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Properties

Label 22.0.70765994783...4496.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{22}\cdot 167^{10}$
Root discriminant $20.48$
Ramified primes $2, 167$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{22}$ (as 22T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 12, 0, -34, 0, 81, 0, -149, 0, 207, 0, -214, 0, 166, 0, -94, 0, 37, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^20 + 37*x^18 - 94*x^16 + 166*x^14 - 214*x^12 + 207*x^10 - 149*x^8 + 81*x^6 - 34*x^4 + 12*x^2 + 1)
 
gp: K = bnfinit(x^22 - 9*x^20 + 37*x^18 - 94*x^16 + 166*x^14 - 214*x^12 + 207*x^10 - 149*x^8 + 81*x^6 - 34*x^4 + 12*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{20} + 37 x^{18} - 94 x^{16} + 166 x^{14} - 214 x^{12} + 207 x^{10} - 149 x^{8} + 81 x^{6} - 34 x^{4} + 12 x^{2} + 1 \)

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:  \(-70765994783241803539463274496=-\,2^{22}\cdot 167^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.48$
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:  $2, 167$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{16} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{19} - \frac{1}{5} a^{17} + \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{3145} a^{20} + \frac{4}{3145} a^{18} - \frac{108}{629} a^{16} - \frac{39}{629} a^{14} + \frac{776}{3145} a^{12} + \frac{439}{3145} a^{10} - \frac{201}{629} a^{8} - \frac{1}{629} a^{6} + \frac{16}{3145} a^{4} + \frac{174}{3145} a^{2} - \frac{300}{629}$, $\frac{1}{3145} a^{21} + \frac{4}{3145} a^{19} - \frac{108}{629} a^{17} - \frac{39}{629} a^{15} + \frac{776}{3145} a^{13} + \frac{439}{3145} a^{11} - \frac{201}{629} a^{9} - \frac{1}{629} a^{7} + \frac{16}{3145} a^{5} + \frac{174}{3145} a^{3} - \frac{300}{629} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( -\frac{27}{185} a^{21} + \frac{262}{185} a^{19} - \frac{229}{37} a^{17} + \frac{609}{37} a^{15} - \frac{5597}{185} a^{13} + \frac{7572}{185} a^{11} - \frac{1566}{37} a^{9} + \frac{1248}{37} a^{7} - \frac{3762}{185} a^{5} + \frac{1777}{185} a^{3} - \frac{114}{37} a \) (order $4$)
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:  \( 743434.07758 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{22}$ (as 22T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 44
The 14 conjugacy class representatives for $D_{22}$
Character table for $D_{22}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 11.1.129891985607.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 22 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $22$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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
$167$167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$