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: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-39471584120695485887249589623\) \(\medspace = -\,23^{21}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.94\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $23^{21/22}\approx 19.944865695037844$ | ||
Ramified primes: | \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Gal(K/\Q) }$: | $22$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
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
$C_{3}$, which has order $3$
Unit group
Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( a \) (order $46$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a^{4}-a^{3}$, $a^{4}-a$, $a^{11}+a^{9}+a^{7}$, $a^{20}-a^{3}$, $a^{16}+a^{6}$, $a^{11}+a^{9}$, $a^{21}-a^{14}+a^{5}$, $a^{11}-a^{8}+a^{5}$, $a^{21}-a^{20}-a^{18}-a^{16}+a^{15}+a^{13}+a^{11}-a^{10}-a^{8}+a^{7}-a^{6}+a^{5}+a^{3}-a^{2}-1$, $a^{16}+a^{8}+1$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 1038656.82438 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
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 $C_{22}$ is not computed |
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 |
---|---|---|---|---|---|---|---|
\(23\) | 23.22.21.17 | $x^{22} + 23$ | $22$ | $1$ | $21$ | 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) | $0$ | $-1$ |