Normalized defining polynomial
\( x^{20} - x^{19} - 9 x^{18} + 31 x^{17} + 55 x^{16} - 48 x^{15} + 189 x^{14} + 352 x^{13} + 1680 x^{12} + \cdots + 40283 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(2351977956823175708448011472615877\)
\(\medspace = 11^{16}\cdot 13^{15}\)
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| Root discriminant: | \(46.62\) |
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| Galois root discriminant: | $11^{4/5}13^{3/4}\approx 46.61993485918934$ | ||
| Ramified primes: |
\(11\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(143=11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{143}(64,·)$, $\chi_{143}(1,·)$, $\chi_{143}(5,·)$, $\chi_{143}(70,·)$, $\chi_{143}(135,·)$, $\chi_{143}(12,·)$, $\chi_{143}(14,·)$, $\chi_{143}(86,·)$, $\chi_{143}(25,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(31,·)$, $\chi_{143}(34,·)$, $\chi_{143}(38,·)$, $\chi_{143}(103,·)$, $\chi_{143}(47,·)$, $\chi_{143}(53,·)$, $\chi_{143}(122,·)$, $\chi_{143}(60,·)$, $\chi_{143}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{10\cdots 33}a^{19}+\frac{13\cdots 47}{33\cdots 11}a^{18}+\frac{10\cdots 46}{10\cdots 33}a^{17}-\frac{68\cdots 46}{33\cdots 11}a^{16}+\frac{51\cdots 30}{33\cdots 11}a^{15}-\frac{14\cdots 29}{33\cdots 11}a^{14}+\frac{31\cdots 05}{10\cdots 33}a^{13}-\frac{61\cdots 01}{33\cdots 11}a^{12}-\frac{44\cdots 87}{10\cdots 33}a^{11}+\frac{14\cdots 52}{33\cdots 11}a^{10}+\frac{17\cdots 82}{10\cdots 33}a^{9}+\frac{11\cdots 09}{33\cdots 11}a^{8}+\frac{19\cdots 25}{10\cdots 33}a^{7}-\frac{71\cdots 10}{33\cdots 11}a^{6}+\frac{12\cdots 80}{33\cdots 11}a^{5}-\frac{33\cdots 78}{10\cdots 33}a^{4}+\frac{15\cdots 50}{33\cdots 11}a^{3}-\frac{15\cdots 39}{33\cdots 11}a^{2}+\frac{21\cdots 38}{10\cdots 33}a-\frac{44\cdots 09}{10\cdots 33}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{61}$, which has order $61$ (assuming GRH) |
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| Narrow class group: | $C_{61}$, which has order $61$ (assuming GRH) |
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| Relative class number: | $61$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{12\cdots 12}{33\cdots 11}a^{19}-\frac{30\cdots 97}{33\cdots 11}a^{18}-\frac{10\cdots 56}{33\cdots 11}a^{17}+\frac{57\cdots 43}{33\cdots 11}a^{16}+\frac{17\cdots 53}{33\cdots 11}a^{15}-\frac{21\cdots 66}{33\cdots 11}a^{14}+\frac{33\cdots 47}{33\cdots 11}a^{13}+\frac{15\cdots 50}{33\cdots 11}a^{12}+\frac{82\cdots 65}{33\cdots 11}a^{11}+\frac{23\cdots 34}{33\cdots 11}a^{10}-\frac{40\cdots 21}{33\cdots 11}a^{9}+\frac{72\cdots 00}{33\cdots 11}a^{8}-\frac{45\cdots 51}{33\cdots 11}a^{7}-\frac{37\cdots 85}{33\cdots 11}a^{6}-\frac{45\cdots 78}{33\cdots 11}a^{5}-\frac{15\cdots 53}{33\cdots 11}a^{4}+\frac{66\cdots 79}{33\cdots 11}a^{3}+\frac{24\cdots 37}{33\cdots 11}a^{2}-\frac{10\cdots 79}{33\cdots 11}a-\frac{10\cdots 06}{33\cdots 11}$, $\frac{38\cdots 12}{33\cdots 11}a^{19}-\frac{34\cdots 76}{33\cdots 11}a^{18}-\frac{33\cdots 65}{33\cdots 11}a^{17}+\frac{11\cdots 04}{33\cdots 11}a^{16}+\frac{20\cdots 15}{33\cdots 11}a^{15}-\frac{12\cdots 45}{33\cdots 11}a^{14}+\frac{89\cdots 25}{33\cdots 11}a^{13}+\frac{15\cdots 17}{33\cdots 11}a^{12}+\frac{70\cdots 69}{33\cdots 11}a^{11}+\frac{18\cdots 05}{33\cdots 11}a^{10}+\frac{23\cdots 56}{33\cdots 11}a^{9}+\frac{81\cdots 47}{33\cdots 11}a^{8}+\frac{12\cdots 63}{33\cdots 11}a^{7}+\frac{12\cdots 21}{33\cdots 11}a^{6}+\frac{23\cdots 54}{33\cdots 11}a^{5}-\frac{12\cdots 51}{33\cdots 11}a^{4}-\frac{25\cdots 31}{33\cdots 11}a^{3}+\frac{23\cdots 17}{33\cdots 11}a^{2}-\frac{11\cdots 41}{33\cdots 11}a-\frac{10\cdots 51}{33\cdots 11}$, $\frac{98\cdots 46}{33\cdots 11}a^{19}-\frac{41\cdots 71}{33\cdots 11}a^{18}-\frac{73\cdots 40}{33\cdots 11}a^{17}+\frac{59\cdots 07}{33\cdots 11}a^{16}-\frac{28\cdots 93}{33\cdots 11}a^{15}-\frac{25\cdots 66}{33\cdots 11}a^{14}+\frac{20\cdots 71}{33\cdots 11}a^{13}-\frac{25\cdots 26}{33\cdots 11}a^{12}+\frac{45\cdots 05}{33\cdots 11}a^{11}-\frac{18\cdots 06}{33\cdots 11}a^{10}-\frac{12\cdots 85}{33\cdots 11}a^{9}-\frac{47\cdots 65}{33\cdots 11}a^{8}-\frac{40\cdots 73}{33\cdots 11}a^{7}-\frac{92\cdots 97}{33\cdots 11}a^{6}-\frac{66\cdots 14}{33\cdots 11}a^{5}-\frac{20\cdots 46}{33\cdots 11}a^{4}+\frac{11\cdots 05}{33\cdots 11}a^{3}+\frac{31\cdots 32}{33\cdots 11}a^{2}-\frac{12\cdots 58}{33\cdots 11}a-\frac{24\cdots 56}{33\cdots 11}$, $\frac{23\cdots 52}{33\cdots 11}a^{19}-\frac{30\cdots 04}{33\cdots 11}a^{18}-\frac{19\cdots 00}{33\cdots 11}a^{17}+\frac{78\cdots 45}{33\cdots 11}a^{16}+\frac{10\cdots 32}{33\cdots 11}a^{15}-\frac{15\cdots 64}{33\cdots 11}a^{14}+\frac{50\cdots 24}{33\cdots 11}a^{13}+\frac{84\cdots 30}{33\cdots 11}a^{12}+\frac{36\cdots 80}{33\cdots 11}a^{11}+\frac{87\cdots 02}{33\cdots 11}a^{10}+\frac{91\cdots 98}{33\cdots 11}a^{9}+\frac{38\cdots 81}{33\cdots 11}a^{8}+\frac{53\cdots 48}{33\cdots 11}a^{7}+\frac{40\cdots 74}{33\cdots 11}a^{6}+\frac{82\cdots 34}{33\cdots 11}a^{5}-\frac{95\cdots 47}{33\cdots 11}a^{4}+\frac{23\cdots 46}{33\cdots 11}a^{3}+\frac{17\cdots 65}{33\cdots 11}a^{2}-\frac{78\cdots 08}{33\cdots 11}a-\frac{21\cdots 82}{33\cdots 11}$, $\frac{74\cdots 10}{57\cdots 03}a^{19}-\frac{11\cdots 67}{57\cdots 03}a^{18}-\frac{90\cdots 45}{57\cdots 03}a^{17}+\frac{10\cdots 95}{19\cdots 01}a^{16}+\frac{19\cdots 84}{19\cdots 01}a^{15}-\frac{59\cdots 57}{19\cdots 01}a^{14}+\frac{83\cdots 16}{57\cdots 03}a^{13}+\frac{73\cdots 67}{57\cdots 03}a^{12}+\frac{27\cdots 44}{19\cdots 01}a^{11}+\frac{10\cdots 61}{57\cdots 03}a^{10}-\frac{78\cdots 31}{57\cdots 03}a^{9}+\frac{14\cdots 21}{19\cdots 01}a^{8}+\frac{16\cdots 14}{57\cdots 03}a^{7}-\frac{76\cdots 42}{57\cdots 03}a^{6}-\frac{17\cdots 38}{57\cdots 03}a^{5}+\frac{39\cdots 67}{19\cdots 01}a^{4}+\frac{69\cdots 20}{19\cdots 01}a^{3}-\frac{40\cdots 30}{57\cdots 03}a^{2}+\frac{57\cdots 75}{57\cdots 03}a-\frac{17\cdots 42}{57\cdots 03}$, $\frac{43\cdots 30}{33\cdots 11}a^{19}-\frac{23\cdots 99}{33\cdots 11}a^{18}+\frac{57\cdots 12}{10\cdots 33}a^{17}+\frac{18\cdots 62}{33\cdots 11}a^{16}-\frac{50\cdots 97}{33\cdots 11}a^{15}+\frac{35\cdots 05}{33\cdots 11}a^{14}+\frac{86\cdots 52}{33\cdots 11}a^{13}-\frac{28\cdots 58}{33\cdots 11}a^{12}+\frac{27\cdots 64}{10\cdots 33}a^{11}-\frac{20\cdots 37}{33\cdots 11}a^{10}+\frac{62\cdots 00}{10\cdots 33}a^{9}-\frac{16\cdots 89}{33\cdots 11}a^{8}-\frac{13\cdots 37}{33\cdots 11}a^{7}+\frac{90\cdots 90}{33\cdots 11}a^{6}-\frac{15\cdots 00}{10\cdots 33}a^{5}+\frac{15\cdots 90}{33\cdots 11}a^{4}+\frac{92\cdots 94}{10\cdots 33}a^{3}-\frac{45\cdots 03}{10\cdots 33}a^{2}+\frac{38\cdots 85}{10\cdots 33}a-\frac{18\cdots 89}{10\cdots 33}$, $\frac{69\cdots 64}{10\cdots 33}a^{19}-\frac{12\cdots 63}{33\cdots 11}a^{18}-\frac{57\cdots 56}{10\cdots 33}a^{17}+\frac{18\cdots 41}{10\cdots 33}a^{16}+\frac{42\cdots 56}{10\cdots 33}a^{15}+\frac{56\cdots 17}{10\cdots 33}a^{14}+\frac{15\cdots 42}{10\cdots 33}a^{13}+\frac{30\cdots 65}{10\cdots 33}a^{12}+\frac{53\cdots 15}{33\cdots 11}a^{11}+\frac{13\cdots 14}{33\cdots 11}a^{10}+\frac{55\cdots 60}{10\cdots 33}a^{9}+\frac{17\cdots 19}{10\cdots 33}a^{8}+\frac{30\cdots 47}{10\cdots 33}a^{7}+\frac{44\cdots 52}{10\cdots 33}a^{6}+\frac{68\cdots 56}{10\cdots 33}a^{5}-\frac{51\cdots 01}{10\cdots 33}a^{4}-\frac{99\cdots 40}{33\cdots 11}a^{3}+\frac{11\cdots 65}{10\cdots 33}a^{2}-\frac{40\cdots 50}{33\cdots 11}a-\frac{43\cdots 98}{10\cdots 33}$, $\frac{15\cdots 77}{33\cdots 11}a^{19}-\frac{72\cdots 94}{10\cdots 33}a^{18}-\frac{39\cdots 36}{10\cdots 33}a^{17}+\frac{17\cdots 40}{10\cdots 33}a^{16}+\frac{16\cdots 88}{10\cdots 33}a^{15}-\frac{37\cdots 66}{10\cdots 33}a^{14}+\frac{11\cdots 82}{10\cdots 33}a^{13}+\frac{35\cdots 60}{33\cdots 11}a^{12}+\frac{68\cdots 23}{10\cdots 33}a^{11}+\frac{17\cdots 16}{10\cdots 33}a^{10}+\frac{10\cdots 11}{10\cdots 33}a^{9}+\frac{75\cdots 45}{10\cdots 33}a^{8}+\frac{78\cdots 32}{10\cdots 33}a^{7}+\frac{10\cdots 51}{33\cdots 11}a^{6}+\frac{17\cdots 31}{10\cdots 33}a^{5}-\frac{36\cdots 82}{10\cdots 33}a^{4}+\frac{90\cdots 05}{10\cdots 33}a^{3}+\frac{60\cdots 37}{10\cdots 33}a^{2}-\frac{26\cdots 97}{10\cdots 33}a+\frac{21\cdots 14}{10\cdots 33}$, $\frac{12\cdots 37}{33\cdots 11}a^{19}-\frac{18\cdots 88}{33\cdots 11}a^{18}-\frac{11\cdots 23}{33\cdots 11}a^{17}+\frac{13\cdots 27}{10\cdots 33}a^{16}+\frac{63\cdots 42}{33\cdots 11}a^{15}-\frac{43\cdots 37}{10\cdots 33}a^{14}+\frac{17\cdots 27}{33\cdots 11}a^{13}+\frac{16\cdots 80}{10\cdots 33}a^{12}+\frac{17\cdots 97}{33\cdots 11}a^{11}+\frac{30\cdots 88}{33\cdots 11}a^{10}+\frac{13\cdots 11}{33\cdots 11}a^{9}+\frac{49\cdots 30}{10\cdots 33}a^{8}+\frac{18\cdots 89}{33\cdots 11}a^{7}-\frac{46\cdots 30}{10\cdots 33}a^{6}-\frac{36\cdots 39}{33\cdots 11}a^{5}-\frac{55\cdots 93}{33\cdots 11}a^{4}+\frac{47\cdots 80}{33\cdots 11}a^{3}+\frac{34\cdots 30}{10\cdots 33}a^{2}-\frac{12\cdots 80}{10\cdots 33}a-\frac{23\cdots 03}{33\cdots 11}$
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| Regulator: | \( 2015201.7242 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 2015201.7242 \cdot 61}{2\cdot\sqrt{2351977956823175708448011472615877}}\cr\approx \mathstrut & 0.12153473698 \end{aligned}\] (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
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-26 -6 \sqrt{13}})\), \(\Q(\zeta_{11})^+\), 10.10.79589952003133.1 |
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 | $20$ | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | $20$ | $20$ | R | R | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | $20$ | $20$ | $20$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | $20$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.4.5.16a1.4 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1600 x^{15} + 6440 x^{14} + 19600 x^{13} + 48360 x^{12} + 122000 x^{11} + 252208 x^{10} + 442800 x^{9} + 706720 x^{8} + 904000 x^{7} + 817760 x^{6} + 498400 x^{5} + 201200 x^{4} + 52800 x^{3} + 8640 x^{2} + 800 x + 43$ | $5$ | $4$ | $16$ | 20T1 | $$[\ ]_{5}^{4}$$ |
|
\(13\)
| 13.5.4.15a1.4 | $x^{20} + 16 x^{16} + 44 x^{15} + 96 x^{12} + 528 x^{11} + 726 x^{10} + 256 x^{8} + 2112 x^{7} + 5808 x^{6} + 5324 x^{5} + 256 x^{4} + 2816 x^{3} + 11616 x^{2} + 21296 x + 14654$ | $4$ | $5$ | $15$ | 20T1 | $$[\ ]_{4}^{5}$$ |