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Properties

Label 22.0.32841739770...3603.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{15}\cdot 4784137656827^{2}$
Root discriminant $30.06$
Ramified primes $3, 4784137656827$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group 22T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 153, -102, 2301, -987, 4957, -2284, 6945, -3124, 6120, -2650, 3922, -1556, 1719, -588, 537, -154, 110, -22, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 14*x^20 - 22*x^19 + 110*x^18 - 154*x^17 + 537*x^16 - 588*x^15 + 1719*x^14 - 1556*x^13 + 3922*x^12 - 2650*x^11 + 6120*x^10 - 3124*x^9 + 6945*x^8 - 2284*x^7 + 4957*x^6 - 987*x^5 + 2301*x^4 - 102*x^3 + 153*x^2 + 9)
 
gp: K = bnfinit(x^22 - 2*x^21 + 14*x^20 - 22*x^19 + 110*x^18 - 154*x^17 + 537*x^16 - 588*x^15 + 1719*x^14 - 1556*x^13 + 3922*x^12 - 2650*x^11 + 6120*x^10 - 3124*x^9 + 6945*x^8 - 2284*x^7 + 4957*x^6 - 987*x^5 + 2301*x^4 - 102*x^3 + 153*x^2 + 9, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 14 x^{20} - 22 x^{19} + 110 x^{18} - 154 x^{17} + 537 x^{16} - 588 x^{15} + 1719 x^{14} - 1556 x^{13} + 3922 x^{12} - 2650 x^{11} + 6120 x^{10} - 3124 x^{9} + 6945 x^{8} - 2284 x^{7} + 4957 x^{6} - 987 x^{5} + 2301 x^{4} - 102 x^{3} + 153 x^{2} + 9 \)

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:  \(-328417397709776899721953240383603=-\,3^{15}\cdot 4784137656827^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.06$
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:  $3, 4784137656827$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{19} + \frac{2}{9} a^{18} + \frac{2}{9} a^{17} + \frac{2}{9} a^{16} - \frac{4}{9} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{2}{9} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{2262805916248746357646306833} a^{21} + \frac{112831343042116330900187656}{2262805916248746357646306833} a^{20} - \frac{50522138475436288506298645}{2262805916248746357646306833} a^{19} + \frac{811737655407381466369299578}{2262805916248746357646306833} a^{18} - \frac{2116806844233454278999013}{10724198655207328709224203} a^{17} + \frac{691415038433011562254590668}{2262805916248746357646306833} a^{16} + \frac{359447253722398671740184806}{754268638749582119215435611} a^{15} + \frac{144597358044405217978600685}{754268638749582119215435611} a^{14} - \frac{285441552355952260294024739}{754268638749582119215435611} a^{13} + \frac{303698194474752399929301748}{2262805916248746357646306833} a^{12} - \frac{780239460358690101645601898}{2262805916248746357646306833} a^{11} + \frac{1091324811403953580430175443}{2262805916248746357646306833} a^{10} + \frac{5270264760202844144023024}{251422879583194039738478537} a^{9} + \frac{277204471372464660616247930}{2262805916248746357646306833} a^{8} - \frac{93544956102475548782948246}{251422879583194039738478537} a^{7} - \frac{821359114051573934633981575}{2262805916248746357646306833} a^{6} - \frac{80733913558430485926404438}{2262805916248746357646306833} a^{5} - \frac{53695409728364164375645626}{251422879583194039738478537} a^{4} - \frac{240697586499374047781308715}{754268638749582119215435611} a^{3} - \frac{230479422346177341685831523}{754268638749582119215435611} a^{2} - \frac{55093964745850845885593892}{251422879583194039738478537} a + \frac{124860235505436232270506331}{251422879583194039738478537}$

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

Class group and class number

$C_{16}$, which has order $16$ (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:  \( -\frac{651010095533006858750918}{2262805916248746357646306833} a^{21} + \frac{19824223622364690837738649}{2262805916248746357646306833} a^{20} - \frac{46310417890235158211897068}{2262805916248746357646306833} a^{19} + \frac{271850361092604752673106082}{2262805916248746357646306833} a^{18} - \frac{2258919037112229634417855}{10724198655207328709224203} a^{17} + \frac{2112388589180142391912411034}{2262805916248746357646306833} a^{16} - \frac{1056730703859112359840058394}{754268638749582119215435611} a^{15} + \frac{1125910646697722445067691226}{251422879583194039738478537} a^{14} - \frac{3919408226557600713404878958}{754268638749582119215435611} a^{13} + \frac{31904387531136396622486345729}{2262805916248746357646306833} a^{12} - \frac{30416154863593231980145962560}{2262805916248746357646306833} a^{11} + \frac{71470174693667180172077255714}{2262805916248746357646306833} a^{10} - \frac{5615881645072470930191454145}{251422879583194039738478537} a^{9} + \frac{109048155087548187051240838079}{2262805916248746357646306833} a^{8} - \frac{19437095215588670734660889912}{754268638749582119215435611} a^{7} + \frac{121256822101961890685434088582}{2262805916248746357646306833} a^{6} - \frac{40978984356780850913551484420}{2262805916248746357646306833} a^{5} + \frac{9254882920684126105202169547}{251422879583194039738478537} a^{4} - \frac{1868788228968458340243262235}{251422879583194039738478537} a^{3} + \frac{12956663028979796458549019929}{754268638749582119215435611} a^{2} - \frac{98480677150375312832796385}{251422879583194039738478537} a + \frac{287081910918833708882533347}{251422879583194039738478537} \) (order $6$)
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:  \( 2614417.99814 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T47:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.11.129171716734329.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 44 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 ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
4784137656827Data not computed