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Properties

Label 22.0.32735795660...4256.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 7^{11}\cdot 23^{21}$
Root discriminant $105.54$
Ramified primes $2, 7, 23$
Class number Not computed
Class group Not computed
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![45478515089, 0, 142932475994, 0, 132723013423, 0, 56881291467, 0, 13543164635, 0, 1969914856, 0, 184003036, 0, 11265492, 0, 449673, 0, 11270, 0, 161, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 161*x^20 + 11270*x^18 + 449673*x^16 + 11265492*x^14 + 184003036*x^12 + 1969914856*x^10 + 13543164635*x^8 + 56881291467*x^6 + 132723013423*x^4 + 142932475994*x^2 + 45478515089)
 
gp: K = bnfinit(x^22 + 161*x^20 + 11270*x^18 + 449673*x^16 + 11265492*x^14 + 184003036*x^12 + 1969914856*x^10 + 13543164635*x^8 + 56881291467*x^6 + 132723013423*x^4 + 142932475994*x^2 + 45478515089, 1)
 

Normalized defining polynomial

\( x^{22} + 161 x^{20} + 11270 x^{18} + 449673 x^{16} + 11265492 x^{14} + 184003036 x^{12} + 1969914856 x^{10} + 13543164635 x^{8} + 56881291467 x^{6} + 132723013423 x^{4} + 142932475994 x^{2} + 45478515089 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-327357956601100418172270186446803699295584256=-\,2^{22}\cdot 7^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.54$
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:  $2, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(644=2^{2}\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{644}(1,·)$, $\chi_{644}(643,·)$, $\chi_{644}(197,·)$, $\chi_{644}(449,·)$, $\chi_{644}(393,·)$, $\chi_{644}(111,·)$, $\chi_{644}(141,·)$, $\chi_{644}(195,·)$, $\chi_{644}(533,·)$, $\chi_{644}(475,·)$, $\chi_{644}(29,·)$, $\chi_{644}(225,·)$, $\chi_{644}(419,·)$, $\chi_{644}(615,·)$, $\chi_{644}(169,·)$, $\chi_{644}(559,·)$, $\chi_{644}(561,·)$, $\chi_{644}(83,·)$, $\chi_{644}(503,·)$, $\chi_{644}(251,·)$, $\chi_{644}(85,·)$, $\chi_{644}(447,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{16807} a^{11}$, $\frac{1}{117649} a^{12}$, $\frac{1}{117649} a^{13}$, $\frac{1}{823543} a^{14}$, $\frac{1}{823543} a^{15}$, $\frac{1}{5764801} a^{16}$, $\frac{1}{5764801} a^{17}$, $\frac{1}{40353607} a^{18}$, $\frac{1}{40353607} a^{19}$, $\frac{1}{282475249} a^{20}$, $\frac{1}{282475249} a^{21}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-161}) \), \(\Q(\zeta_{23})^+\)

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 ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
7Data not computed
23Data not computed