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Properties

Label 22.22.3111193701...8032.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{24}\cdot 3^{11}\cdot 11^{27}\cdot 41^{8}$
Root discriminant $270.08$
Ramified primes $2, 3, 11, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T14

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1841628392, -10348950480, -12622704501, 16385057678, 39169758474, 8705909278, -24242610414, -13937355960, 4465613977, 4865875674, 142254739, -695875072, -112009051, 47695890, 11663817, -1673936, -561253, 29062, 14278, -198, -187, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 187*x^20 - 198*x^19 + 14278*x^18 + 29062*x^17 - 561253*x^16 - 1673936*x^15 + 11663817*x^14 + 47695890*x^13 - 112009051*x^12 - 695875072*x^11 + 142254739*x^10 + 4865875674*x^9 + 4465613977*x^8 - 13937355960*x^7 - 24242610414*x^6 + 8705909278*x^5 + 39169758474*x^4 + 16385057678*x^3 - 12622704501*x^2 - 10348950480*x - 1841628392)
 
gp: K = bnfinit(x^22 - 187*x^20 - 198*x^19 + 14278*x^18 + 29062*x^17 - 561253*x^16 - 1673936*x^15 + 11663817*x^14 + 47695890*x^13 - 112009051*x^12 - 695875072*x^11 + 142254739*x^10 + 4865875674*x^9 + 4465613977*x^8 - 13937355960*x^7 - 24242610414*x^6 + 8705909278*x^5 + 39169758474*x^4 + 16385057678*x^3 - 12622704501*x^2 - 10348950480*x - 1841628392, 1)
 

Normalized defining polynomial

\( x^{22} - 187 x^{20} - 198 x^{19} + 14278 x^{18} + 29062 x^{17} - 561253 x^{16} - 1673936 x^{15} + 11663817 x^{14} + 47695890 x^{13} - 112009051 x^{12} - 695875072 x^{11} + 142254739 x^{10} + 4865875674 x^{9} + 4465613977 x^{8} - 13937355960 x^{7} - 24242610414 x^{6} + 8705909278 x^{5} + 39169758474 x^{4} + 16385057678 x^{3} - 12622704501 x^{2} - 10348950480 x - 1841628392 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(311119370117831523477714234566730921080649451969708032=2^{24}\cdot 3^{11}\cdot 11^{27}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $270.08$
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:  $2, 3, 11, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{5}{11}$, $\frac{1}{44} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{5}{22} a$, $\frac{1}{44} a^{13} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{3}{11} a^{2} - \frac{1}{2} a$, $\frac{1}{88} a^{14} - \frac{1}{88} a^{12} - \frac{1}{44} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{5}{44} a^{3} - \frac{1}{8} a^{2} + \frac{3}{22} a + \frac{3}{11}$, $\frac{1}{88} a^{15} - \frac{1}{88} a^{13} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{3}{22} a^{4} + \frac{3}{8} a^{3} + \frac{17}{44} a^{2}$, $\frac{1}{176} a^{16} - \frac{1}{88} a^{13} - \frac{1}{176} a^{12} - \frac{1}{44} a^{11} - \frac{1}{16} a^{10} + \frac{1}{8} a^{9} - \frac{3}{16} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{19}{44} a^{5} - \frac{5}{16} a^{4} - \frac{3}{8} a^{3} - \frac{31}{176} a^{2} - \frac{2}{11} a - \frac{5}{22}$, $\frac{1}{352} a^{17} - \frac{1}{352} a^{16} - \frac{1}{176} a^{14} + \frac{1}{352} a^{13} - \frac{3}{352} a^{12} - \frac{7}{352} a^{11} - \frac{5}{32} a^{10} + \frac{3}{32} a^{9} + \frac{5}{32} a^{8} + \frac{1}{8} a^{7} - \frac{39}{176} a^{6} - \frac{131}{352} a^{5} - \frac{9}{32} a^{4} - \frac{141}{352} a^{3} + \frac{175}{352} a^{2} - \frac{1}{44} a - \frac{17}{44}$, $\frac{1}{704} a^{18} - \frac{1}{704} a^{16} + \frac{1}{352} a^{15} + \frac{3}{704} a^{14} + \frac{1}{352} a^{13} + \frac{1}{352} a^{12} + \frac{5}{352} a^{11} - \frac{5}{32} a^{10} + \frac{3}{16} a^{9} + \frac{5}{64} a^{8} + \frac{71}{352} a^{7} - \frac{11}{64} a^{6} - \frac{27}{352} a^{5} + \frac{15}{44} a^{4} + \frac{147}{352} a^{3} - \frac{189}{704} a^{2} - \frac{7}{44} a - \frac{37}{88}$, $\frac{1}{15488} a^{19} + \frac{3}{15488} a^{18} - \frac{5}{15488} a^{17} + \frac{43}{15488} a^{16} + \frac{81}{15488} a^{15} + \frac{43}{15488} a^{14} - \frac{37}{3872} a^{13} - \frac{13}{3872} a^{12} + \frac{47}{3872} a^{11} + \frac{69}{704} a^{10} - \frac{171}{1408} a^{9} - \frac{793}{15488} a^{8} - \frac{1631}{15488} a^{7} - \frac{633}{15488} a^{6} + \frac{3669}{7744} a^{5} + \frac{3309}{7744} a^{4} + \frac{4049}{15488} a^{3} + \frac{4669}{15488} a^{2} - \frac{175}{1936} a - \frac{315}{1936}$, $\frac{1}{30976} a^{20} - \frac{7}{15488} a^{18} - \frac{15}{15488} a^{17} + \frac{5}{3872} a^{16} + \frac{19}{3872} a^{15} - \frac{101}{30976} a^{14} - \frac{3}{1936} a^{13} - \frac{3}{968} a^{12} - \frac{271}{15488} a^{11} - \frac{145}{2816} a^{10} - \frac{963}{15488} a^{9} - \frac{93}{704} a^{8} + \frac{97}{7744} a^{7} - \frac{7131}{30976} a^{6} + \frac{2905}{7744} a^{5} - \frac{3573}{30976} a^{4} - \frac{7215}{15488} a^{3} + \frac{12137}{30976} a^{2} - \frac{291}{1936} a + \frac{1737}{3872}$, $\frac{1}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{21} + \frac{9495311078511639967325350034155313135186821251063941053470904129573}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{20} - \frac{2419927823639751647843815817719381227660938539405117843864528393253}{183946430498665082646515535680545086233569472718585871692637177076615936} a^{19} + \frac{3944325009966719170691355758078945311461103332946972547919382636365}{8361201386302958302114342530933867556071339669026630531483508048937088} a^{18} + \frac{132300920001143514032966481710941864741995143396742321710702234753417}{551839291495995247939546607041635258700708418155757615077911531229847808} a^{17} + \frac{296886632894311503330786243610225633262218564637864286012818440502275}{137959822873998811984886651760408814675177104538939403769477882807461952} a^{16} + \frac{3212894194231872612060606135806397026439560466440063782629225980352019}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{15} - \frac{3512059581932111497528732479402678199913507336640584155033743381903169}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{14} + \frac{37803012037827136175011418761230222502916208146591900895110678148259}{4057641849235259176026077992953200431622856015851158934396408317866528} a^{13} + \frac{72666052579769077483375482489244163696315666349427313356838499270803}{50167208317817749812686055185603205336428038014159783188901048293622528} a^{12} - \frac{24767772129166846857992191758378658251521903242894657663280525738788817}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{11} - \frac{87380673629593785344414018598014891151428211823852342863850469792351151}{367892860997330165293031071361090172467138945437171743385274354153231872} a^{10} + \frac{113847483939171978304187293702421652375164510550482954071561296774611171}{551839291495995247939546607041635258700708418155757615077911531229847808} a^{9} + \frac{1599672622334635161501974728973707048756611190431837868723315297006993}{68979911436999405992443325880204407337588552269469701884738941403730976} a^{8} - \frac{4861736776526160015795806138051275278481802713717426693183716796798421}{100334416635635499625372110371206410672856076028319566377802096587245056} a^{7} + \frac{103402549336661819844985735534545327612394249926898724365855244680909349}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{6} - \frac{64105857142289496596600750805211788300680373757576788700507444354559329}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{5} + \frac{129801518012051692406331526501869608809088122178653316542639069792606435}{367892860997330165293031071361090172467138945437171743385274354153231872} a^{4} - \frac{157382773902863075066668234498146976288929025674434607666094867976374079}{367892860997330165293031071361090172467138945437171743385274354153231872} a^{3} + \frac{11566728573566752540247974042495362965853509192630622770035585494248117}{1103678582991990495879093214083270517401416836311515230155823062459695616} a^{2} - \frac{3087356296451295341229791585115247641126413882429051576551166794681231}{12541802079454437453171513796400801334107009503539945797225262073405632} a - \frac{16099939572739733435660092292165276065343484752291216257484327288050235}{137959822873998811984886651760408814675177104538939403769477882807461952}$

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:  $21$
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:  \( 800234332751000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T14:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1320
The 13 conjugacy class representatives for t22n14
Character table for t22n14

Intermediate fields

\(\Q(\sqrt{33}) \)

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

Sibling fields

Degree 12 sibling: data not computed
Degree 24 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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
41Data not computed