# Properties

 Label 22.0.394...623.1 Degree $22$ Signature $[0, 11]$ Discriminant $-3.947\times 10^{28}$ Root discriminant $$19.94$$ Ramified prime see page Class number $3$ Class group $[3]$ Galois group $C_{22}$ (as 22T1)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining 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: $$-39471584120695485887249589623$$ -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: $$23$$ 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: $$a$$ a  (order $46$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $a^{4}-a^{3}$, $a^{4}-a$, $a^{11}+a^{9}+a^{7}$, $a^{20}-a^{3}$, $a^{16}+a^{6}$, $a^{11}+a^{9}$, $a^{21}-a^{14}+a^{5}$, $a^{11}-a^{8}+a^{5}$, $a^{21}-a^{20}-a^{18}-a^{16}+a^{15}+a^{13}+a^{11}-a^{10}-a^{8}+a^{7}-a^{6}+a^{5}+a^{3}-a^{2}-1$, $a^{16}+a^{8}+1$ 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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 1038656.82438 \cdot 3}{46\sqrt{39471584120695485887249589623}}\approx 0.205433741423$

## Galois group

$C_{22}$ (as 22T1):

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

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

$p$LabelPolynomial $e$ $f$ $c$ 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 *.