Normalized defining polynomial
\( x^{20} + 60 x^{18} + 1530 x^{16} + 21600 x^{14} + 184275 x^{12} + 973215 x^{10} + 3134700 x^{8} + \cdots + 295245 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(180203247070312500000000000000000000\)
\(\medspace = 2^{20}\cdot 3^{10}\cdot 5^{35}\)
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| Root discriminant: | \(57.91\) |
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| Galois root discriminant: | $2\cdot 3^{1/2}5^{7/4}\approx 57.91460926441345$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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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: | \(300=2^{2}\cdot 3\cdot 5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(263,·)$, $\chi_{300}(203,·)$, $\chi_{300}(143,·)$, $\chi_{300}(83,·)$, $\chi_{300}(23,·)$, $\chi_{300}(287,·)$, $\chi_{300}(289,·)$, $\chi_{300}(227,·)$, $\chi_{300}(229,·)$, $\chi_{300}(167,·)$, $\chi_{300}(169,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(181,·)$, $\chi_{300}(241,·)$, $\chi_{300}(121,·)$, $\chi_{300}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4322}$, which has order $8644$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{4322}$, which has order $8644$ (assuming GRH) |
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| Relative class number: | $8644$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{1}{243}a^{10}+\frac{10}{81}a^{8}+\frac{35}{27}a^{6}+\frac{50}{9}a^{4}+\frac{25}{3}a^{2}+2$, $\frac{1}{2187}a^{14}+\frac{14}{729}a^{12}+\frac{77}{243}a^{10}+\frac{70}{27}a^{8}+\frac{98}{9}a^{6}+\frac{196}{9}a^{4}+\frac{49}{3}a^{2}+2$, $\frac{1}{19683}a^{18}+\frac{17}{6561}a^{16}+\frac{118}{2187}a^{14}+\frac{143}{243}a^{12}+\frac{871}{243}a^{10}+\frac{976}{81}a^{8}+\frac{566}{27}a^{6}+\frac{157}{9}a^{4}+7a^{2}+2$, $\frac{1}{6561}a^{16}+\frac{16}{2187}a^{14}+\frac{104}{729}a^{12}+\frac{353}{243}a^{10}+\frac{223}{27}a^{8}+\frac{700}{27}a^{6}+\frac{124}{3}a^{4}+27a^{2}+3$, $\frac{1}{19683}a^{18}+\frac{2}{729}a^{16}+\frac{5}{81}a^{14}+\frac{547}{729}a^{12}+\frac{433}{81}a^{10}+\frac{1837}{81}a^{8}+\frac{502}{9}a^{6}+\frac{665}{9}a^{4}+\frac{134}{3}a^{2}+8$, $\frac{1}{81}a^{8}+\frac{8}{27}a^{6}+\frac{20}{9}a^{4}+\frac{16}{3}a^{2}+2$, $\frac{1}{729}a^{12}+\frac{4}{81}a^{10}+\frac{2}{3}a^{8}+\frac{112}{27}a^{6}+\frac{35}{3}a^{4}+12a^{2}+2$, $\frac{1}{9}a^{4}+\frac{4}{3}a^{2}+2$, $\frac{1}{27}a^{6}+\frac{2}{3}a^{4}+3a^{2}+2$
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| Regulator: | \( 161406.837641 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 161406.837641 \cdot 8644}{2\cdot\sqrt{180203247070312500000000000000000000}}\cr\approx \mathstrut & 0.157588335615 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30 +6 \sqrt{5}})\), 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 | R | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 | $$[\ ]_{2}^{10}$$ |
|
\(5\)
| 5.1.20.35a1.500 | $x^{20} + 5 x^{16} + 120$ | $20$ | $1$ | $35$ | 20T1 | not computed |