Normalized defining polynomial
\( x^{20} + 55 x^{18} + 1265 x^{16} + 15950 x^{14} + 121550 x^{12} + 581625 x^{10} + 1754500 x^{8} + \cdots + 378125 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(177917354031751407392000000000000000\)
\(\medspace = 2^{20}\cdot 5^{15}\cdot 11^{18}\)
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| Root discriminant: | \(57.88\) |
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| Galois root discriminant: | $2\cdot 5^{3/4}11^{9/10}\approx 57.87765351369302$ | ||
| 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: | \(220=2^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(43,·)$, $\chi_{220}(69,·)$, $\chi_{220}(7,·)$, $\chi_{220}(9,·)$, $\chi_{220}(183,·)$, $\chi_{220}(141,·)$, $\chi_{220}(63,·)$, $\chi_{220}(81,·)$, $\chi_{220}(83,·)$, $\chi_{220}(87,·)$, $\chi_{220}(89,·)$, $\chi_{220}(167,·)$, $\chi_{220}(169,·)$, $\chi_{220}(107,·)$, $\chi_{220}(49,·)$, $\chi_{220}(181,·)$, $\chi_{220}(201,·)$, $\chi_{220}(123,·)$, $\chi_{220}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{275}a^{10}$, $\frac{1}{275}a^{11}$, $\frac{1}{1375}a^{12}$, $\frac{1}{1375}a^{13}$, $\frac{1}{1375}a^{14}$, $\frac{1}{1375}a^{15}$, $\frac{1}{6875}a^{16}$, $\frac{1}{6875}a^{17}$, $\frac{1}{75625}a^{18}-\frac{4}{275}a^{8}$, $\frac{1}{75625}a^{19}-\frac{4}{275}a^{9}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{2762}$, which has order $11048$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{2762}$, which has order $11048$ (assuming GRH) |
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| Relative class number: | $11048$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{7}{15125}a^{18}+\frac{32}{1375}a^{16}+\frac{653}{1375}a^{14}+\frac{1403}{275}a^{12}+\frac{1726}{55}a^{10}+\frac{6198}{55}a^{8}+\frac{1153}{5}a^{6}+251a^{4}+128a^{2}+22$, $\frac{3}{6875}a^{18}+\frac{152}{6875}a^{16}+\frac{57}{125}a^{14}+\frac{1368}{275}a^{12}+\frac{8607}{275}a^{10}+\frac{2907}{25}a^{8}+\frac{1254}{5}a^{6}+\frac{1482}{5}a^{4}+171a^{2}+37$, $\frac{1}{6875}a^{18}+\frac{52}{6875}a^{16}+\frac{222}{1375}a^{14}+\frac{507}{275}a^{12}+\frac{678}{55}a^{10}+\frac{1242}{25}a^{8}+\frac{597}{5}a^{6}+\frac{808}{5}a^{4}+106a^{2}+23$, $\frac{2}{6875}a^{18}+\frac{104}{6875}a^{16}+\frac{444}{1375}a^{14}+\frac{5067}{1375}a^{12}+\frac{6756}{275}a^{10}+\frac{2452}{25}a^{8}+\frac{1152}{5}a^{6}+\frac{1496}{5}a^{4}+188a^{2}+40$, $\frac{1}{6875}a^{18}+\frac{48}{6875}a^{16}+\frac{184}{1375}a^{14}+\frac{1816}{1375}a^{12}+\frac{1992}{275}a^{10}+\frac{556}{25}a^{8}+\frac{182}{5}a^{6}+28a^{4}+8a^{2}+1$, $\frac{3}{15125}a^{18}+\frac{64}{6875}a^{16}+\frac{236}{1375}a^{14}+\frac{2166}{1375}a^{12}+\frac{2038}{275}a^{10}+\frac{4142}{275}a^{8}-4a^{6}-\frac{314}{5}a^{4}-77a^{2}-23$, $\frac{4}{75625}a^{18}+\frac{17}{6875}a^{16}+\frac{63}{1375}a^{14}+\frac{599}{1375}a^{12}+\frac{646}{275}a^{10}+\frac{2173}{275}a^{8}+\frac{97}{5}a^{6}+\frac{184}{5}a^{4}+39a^{2}+11$, $\frac{2}{75625}a^{18}+\frac{8}{6875}a^{16}+\frac{26}{1375}a^{14}+\frac{7}{55}a^{12}+\frac{23}{275}a^{10}-\frac{987}{275}a^{8}-\frac{101}{5}a^{6}-\frac{227}{5}a^{4}-43a^{2}-15$, $\frac{42}{75625}a^{18}+\frac{196}{6875}a^{16}+\frac{824}{1375}a^{14}+\frac{9243}{1375}a^{12}+\frac{2423}{55}a^{10}+\frac{47737}{275}a^{8}+\frac{2032}{5}a^{6}+\frac{2671}{5}a^{4}+345a^{2}+78$
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| Regulator: | \( 140644.599182 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 11048}{2\cdot\sqrt{177917354031751407392000000000000000}}\cr\approx \mathstrut & 0.176630727468 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110 +22 \sqrt{5}})\), \(\Q(\zeta_{11})^+\), 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 | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(2\)
| 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 |
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\(5\)
| 5.5.4.15a1.4 | $x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 86$ | $4$ | $5$ | $15$ | 20T1 | not computed |
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\(11\)
| 11.1.10.9a1.10 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |
| 11.1.10.9a1.10 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |