Normalized defining polynomial
\( x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} + \cdots + 1 \)
Invariants
| Degree: | $22$ |
| |
| Signature: | $[0, 11]$ |
| |
| Discriminant: |
\(-39471584120695485887249589623\)
\(\medspace = -\,23^{21}\)
|
| |
| Root discriminant: | \(19.94\) |
| |
| Galois root discriminant: | $23^{21/22}\approx 19.944865695037844$ | ||
| Ramified primes: |
\(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{22}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{23}(1,·)$, $\chi_{23}(2,·)$, $\chi_{23}(3,·)$, $\chi_{23}(4,·)$, $\chi_{23}(5,·)$, $\chi_{23}(6,·)$, $\chi_{23}(7,·)$, $\chi_{23}(8,·)$, $\chi_{23}(9,·)$, $\chi_{23}(10,·)$, $\chi_{23}(11,·)$, $\chi_{23}(12,·)$, $\chi_{23}(13,·)$, $\chi_{23}(14,·)$, $\chi_{23}(15,·)$, $\chi_{23}(16,·)$, $\chi_{23}(17,·)$, $\chi_{23}(18,·)$, $\chi_{23}(19,·)$, $\chi_{23}(20,·)$, $\chi_{23}(21,·)$, $\chi_{23}(22,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{1024}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ |
| |
| Narrow class group: | $C_{3}$, which has order $3$ |
| |
| Relative class number: | $3$ |
Unit group
| Rank: | $10$ |
| |
| Torsion generator: |
\( a \)
(order $46$)
|
| |
| Fundamental units: |
$a^{4}-a^{3}$, $a^{4}-a$, $a^{11}+a^{9}+a^{7}$, $a^{20}-a^{3}$, $a^{16}+a^{6}$, $a^{11}+a^{9}$, $a^{21}-a^{14}+a^{5}$, $a^{11}-a^{8}+a^{5}$, $a^{21}-a^{20}-a^{18}-a^{16}+a^{15}+a^{13}+a^{11}-a^{10}-a^{8}+a^{7}-a^{6}+a^{5}+a^{3}-a^{2}-1$, $a^{16}+a^{8}+1$
|
| |
| Regulator: | \( 1038656.82438 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 1038656.82438 \cdot 3}{46\cdot\sqrt{39471584120695485887249589623}}\cr\approx \mathstrut & 0.205433741423 \end{aligned}\]
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ |
Intermediate fields
| \(\Q(\sqrt{-23}) \), \(\Q(\zeta_{23})^+\) |
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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | $22$ | $22$ | R | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/47.1.0.1}{1} }^{22}$ | $22$ | ${\href{/padicField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.1.22.21a1.1 | $x^{22} + 23$ | $22$ | $1$ | $21$ | 22T1 | $$[\ ]_{22}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.23.11t1.a.a | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.a | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.b | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.b | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.c | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.c | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.d | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.d | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.e | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.e | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.f | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.f | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.g | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.g | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.h | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.h | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.i | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.i | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
| * | 1.23.11t1.a.j | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.22t1.a.j | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.