Normalized defining polynomial
\( x^{20} + 40 x^{18} + 680 x^{16} + 6400 x^{14} + 36400 x^{12} + 128160 x^{10} + 275200 x^{8} + \cdots + 5120 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(3125000000000000000000000000000000\)
\(\medspace = 2^{30}\cdot 5^{35}\)
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| Root discriminant: | \(47.29\) |
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| Galois root discriminant: | $2^{3/2}5^{7/4}\approx 47.28708045015879$ | ||
| Ramified primes: |
\(2\), \(5\)
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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: | \(200=2^{3}\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(197,·)$, $\chi_{200}(129,·)$, $\chi_{200}(9,·)$, $\chi_{200}(77,·)$, $\chi_{200}(13,·)$, $\chi_{200}(81,·)$, $\chi_{200}(89,·)$, $\chi_{200}(93,·)$, $\chi_{200}(133,·)$, $\chi_{200}(161,·)$, $\chi_{200}(37,·)$, $\chi_{200}(41,·)$, $\chi_{200}(173,·)$, $\chi_{200}(157,·)$, $\chi_{200}(49,·)$, $\chi_{200}(53,·)$, $\chi_{200}(169,·)$, $\chi_{200}(121,·)$, $\chi_{200}(117,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{1202}$, which has order $1202$ (assuming GRH) |
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| Narrow class group: | $C_{1202}$, which has order $1202$ (assuming GRH) |
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| Relative class number: | $1202$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{1}{32}a^{10}+\frac{5}{8}a^{8}+\frac{35}{8}a^{6}+\frac{25}{2}a^{4}+\frac{25}{2}a^{2}+2$, $\frac{1}{4}a^{4}+2a^{2}+2$, $\frac{1}{512}a^{18}+\frac{17}{256}a^{16}+\frac{59}{64}a^{14}+\frac{429}{64}a^{12}+\frac{871}{32}a^{10}+61a^{8}+\frac{283}{4}a^{6}+\frac{157}{4}a^{4}+\frac{21}{2}a^{2}+2$, $\frac{1}{128}a^{14}+\frac{13}{64}a^{12}+\frac{33}{16}a^{10}+\frac{165}{16}a^{8}+\frac{105}{4}a^{6}+\frac{127}{4}a^{4}+\frac{31}{2}a^{2}+2$, $\frac{1}{512}a^{18}+\frac{9}{128}a^{16}+\frac{135}{128}a^{14}+\frac{547}{64}a^{12}+\frac{1299}{32}a^{10}+\frac{1837}{16}a^{8}+\frac{753}{4}a^{6}+\frac{665}{4}a^{4}+67a^{2}+8$, $\frac{1}{256}a^{16}+\frac{1}{8}a^{14}+\frac{13}{8}a^{12}+\frac{353}{32}a^{10}+\frac{669}{16}a^{8}+\frac{175}{2}a^{6}+93a^{4}+\frac{81}{2}a^{2}+3$, $\frac{1}{16}a^{8}+a^{6}+5a^{4}+8a^{2}+2$, $\frac{1}{64}a^{12}+\frac{3}{8}a^{10}+\frac{27}{8}a^{8}+14a^{6}+\frac{105}{4}a^{4}+18a^{2}+2$, $\frac{1}{128}a^{14}+\frac{7}{32}a^{12}+\frac{77}{32}a^{10}+\frac{105}{8}a^{8}+\frac{147}{4}a^{6}+49a^{4}+\frac{49}{2}a^{2}+2$
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| Regulator: | \( 161406.837641 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 161406.837641 \cdot 1202}{2\cdot\sqrt{3125000000000000000000000000000000}}\cr\approx \mathstrut & 0.166406418457 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5 + \sqrt{5}})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\) |
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 | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|
\(5\)
| 5.1.20.35a1.500 | $x^{20} + 5 x^{16} + 120$ | $20$ | $1$ | $35$ | 20T1 | not computed |