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Properties

Label 22.0.49785181124...2411.3
Degree $22$
Signature $[0, 11]$
Discriminant $-\,11^{41}$
Root discriminant $87.25$
Ramified prime $11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{11}\times D_{11}$ (as 22T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2001481, 459635, 4041697, 545116, 2668061, 2092728, 548185, 873180, 1065273, -259182, 372922, 287634, -71324, -24948, 18073, -7216, 2673, 220, -66, -44, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 44*x^19 - 66*x^18 + 220*x^17 + 2673*x^16 - 7216*x^15 + 18073*x^14 - 24948*x^13 - 71324*x^12 + 287634*x^11 + 372922*x^10 - 259182*x^9 + 1065273*x^8 + 873180*x^7 + 548185*x^6 + 2092728*x^5 + 2668061*x^4 + 545116*x^3 + 4041697*x^2 + 459635*x + 2001481)
 
gp: K = bnfinit(x^22 - 44*x^19 - 66*x^18 + 220*x^17 + 2673*x^16 - 7216*x^15 + 18073*x^14 - 24948*x^13 - 71324*x^12 + 287634*x^11 + 372922*x^10 - 259182*x^9 + 1065273*x^8 + 873180*x^7 + 548185*x^6 + 2092728*x^5 + 2668061*x^4 + 545116*x^3 + 4041697*x^2 + 459635*x + 2001481, 1)
 

Normalized defining polynomial

\( x^{22} - 44 x^{19} - 66 x^{18} + 220 x^{17} + 2673 x^{16} - 7216 x^{15} + 18073 x^{14} - 24948 x^{13} - 71324 x^{12} + 287634 x^{11} + 372922 x^{10} - 259182 x^{9} + 1065273 x^{8} + 873180 x^{7} + 548185 x^{6} + 2092728 x^{5} + 2668061 x^{4} + 545116 x^{3} + 4041697 x^{2} + 459635 x + 2001481 \)

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:  \(-4978518112499354698647829163838661251242411=-\,11^{41}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.25$
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:  $11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{21} - \frac{170683639013744981986790160375544641291178843203496973846005784676323607485067948914}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{20} + \frac{520039526782545479071745061433929053006247301445571303837563242709299800951930700900}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{19} + \frac{618324097275195883749287061560914084569159828087316574907462989607304748658628631413}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{18} + \frac{276961749075946533472095556185700307970969109290313245432653627764203376583323533138}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{17} + \frac{482135472414668305508044589667966895605010197225416506915742876979732809865332650954}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{16} - \frac{2071845882318143924495583838836220998394302132610965341123217060142600785989611248}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{15} + \frac{844745432783037036751654622501651179983936823829516284731653194708548127386971212278}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{14} + \frac{550638583957512060440448940517536467407081116867590356075692477476530717833477957602}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{13} - \frac{852525565374643932736926095995395577970436192966773148658823764170778532265219966753}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{12} - \frac{801184453704967951694758348056993825512124494718861233061051502626499381597664226542}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{11} - \frac{242348814681167336380933045573640599814128376214839707259579081237988672567911417343}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{10} - \frac{158612273371037871327031345227005168680426275451796372282084957842705281649038021092}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{9} - \frac{150718854548405721556321328258423616169552601424538488581103677635320534796712817673}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{8} - \frac{175255246093797769593614159442509926581109665098977787481476248253738009026167930083}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{7} + \frac{53560795191141275084243893008284845781396775628999156591146383789335194269582588096}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{6} - \frac{477135652387315488803567467702322358955668364105992109796718822255538697357022936142}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{5} - \frac{836361333699466051211711116618483759789573285272418027795165164069863195301512913806}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{4} - \frac{68787074692825940386089237137397437565809803878750637077096412571234081943050309901}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{3} - \frac{37441670885217480354682158123266623316076503623778675684701941866142689628008968914}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a^{2} + \frac{66640909635399720347350419718852899731478400507329160709366767041271946124557871264}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767} a - \frac{644573752478133155335121823606868066240801528581014748829512202627005912424751430631}{1727272103458797706759726583775285427791324092182186772067668938400582017899136971767}$

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:  \( -1 \) (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:  \( 1205305578130 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{11}\times D_{11}$ (as 22T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 242
The 77 conjugacy class representatives for $C_{11}\times D_{11}$ are not computed
Character table for $C_{11}\times D_{11}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{11}$ $22$ $22$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
11Data not computed