\frac{15\cdots 00}{77\cdots 81}a^{3}+\frac{53\cdots 60}{77\cdots 81}a^{2}-\frac{24\cdots 70}{77\cdots 81}a+\frac{71\cdots 14}{77\cdots 81}$, $\frac{18\cdots 04}{23\cdots 43}a^{19}+\frac{55\cdots 80}{23\cdots 43}a^{18}-\frac{11\cdots 76}{23\cdots 43}a^{17}+\frac{33\cdots 65}{23\cdots 43}a^{16}-\frac{26\cdots 72}{77\cdots 81}a^{15}+\frac{99\cdots 46}{23\cdots 43}a^{14}-\frac{22\cdots 80}{23\cdots 43}a^{13}+\frac{18\cdots 16}{23\cdots 43}a^{12}-\frac{36\cdots 72}{23\cdots 43}a^{11}+\frac{22\cdots 86}{23\cdots 43}a^{10}-\frac{38\cdots 68}{23\cdots 43}a^{9}+\frac{67\cdots 42}{77\cdots 81}a^{8}-\frac{28\cdots 08}{23\cdots 43}a^{7}+\frac{43\cdots 72}{77\cdots 81}a^{6}-\frac{48\cdots 60}{77\cdots 81}a^{5}+\frac{59\cdots 24}{23\cdots 43}a^{4}-\frac{48\cdots 16}{23\cdots 43}a^{3}+\frac{17\cdots 34}{23\cdots 43}a^{2}-\frac{26\cdots 44}{77\cdots 81}a+\frac{27\cdots 29}{23\cdots 43}$, $\frac{22\cdots 44}{23\cdots 43}a^{19}-\frac{13\cdots 20}{23\cdots 43}a^{18}+\frac{89\cdots 44}{23\cdots 43}a^{17}-\frac{40\cdots 90}{23\cdots 43}a^{16}+\frac{28\cdots 84}{77\cdots 81}a^{15}-\frac{42\cdots 34}{23\cdots 43}a^{14}-\frac{21\cdots 80}{23\cdots 43}a^{13}+\frac{35\cdots 26}{23\cdots 43}a^{12}-\frac{80\cdots 16}{23\cdots 43}a^{11}+\frac{15\cdots 98}{23\cdots 43}a^{10}-\frac{12\cdots 88}{23\cdots 43}a^{9}+\frac{69\cdots 27}{77\cdots 81}a^{8}-\frac{11\cdots 88}{23\cdots 43}a^{7}+\frac{55\cdots 72}{77\cdots 81}a^{6}-\frac{23\cdots 68}{77\cdots 81}a^{5}+\frac{91\cdots 74}{23\cdots 43}a^{4}-\frac{29\cdots 76}{23\cdots 43}a^{3}+\frac{30\cdots 12}{23\cdots 43}a^{2}-\frac{18\cdots 08}{77\cdots 81}a+\frac{34\cdots 96}{23\cdots 43}$, $\frac{38\cdots 44}{23\cdots 43}a^{19}+\frac{16\cdots 10}{23\cdots 43}a^{18}-\frac{13\cdots 26}{23\cdots 43}a^{17}+\frac{75\cdots 15}{23\cdots 43}a^{16}-\frac{21\cdots 72}{77\cdots 81}a^{15}+\frac{18\cdots 76}{23\cdots 43}a^{14}-\frac{12\cdots 00}{23\cdots 43}a^{13}+\frac{27\cdots 01}{23\cdots 43}a^{12}-\frac{10\cdots 52}{23\cdots 43}a^{11}+\frac{28\cdots 09}{23\cdots 43}a^{10}-\frac{10\cdots 78}{23\cdots 43}a^{9}+\frac{69\cdots 02}{77\cdots 81}a^{8}+\frac{77\cdots 12}{23\cdots 43}a^{7}+\frac{38\cdots 37}{77\cdots 81}a^{6}+\frac{33\cdots 14}{77\cdots 81}a^{5}+\frac{44\cdots 84}{23\cdots 43}a^{4}+\frac{72\cdots 54}{23\cdots 43}a^{3}+\frac{11\cdots 79}{23\cdots 43}a^{2}+\frac{56\cdots 96}{77\cdots 81}a+\frac{50\cdots 36}{23\cdots 43}$, $\frac{11\cdots 94}{23\cdots 43}a^{19}+\frac{34\cdots 77}{23\cdots 43}a^{18}-\frac{51\cdots 96}{23\cdots 43}a^{17}+\frac{21\cdots 67}{23\cdots 43}a^{16}-\frac{40\cdots 12}{23\cdots 43}a^{15}+\frac{21\cdots 32}{77\cdots 81}a^{14}-\frac{11\cdots 42}{23\cdots 43}a^{13}+\frac{11\cdots 53}{23\cdots 43}a^{12}-\frac{18\cdots 76}{23\cdots 43}a^{11}+\frac{15\cdots 22}{23\cdots 43}a^{10}-\frac{65\cdots 70}{77\cdots 81}a^{9}+\frac{13\cdots 37}{23\cdots 43}a^{8}-\frac{47\cdots 80}{77\cdots 81}a^{7}+\frac{29\cdots 54}{77\cdots 81}a^{6}-\frac{25\cdots 20}{77\cdots 81}a^{5}+\frac{41\cdots 36}{23\cdots 43}a^{4}-\frac{87\cdots 64}{77\cdots 81}a^{3}+\frac{12\cdots 41}{23\cdots 43}a^{2}-\frac{43\cdots 34}{23\cdots 43}a+\frac{61\cdots 39}{77\cdots 81}$
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| Regulator: | \( 140644.599182 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 318100}{2\cdot\sqrt{86825139158850321116448000000000000000}}\cr\approx \mathstrut & 0.230214499275 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30 +6 \sqrt{5}})\), \(\Q(\zeta_{11})^+\), 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 | R | R | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.2.20a1.2 | $x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 5 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 7 x^{6} + 10 x^{5} + 3 x^{4} + 6 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $10$ | $20$ | 20T1 | not computed |
|
\(3\)
| 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 |
|
\(11\)
| 11.1.5.4a1.1 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
| 11.1.5.4a1.1 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ | |
| 11.1.5.4a1.1 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ | |
| 11.1.5.4a1.1 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |