Normalized defining polynomial
\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(0, 10)$ |
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| Discriminant: |
\(328307557444402776721569\)
\(\medspace = 3^{10}\cdot 11^{18}\)
|
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| Root discriminant: | \(14.99\) |
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| Galois root discriminant: | $3^{1/2}11^{9/10}\approx 14.990428386414978$ | ||
| Ramified primes: |
\(3\), \(11\)
|
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_{10}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(33=3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{33}(1,·)$, $\chi_{33}(2,·)$, $\chi_{33}(4,·)$, $\chi_{33}(5,·)$, $\chi_{33}(7,·)$, $\chi_{33}(8,·)$, $\chi_{33}(10,·)$, $\chi_{33}(13,·)$, $\chi_{33}(14,·)$, $\chi_{33}(16,·)$, $\chi_{33}(17,·)$, $\chi_{33}(19,·)$, $\chi_{33}(20,·)$, $\chi_{33}(23,·)$, $\chi_{33}(25,·)$, $\chi_{33}(26,·)$, $\chi_{33}(28,·)$, $\chi_{33}(29,·)$, $\chi_{33}(31,·)$, $\chi_{33}(32,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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| Relative class number: | $1$ |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -a \)
(order $66$)
|
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| Fundamental units: |
$a^{17}+a^{8}$, $a^{19}-a^{18}+a^{8}$, $a^{19}-a^{9}+a^{8}-a^{6}-a^{3}-1$, $a^{19}+a^{13}+a^{8}-a^{3}+a^{2}$, $a^{14}-a^{13}$, $a^{12}+a^{6}+a$, $a^{19}-a^{17}-a^{15}+a^{13}+a^{10}-a^{9}-a^{6}+a^{2}-1$, $a^{16}-a^{9}$, $a^{18}-a^{16}+a^{12}-a^{8}+a^{7}+a^{6}-a^{5}-a^{2}+a+1$
|
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| Regulator: | \( 62791.3897584 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 62791.3897584 \cdot 1}{66\cdot\sqrt{328307557444402776721569}}\cr\approx \mathstrut & 0.159226153361 \end{aligned}\]
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 10.0.52089208083.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{33})^+\) |
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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.5.2.5a1.2 | $x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
| 3.5.2.5a1.2 | $x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ | |
|
\(11\)
| 11.2.10.18a1.2 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35840 x + 1035$ | $10$ | $2$ | $18$ | 20T3 | $$[\ ]_{10}^{2}$$ |