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Properties

Label 22.2.82382460570...8125.1
Degree $22$
Signature $[2, 10]$
Discriminant $5^{11}\cdot 167^{10}$
Root discriminant $22.90$
Ramified primes $5, 167$
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, 2, 16, 44, 87, 141, 259, 309, 628, 311, 930, -47, 966, -312, 710, -162, 290, -46, 83, -11, 14, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 14*x^20 - 11*x^19 + 83*x^18 - 46*x^17 + 290*x^16 - 162*x^15 + 710*x^14 - 312*x^13 + 966*x^12 - 47*x^11 + 930*x^10 + 311*x^9 + 628*x^8 + 309*x^7 + 259*x^6 + 141*x^5 + 87*x^4 + 44*x^3 + 16*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^22 - x^21 + 14*x^20 - 11*x^19 + 83*x^18 - 46*x^17 + 290*x^16 - 162*x^15 + 710*x^14 - 312*x^13 + 966*x^12 - 47*x^11 + 930*x^10 + 311*x^9 + 628*x^8 + 309*x^7 + 259*x^6 + 141*x^5 + 87*x^4 + 44*x^3 + 16*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 14 x^{20} - 11 x^{19} + 83 x^{18} - 46 x^{17} + 290 x^{16} - 162 x^{15} + 710 x^{14} - 312 x^{13} + 966 x^{12} - 47 x^{11} + 930 x^{10} + 311 x^{9} + 628 x^{8} + 309 x^{7} + 259 x^{6} + 141 x^{5} + 87 x^{4} + 44 x^{3} + 16 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(823824605709428474533642578125=5^{11}\cdot 167^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.90$
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:  $5, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{221} a^{20} + \frac{1}{221} a^{19} + \frac{20}{221} a^{18} + \frac{33}{221} a^{17} + \frac{8}{221} a^{16} + \frac{6}{13} a^{15} + \frac{84}{221} a^{14} - \frac{28}{221} a^{13} + \frac{106}{221} a^{12} + \frac{9}{221} a^{11} + \frac{82}{221} a^{10} - \frac{4}{13} a^{9} + \frac{1}{13} a^{8} + \frac{73}{221} a^{7} - \frac{42}{221} a^{6} + \frac{75}{221} a^{5} + \frac{20}{221} a^{4} + \frac{3}{17} a^{3} + \frac{24}{221} a^{2} + \frac{27}{221} a - \frac{55}{221}$, $\frac{1}{363751046168327874232423} a^{21} - \frac{553875068378813795126}{363751046168327874232423} a^{20} + \frac{3708084132289217260126}{27980849705255990325571} a^{19} - \frac{149390517050364495621277}{363751046168327874232423} a^{18} - \frac{159215562426782283490239}{363751046168327874232423} a^{17} - \frac{2752306539813749488017}{363751046168327874232423} a^{16} + \frac{45503668889275985437277}{363751046168327874232423} a^{15} + \frac{150649040811131124820345}{363751046168327874232423} a^{14} + \frac{87896429328997191258242}{363751046168327874232423} a^{13} - \frac{80832724581025805910683}{363751046168327874232423} a^{12} - \frac{86730541138695854273231}{363751046168327874232423} a^{11} + \frac{102853577677487334098149}{363751046168327874232423} a^{10} + \frac{5445835402258963003854}{21397120362842816131319} a^{9} - \frac{94097871916555726004195}{363751046168327874232423} a^{8} - \frac{96713141085311424540575}{363751046168327874232423} a^{7} + \frac{6808802376769499193475}{21397120362842816131319} a^{6} + \frac{10142578341374300083368}{363751046168327874232423} a^{5} + \frac{172402632506084771901945}{363751046168327874232423} a^{4} - \frac{79491457502162615717992}{363751046168327874232423} a^{3} - \frac{85376667923377247641066}{363751046168327874232423} a^{2} - \frac{5581659833818637196941}{15815262876883820618801} a - \frac{55303068791643883276604}{363751046168327874232423}$

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:  $11$
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:  \( 1079623.65534 \) (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

\(\Q(\sqrt{5}) \), 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 $22$ $22$ R $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$