Normalized defining polynomial
\( x^{20} - 4 x^{19} + 40 x^{18} - 124 x^{17} + 815 x^{16} - 2164 x^{15} + 10834 x^{14} - 24620 x^{13} + \cdots + 90594151 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(1505680748169532571648000000000000000\)
\(\medspace = 2^{30}\cdot 5^{15}\cdot 11^{16}\)
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| Root discriminant: | \(64.40\) |
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| Galois root discriminant: | $2^{3/2}5^{3/4}11^{4/5}\approx 64.4001152950292$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
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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: | \(440=2^{3}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(133,·)$, $\chi_{440}(201,·)$, $\chi_{440}(397,·)$, $\chi_{440}(333,·)$, $\chi_{440}(81,·)$, $\chi_{440}(37,·)$, $\chi_{440}(213,·)$, $\chi_{440}(89,·)$, $\chi_{440}(93,·)$, $\chi_{440}(361,·)$, $\chi_{440}(289,·)$, $\chi_{440}(357,·)$, $\chi_{440}(401,·)$, $\chi_{440}(169,·)$, $\chi_{440}(157,·)$, $\chi_{440}(49,·)$, $\chi_{440}(53,·)$, $\chi_{440}(9,·)$, $\chi_{440}(317,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{12\cdots 82}a^{18}-\frac{43\cdots 93}{12\cdots 82}a^{17}-\frac{57\cdots 87}{12\cdots 82}a^{16}+\frac{53\cdots 17}{64\cdots 41}a^{15}-\frac{21\cdots 39}{12\cdots 82}a^{14}-\frac{32\cdots 89}{12\cdots 82}a^{13}-\frac{62\cdots 49}{64\cdots 41}a^{12}-\frac{12\cdots 33}{12\cdots 82}a^{11}+\frac{17\cdots 87}{64\cdots 41}a^{10}-\frac{70\cdots 39}{64\cdots 41}a^{9}-\frac{51\cdots 25}{12\cdots 82}a^{8}-\frac{18\cdots 31}{64\cdots 41}a^{7}-\frac{55\cdots 83}{12\cdots 82}a^{6}-\frac{21\cdots 83}{12\cdots 82}a^{5}+\frac{52\cdots 09}{64\cdots 41}a^{4}-\frac{30\cdots 11}{64\cdots 41}a^{3}-\frac{54\cdots 23}{12\cdots 82}a^{2}+\frac{17\cdots 70}{64\cdots 41}a-\frac{49\cdots 53}{11\cdots 98}$, $\frac{1}{11\cdots 42}a^{19}+\frac{24\cdots 41}{11\cdots 42}a^{18}+\frac{18\cdots 86}{59\cdots 21}a^{17}+\frac{15\cdots 95}{11\cdots 42}a^{16}+\frac{10\cdots 02}{59\cdots 21}a^{15}+\frac{12\cdots 43}{11\cdots 42}a^{14}+\frac{88\cdots 83}{59\cdots 21}a^{13}+\frac{24\cdots 39}{11\cdots 42}a^{12}-\frac{97\cdots 71}{11\cdots 42}a^{11}-\frac{36\cdots 64}{59\cdots 21}a^{10}+\frac{16\cdots 96}{59\cdots 21}a^{9}+\frac{10\cdots 57}{59\cdots 21}a^{8}-\frac{28\cdots 06}{59\cdots 21}a^{7}-\frac{28\cdots 39}{59\cdots 21}a^{6}+\frac{16\cdots 76}{59\cdots 21}a^{5}+\frac{19\cdots 55}{59\cdots 21}a^{4}+\frac{47\cdots 19}{11\cdots 42}a^{3}-\frac{40\cdots 99}{11\cdots 42}a^{2}+\frac{22\cdots 01}{11\cdots 42}a+\frac{24\cdots 91}{54\cdots 69}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{22082}$, which has order $22082$ (assuming GRH) |
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| Narrow class group: | $C_{22082}$, which has order $22082$ (assuming GRH) |
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| Relative class number: | $22082$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{404924790182}{64\cdots 41}a^{19}+\frac{1764075181645}{64\cdots 41}a^{18}+\frac{1618421850242}{64\cdots 41}a^{17}+\frac{107520892245065}{12\cdots 82}a^{16}-\frac{11594532537538}{64\cdots 41}a^{15}+\frac{911820723687953}{64\cdots 41}a^{14}-\frac{875504250517845}{64\cdots 41}a^{13}+\frac{20\cdots 91}{12\cdots 82}a^{12}-\frac{94\cdots 06}{64\cdots 41}a^{11}+\frac{79\cdots 03}{64\cdots 41}a^{10}-\frac{73\cdots 84}{64\cdots 41}a^{9}+\frac{94\cdots 61}{12\cdots 82}a^{8}-\frac{36\cdots 84}{64\cdots 41}a^{7}+\frac{21\cdots 53}{64\cdots 41}a^{6}-\frac{14\cdots 39}{64\cdots 41}a^{5}+\frac{71\cdots 12}{64\cdots 41}a^{4}-\frac{33\cdots 38}{64\cdots 41}a^{3}+\frac{31\cdots 29}{12\cdots 82}a^{2}-\frac{41\cdots 36}{64\cdots 41}a+\frac{10\cdots 53}{11\cdots 98}$, $\frac{32\cdots 80}{59\cdots 21}a^{19}+\frac{11\cdots 90}{59\cdots 21}a^{18}-\frac{20\cdots 20}{59\cdots 21}a^{17}+\frac{38\cdots 90}{59\cdots 21}a^{16}-\frac{66\cdots 76}{59\cdots 21}a^{15}+\frac{72\cdots 70}{59\cdots 21}a^{14}-\frac{11\cdots 40}{59\cdots 21}a^{13}+\frac{89\cdots 90}{59\cdots 21}a^{12}-\frac{11\cdots 00}{59\cdots 21}a^{11}+\frac{77\cdots 30}{59\cdots 21}a^{10}-\frac{83\cdots 20}{59\cdots 21}a^{9}+\frac{50\cdots 90}{59\cdots 21}a^{8}-\frac{43\cdots 60}{59\cdots 21}a^{7}+\frac{25\cdots 40}{59\cdots 21}a^{6}-\frac{16\cdots 12}{59\cdots 21}a^{5}+\frac{98\cdots 75}{59\cdots 21}a^{4}-\frac{42\cdots 40}{59\cdots 21}a^{3}+\frac{23\cdots 20}{59\cdots 21}a^{2}-\frac{52\cdots 20}{59\cdots 21}a+\frac{31\cdots 60}{54\cdots 69}$, $\frac{31\cdots 04}{59\cdots 21}a^{19}+\frac{72\cdots 60}{59\cdots 21}a^{18}-\frac{51\cdots 16}{59\cdots 21}a^{17}+\frac{24\cdots 75}{59\cdots 21}a^{16}-\frac{22\cdots 52}{59\cdots 21}a^{15}+\frac{46\cdots 96}{59\cdots 21}a^{14}-\frac{41\cdots 80}{59\cdots 21}a^{13}+\frac{58\cdots 96}{59\cdots 21}a^{12}-\frac{43\cdots 44}{59\cdots 21}a^{11}+\frac{51\cdots 12}{59\cdots 21}a^{10}-\frac{29\cdots 08}{59\cdots 21}a^{9}+\frac{33\cdots 31}{59\cdots 21}a^{8}-\frac{13\cdots 88}{59\cdots 21}a^{7}+\frac{16\cdots 56}{59\cdots 21}a^{6}-\frac{35\cdots 40}{59\cdots 21}a^{5}+\frac{61\cdots 24}{59\cdots 21}a^{4}-\frac{10\cdots 16}{59\cdots 21}a^{3}+\frac{14\cdots 08}{59\cdots 21}a^{2}+\frac{60\cdots 72}{59\cdots 21}a+\frac{12\cdots 31}{54\cdots 69}$, $\frac{13\cdots 96}{59\cdots 21}a^{19}+\frac{21\cdots 80}{59\cdots 21}a^{18}-\frac{12\cdots 04}{59\cdots 21}a^{17}+\frac{10\cdots 00}{59\cdots 21}a^{16}-\frac{46\cdots 08}{59\cdots 21}a^{15}+\frac{21\cdots 24}{59\cdots 21}a^{14}-\frac{85\cdots 80}{59\cdots 21}a^{13}+\frac{29\cdots 79}{59\cdots 21}a^{12}-\frac{98\cdots 76}{59\cdots 21}a^{11}+\frac{26\cdots 24}{59\cdots 21}a^{10}-\frac{78\cdots 32}{59\cdots 21}a^{9}+\frac{17\cdots 89}{59\cdots 21}a^{8}-\frac{44\cdots 12}{59\cdots 21}a^{7}+\frac{86\cdots 74}{59\cdots 21}a^{6}-\frac{18\cdots 24}{59\cdots 21}a^{5}+\frac{32\cdots 01}{59\cdots 21}a^{4}-\frac{53\cdots 84}{59\cdots 21}a^{3}+\frac{75\cdots 92}{59\cdots 21}a^{2}-\frac{69\cdots 92}{59\cdots 21}a+\frac{85\cdots 02}{54\cdots 69}$, $\frac{81\cdots 76}{59\cdots 21}a^{19}+\frac{37\cdots 30}{59\cdots 21}a^{18}-\frac{15\cdots 04}{59\cdots 21}a^{17}+\frac{13\cdots 15}{59\cdots 21}a^{16}-\frac{43\cdots 24}{59\cdots 21}a^{15}+\frac{25\cdots 74}{59\cdots 21}a^{14}-\frac{69\cdots 60}{59\cdots 21}a^{13}+\frac{31\cdots 94}{59\cdots 21}a^{12}-\frac{72\cdots 56}{59\cdots 21}a^{11}+\frac{26\cdots 18}{59\cdots 21}a^{10}-\frac{53\cdots 12}{59\cdots 21}a^{9}+\frac{17\cdots 59}{59\cdots 21}a^{8}-\frac{30\cdots 72}{59\cdots 21}a^{7}+\frac{88\cdots 84}{59\cdots 21}a^{6}-\frac{13\cdots 72}{59\cdots 21}a^{5}+\frac{36\cdots 51}{59\cdots 21}a^{4}-\frac{42\cdots 24}{59\cdots 21}a^{3}+\frac{91\cdots 12}{59\cdots 21}a^{2}-\frac{58\cdots 92}{59\cdots 21}a+\frac{14\cdots 60}{54\cdots 69}$, $\frac{51\cdots 82}{59\cdots 21}a^{19}+\frac{11\cdots 65}{59\cdots 21}a^{18}-\frac{31\cdots 38}{59\cdots 21}a^{17}+\frac{86\cdots 75}{11\cdots 42}a^{16}-\frac{10\cdots 38}{59\cdots 21}a^{15}+\frac{84\cdots 93}{59\cdots 21}a^{14}-\frac{18\cdots 85}{59\cdots 21}a^{13}+\frac{21\cdots 91}{11\cdots 42}a^{12}-\frac{20\cdots 86}{59\cdots 21}a^{11}+\frac{85\cdots 07}{54\cdots 69}a^{10}-\frac{15\cdots 04}{59\cdots 21}a^{9}+\frac{12\cdots 81}{11\cdots 42}a^{8}-\frac{82\cdots 44}{59\cdots 21}a^{7}+\frac{29\cdots 13}{59\cdots 21}a^{6}-\frac{33\cdots 03}{59\cdots 21}a^{5}+\frac{10\cdots 72}{59\cdots 21}a^{4}-\frac{91\cdots 98}{59\cdots 21}a^{3}+\frac{51\cdots 89}{11\cdots 42}a^{2}-\frac{11\cdots 16}{59\cdots 21}a+\frac{66\cdots 73}{10\cdots 38}$, $\frac{19\cdots 78}{59\cdots 21}a^{19}+\frac{44\cdots 05}{59\cdots 21}a^{18}-\frac{26\cdots 22}{59\cdots 21}a^{17}+\frac{37\cdots 25}{11\cdots 42}a^{16}-\frac{82\cdots 86}{59\cdots 21}a^{15}+\frac{37\cdots 97}{59\cdots 21}a^{14}-\frac{14\cdots 05}{59\cdots 21}a^{13}+\frac{95\cdots 99}{11\cdots 42}a^{12}-\frac{15\cdots 42}{59\cdots 21}a^{11}+\frac{41\cdots 51}{59\cdots 21}a^{10}-\frac{12\cdots 96}{59\cdots 21}a^{9}+\frac{54\cdots 19}{11\cdots 42}a^{8}-\frac{69\cdots 56}{59\cdots 21}a^{7}+\frac{13\cdots 57}{59\cdots 21}a^{6}-\frac{29\cdots 63}{59\cdots 21}a^{5}+\frac{47\cdots 48}{59\cdots 21}a^{4}-\frac{90\cdots 82}{59\cdots 21}a^{3}+\frac{21\cdots 73}{11\cdots 42}a^{2}-\frac{12\cdots 88}{59\cdots 21}a+\frac{41\cdots 11}{10\cdots 38}$, $\frac{66\cdots 82}{59\cdots 21}a^{19}+\frac{22\cdots 25}{59\cdots 21}a^{18}-\frac{13\cdots 18}{59\cdots 21}a^{17}+\frac{17\cdots 25}{11\cdots 42}a^{16}-\frac{35\cdots 78}{59\cdots 21}a^{15}+\frac{15\cdots 73}{59\cdots 21}a^{14}-\frac{58\cdots 25}{59\cdots 21}a^{13}+\frac{36\cdots 41}{11\cdots 42}a^{12}-\frac{59\cdots 66}{59\cdots 21}a^{11}+\frac{15\cdots 27}{59\cdots 21}a^{10}-\frac{44\cdots 64}{59\cdots 21}a^{9}+\frac{18\cdots 41}{11\cdots 42}a^{8}-\frac{25\cdots 44}{59\cdots 21}a^{7}+\frac{43\cdots 83}{59\cdots 21}a^{6}-\frac{11\cdots 39}{59\cdots 21}a^{5}+\frac{15\cdots 47}{59\cdots 21}a^{4}-\frac{37\cdots 98}{59\cdots 21}a^{3}+\frac{66\cdots 89}{11\cdots 42}a^{2}-\frac{53\cdots 96}{59\cdots 21}a+\frac{13\cdots 69}{10\cdots 38}$, $\frac{75\cdots 30}{59\cdots 21}a^{19}+\frac{58\cdots 86}{59\cdots 21}a^{18}-\frac{15\cdots 65}{59\cdots 21}a^{17}+\frac{23\cdots 01}{59\cdots 21}a^{16}-\frac{30\cdots 90}{59\cdots 21}a^{15}+\frac{97\cdots 25}{11\cdots 42}a^{14}-\frac{70\cdots 01}{59\cdots 21}a^{13}+\frac{12\cdots 77}{11\cdots 42}a^{12}-\frac{84\cdots 66}{59\cdots 21}a^{11}+\frac{57\cdots 14}{59\cdots 21}a^{10}-\frac{67\cdots 61}{59\cdots 21}a^{9}+\frac{76\cdots 51}{11\cdots 42}a^{8}-\frac{37\cdots 86}{59\cdots 21}a^{7}+\frac{38\cdots 41}{11\cdots 42}a^{6}-\frac{16\cdots 00}{59\cdots 21}a^{5}+\frac{14\cdots 87}{11\cdots 42}a^{4}-\frac{49\cdots 18}{59\cdots 21}a^{3}+\frac{17\cdots 24}{59\cdots 21}a^{2}-\frac{66\cdots 03}{59\cdots 21}a+\frac{20\cdots 67}{54\cdots 69}$
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| Regulator: | \( 140644.599182 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 22082}{2\cdot\sqrt{1505680748169532571648000000000000000}}\cr\approx \mathstrut & 0.121356684930 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5 + \sqrt{5}})\), \(\Q(\zeta_{11})^+\), 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 | $20$ | R | $20$ | R | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.2.30a1.114 | $x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 7 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 9 x^{6} + 16 x^{5} + 3 x^{4} + 8 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $10$ | $30$ | 20T1 | not computed |
|
\(5\)
| 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 |
|
\(11\)
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |