Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp: K = bnfinit(x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]);
\( x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: |
\(-39471584120695485887249589623\)
\(\medspace = -\,23^{21}\)
| sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | \(19.94\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: |
\(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $\card{ \Gal(K/\Q) }$: | $22$ | ||
| 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. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: |
\( a \)
(order $46$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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!f(g): g in Generators(UK)];
| |
| Regulator: | \( 1038656.82438 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| 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.
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.22.21.17 | $x^{22} + 47104$ | $22$ | $1$ | $21$ | 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 *.