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Properties

Label 22.6.40980582781...8729.3
Degree $22$
Signature $[6, 8]$
Discriminant $23^{21}\cdot 47^{3}$
Root discriminant $33.72$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7499, -19173, 54660, -81220, 57828, -17265, -25274, 40783, -21998, 1474, 6435, -5082, 1642, 58, 108, -224, -15, 95, -88, 36, 2, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 2*x^20 + 36*x^19 - 88*x^18 + 95*x^17 - 15*x^16 - 224*x^15 + 108*x^14 + 58*x^13 + 1642*x^12 - 5082*x^11 + 6435*x^10 + 1474*x^9 - 21998*x^8 + 40783*x^7 - 25274*x^6 - 17265*x^5 + 57828*x^4 - 81220*x^3 + 54660*x^2 - 19173*x + 7499)
 
gp: K = bnfinit(x^22 - 5*x^21 + 2*x^20 + 36*x^19 - 88*x^18 + 95*x^17 - 15*x^16 - 224*x^15 + 108*x^14 + 58*x^13 + 1642*x^12 - 5082*x^11 + 6435*x^10 + 1474*x^9 - 21998*x^8 + 40783*x^7 - 25274*x^6 - 17265*x^5 + 57828*x^4 - 81220*x^3 + 54660*x^2 - 19173*x + 7499, 1)
 

Normalized defining polynomial

\( x^{22} - 5 x^{21} + 2 x^{20} + 36 x^{19} - 88 x^{18} + 95 x^{17} - 15 x^{16} - 224 x^{15} + 108 x^{14} + 58 x^{13} + 1642 x^{12} - 5082 x^{11} + 6435 x^{10} + 1474 x^{9} - 21998 x^{8} + 40783 x^{7} - 25274 x^{6} - 17265 x^{5} + 57828 x^{4} - 81220 x^{3} + 54660 x^{2} - 19173 x + 7499 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4098058278162967431271914143428729=23^{21}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.72$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{137} a^{20} - \frac{47}{137} a^{19} + \frac{6}{137} a^{18} + \frac{36}{137} a^{17} + \frac{6}{137} a^{16} + \frac{26}{137} a^{15} - \frac{49}{137} a^{14} - \frac{66}{137} a^{13} - \frac{52}{137} a^{12} + \frac{57}{137} a^{11} + \frac{34}{137} a^{10} - \frac{21}{137} a^{9} - \frac{68}{137} a^{8} - \frac{58}{137} a^{7} + \frac{3}{137} a^{6} - \frac{30}{137} a^{5} - \frac{58}{137} a^{4} + \frac{20}{137} a^{3} - \frac{2}{137} a^{2} + \frac{24}{137} a + \frac{52}{137}$, $\frac{1}{602007280861078150977490609392459798421921596512429} a^{21} - \frac{489257835740611558034983155774744251932749410459}{602007280861078150977490609392459798421921596512429} a^{20} + \frac{148370711526196787800711668790827885157135175058658}{602007280861078150977490609392459798421921596512429} a^{19} - \frac{68404341717511164740800106135139988137135084860946}{602007280861078150977490609392459798421921596512429} a^{18} + \frac{193101633981938345406092173986716636458711337483625}{602007280861078150977490609392459798421921596512429} a^{17} + \frac{124727868228380105745404877533234890604849368282149}{602007280861078150977490609392459798421921596512429} a^{16} + \frac{179371586155840228993875876221977892165242997388251}{602007280861078150977490609392459798421921596512429} a^{15} + \frac{236047285995175975480016472556413453575550514987852}{602007280861078150977490609392459798421921596512429} a^{14} + \frac{189460879922027419847936544454289435482796281306567}{602007280861078150977490609392459798421921596512429} a^{13} + \frac{284493561098021532549902751558862950597703882867929}{602007280861078150977490609392459798421921596512429} a^{12} - \frac{32254313753950253165268402013424214975521049217286}{602007280861078150977490609392459798421921596512429} a^{11} + \frac{149344323800938044037007995434059144842975438592019}{602007280861078150977490609392459798421921596512429} a^{10} - \frac{33627973749550140142215729209742193833508847512818}{602007280861078150977490609392459798421921596512429} a^{9} + \frac{44049219550753665062142266285449073384280253375665}{602007280861078150977490609392459798421921596512429} a^{8} + \frac{137928535125520475424924168344253588975460016511496}{602007280861078150977490609392459798421921596512429} a^{7} - \frac{199427085951746056064501350149943705030622358293326}{602007280861078150977490609392459798421921596512429} a^{6} + \frac{1892572798296037776232219926405803340949049470764}{602007280861078150977490609392459798421921596512429} a^{5} + \frac{91409761680288535093146018911617067501513354876206}{602007280861078150977490609392459798421921596512429} a^{4} + \frac{10549006215688284438567846526306656394944613684329}{602007280861078150977490609392459798421921596512429} a^{3} + \frac{190975913208948743787450637679872134868535928416247}{602007280861078150977490609392459798421921596512429} a^{2} - \frac{140983097762892517560895707717728626445160322071312}{602007280861078150977490609392459798421921596512429} a - \frac{117987770930008061007947631614922997081551928166713}{602007280861078150977490609392459798421921596512429}$

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:  $13$
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:  \( 132400558.372 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

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

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
23Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$