Normalized defining polynomial
\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(0, 10)$ |
| |
| Discriminant: |
\(5829995856912430117421056\)
\(\medspace = 2^{20}\cdot 11^{18}\)
|
| |
| Root discriminant: | \(17.31\) |
| |
| Galois root discriminant: | $2\cdot 11^{9/10}\approx 17.309455728328988$ | ||
| Ramified primes: |
\(2\), \(11\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_{10}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(44=2^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{44}(1,·)$, $\chi_{44}(3,·)$, $\chi_{44}(5,·)$, $\chi_{44}(7,·)$, $\chi_{44}(9,·)$, $\chi_{44}(13,·)$, $\chi_{44}(15,·)$, $\chi_{44}(17,·)$, $\chi_{44}(19,·)$, $\chi_{44}(21,·)$, $\chi_{44}(23,·)$, $\chi_{44}(25,·)$, $\chi_{44}(27,·)$, $\chi_{44}(29,·)$, $\chi_{44}(31,·)$, $\chi_{44}(35,·)$, $\chi_{44}(37,·)$, $\chi_{44}(39,·)$, $\chi_{44}(41,·)$, $\chi_{44}(43,·)$$\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 $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
| |
| Relative class number: | $1$ |
Unit group
| Rank: | $9$ |
| |
| Torsion generator: |
\( a \)
(order $44$)
|
| |
| Fundamental units: |
$a^{18}+a^{10}$, $a^{16}-a^{10}$, $a^{18}+a^{14}$, $a^{14}-a^{12}+a^{10}$, $a^{16}-a$, $a^{18}-a^{17}$, $a^{10}-a^{7}$, $a^{10}+a$, $a^{14}-a^{9}$
|
| |
| Regulator: | \( 140601.245383 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140601.245383 \cdot 1}{44\cdot\sqrt{5829995856912430117421056}}\cr\approx \mathstrut & 0.126911521724 \end{aligned}\]
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
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{11}) \), \(\Q(i, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 10.0.219503494144.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{44})^+\) |
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 | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.2.20a1.1 | $x^{20} + 2 x^{16} + 2 x^{15} + 2 x^{13} + 3 x^{12} + 4 x^{11} + 5 x^{10} + 2 x^{9} + 4 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 3 x^{4} + 6 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $10$ | $20$ | 20T3 | $$[2]^{10}$$ |
|
\(11\)
| 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 | $$[\ ]_{10}^{2}$$ |