Normalized defining polynomial
\( x^{22} - 42 x^{20} + 760 x^{18} - 7752 x^{16} + 48960 x^{14} - 198016 x^{12} + 512512 x^{10} - 823680 x^{8} + 768768 x^{6} - 366080 x^{4} + 67584 x^{2} - 2048 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[22, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(14741666340843480753092741810452692992=2^{33}\cdot 23^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.92$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(184=2^{3}\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{184}(1,·)$, $\chi_{184}(133,·)$, $\chi_{184}(9,·)$, $\chi_{184}(13,·)$, $\chi_{184}(77,·)$, $\chi_{184}(141,·)$, $\chi_{184}(81,·)$, $\chi_{184}(85,·)$, $\chi_{184}(105,·)$, $\chi_{184}(25,·)$, $\chi_{184}(29,·)$, $\chi_{184}(101,·)$, $\chi_{184}(165,·)$, $\chi_{184}(177,·)$, $\chi_{184}(41,·)$, $\chi_{184}(173,·)$, $\chi_{184}(93,·)$, $\chi_{184}(49,·)$, $\chi_{184}(117,·)$, $\chi_{184}(73,·)$, $\chi_{184}(121,·)$, $\chi_{184}(169,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $21$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 354187150112 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{23})^+\) |
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 | R | $22$ | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ | $22$ | $22$ |
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 | Data not computed | ||||||
| $23$ | 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} - 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |