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Properties

Label 22.2.49913150649...5501.1
Degree $22$
Signature $[2, 10]$
Discriminant $3^{11}\cdot 167^{11}$
Root discriminant $22.38$
Ramified primes $3, 167$
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![-5, 39, -4, -826, -351, 168, 1528, 1215, 78, -3315, 1821, -793, 792, -686, 497, -228, 63, -14, 4, -4, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5)
 
gp: K = bnfinit(x^22 - 2*x^21 + 5*x^20 - 4*x^19 + 4*x^18 - 14*x^17 + 63*x^16 - 228*x^15 + 497*x^14 - 686*x^13 + 792*x^12 - 793*x^11 + 1821*x^10 - 3315*x^9 + 78*x^8 + 1215*x^7 + 1528*x^6 + 168*x^5 - 351*x^4 - 826*x^3 - 4*x^2 + 39*x - 5, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 5 x^{20} - 4 x^{19} + 4 x^{18} - 14 x^{17} + 63 x^{16} - 228 x^{15} + 497 x^{14} - 686 x^{13} + 792 x^{12} - 793 x^{11} + 1821 x^{10} - 3315 x^{9} + 78 x^{8} + 1215 x^{7} + 1528 x^{6} + 168 x^{5} - 351 x^{4} - 826 x^{3} - 4 x^{2} + 39 x - 5 \)

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:  \(499131506491608208145638755501=3^{11}\cdot 167^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.38$
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:  $3, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{13} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{25} a^{19} - \frac{2}{25} a^{16} + \frac{1}{25} a^{14} + \frac{11}{25} a^{13} + \frac{4}{25} a^{12} - \frac{8}{25} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{9}{25} a^{8} - \frac{12}{25} a^{7} - \frac{4}{25} a^{6} - \frac{3}{25} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{8}{25} a^{2} + \frac{8}{25} a - \frac{2}{5}$, $\frac{1}{25} a^{20} - \frac{2}{25} a^{17} + \frac{1}{25} a^{15} + \frac{1}{25} a^{14} + \frac{9}{25} a^{13} - \frac{3}{25} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{6}{25} a^{9} + \frac{8}{25} a^{8} + \frac{1}{25} a^{7} + \frac{2}{25} a^{6} + \frac{12}{25} a^{3} - \frac{12}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{74790684859346257243233256544712175} a^{21} + \frac{857988106759387599020154125064398}{74790684859346257243233256544712175} a^{20} - \frac{1351596222449286087852452617894714}{74790684859346257243233256544712175} a^{19} - \frac{6425853310328055129727887215841742}{74790684859346257243233256544712175} a^{18} + \frac{6762379571289284332427276884370789}{74790684859346257243233256544712175} a^{17} + \frac{4285205750638124361824024113672254}{74790684859346257243233256544712175} a^{16} - \frac{343724335705705181224297619438201}{74790684859346257243233256544712175} a^{15} + \frac{6384205603522217744360596241741933}{74790684859346257243233256544712175} a^{14} - \frac{1884280585364836706156005369070234}{14958136971869251448646651308942435} a^{13} - \frac{5541952319797508302485050839887762}{14958136971869251448646651308942435} a^{12} + \frac{24156897118986399242142402272294607}{74790684859346257243233256544712175} a^{11} - \frac{24617354741228346827965687965922574}{74790684859346257243233256544712175} a^{10} - \frac{17389026836853248890735358775663474}{74790684859346257243233256544712175} a^{9} + \frac{31394568081591371179455896303210266}{74790684859346257243233256544712175} a^{8} - \frac{5013405849484694349500907006304707}{74790684859346257243233256544712175} a^{7} + \frac{28237137326518672349767106153662047}{74790684859346257243233256544712175} a^{6} - \frac{3133137785359092594959939229613048}{74790684859346257243233256544712175} a^{5} - \frac{12675044360170921986291637043939873}{74790684859346257243233256544712175} a^{4} - \frac{30658995272537926911993162516138161}{74790684859346257243233256544712175} a^{3} + \frac{31458125394494544024561830175617056}{74790684859346257243233256544712175} a^{2} + \frac{18174525640623119556619899137665273}{74790684859346257243233256544712175} a + \frac{6970655187218301619405778577489518}{14958136971869251448646651308942435}$

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:  \( 1424457.4515 \) (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

\(\Q(\sqrt{501}) \), 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 $22$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
167Data not computed