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Properties

Label 22.0.65701380384...9543.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,7^{11}\cdot 67^{20}$
Root discriminant $120.95$
Ramified primes $7, 67$
Class number $420607$ (GRH)
Class group $[420607]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50167427, 59497518, 73795185, 60447125, 51453681, 33250364, 20867753, 8707748, 3092873, 730124, 178157, -220921, -151828, -4617, 43511, 10860, -3685, -2061, 231, 159, -3, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 - 3*x^20 + 159*x^19 + 231*x^18 - 2061*x^17 - 3685*x^16 + 10860*x^15 + 43511*x^14 - 4617*x^13 - 151828*x^12 - 220921*x^11 + 178157*x^10 + 730124*x^9 + 3092873*x^8 + 8707748*x^7 + 20867753*x^6 + 33250364*x^5 + 51453681*x^4 + 60447125*x^3 + 73795185*x^2 + 59497518*x + 50167427)
 
gp: K = bnfinit(x^22 - 9*x^21 - 3*x^20 + 159*x^19 + 231*x^18 - 2061*x^17 - 3685*x^16 + 10860*x^15 + 43511*x^14 - 4617*x^13 - 151828*x^12 - 220921*x^11 + 178157*x^10 + 730124*x^9 + 3092873*x^8 + 8707748*x^7 + 20867753*x^6 + 33250364*x^5 + 51453681*x^4 + 60447125*x^3 + 73795185*x^2 + 59497518*x + 50167427, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} - 3 x^{20} + 159 x^{19} + 231 x^{18} - 2061 x^{17} - 3685 x^{16} + 10860 x^{15} + 43511 x^{14} - 4617 x^{13} - 151828 x^{12} - 220921 x^{11} + 178157 x^{10} + 730124 x^{9} + 3092873 x^{8} + 8707748 x^{7} + 20867753 x^{6} + 33250364 x^{5} + 51453681 x^{4} + 60447125 x^{3} + 73795185 x^{2} + 59497518 x + 50167427 \)

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:  \(-6570138038458798225574551839115963943843569543=-\,7^{11}\cdot 67^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.95$
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:  $7, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(469=7\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{469}(64,·)$, $\chi_{469}(1,·)$, $\chi_{469}(260,·)$, $\chi_{469}(265,·)$, $\chi_{469}(202,·)$, $\chi_{469}(76,·)$, $\chi_{469}(461,·)$, $\chi_{469}(15,·)$, $\chi_{469}(216,·)$, $\chi_{469}(148,·)$, $\chi_{469}(22,·)$, $\chi_{469}(344,·)$, $\chi_{469}(92,·)$, $\chi_{469}(349,·)$, $\chi_{469}(223,·)$, $\chi_{469}(225,·)$, $\chi_{469}(293,·)$, $\chi_{469}(426,·)$, $\chi_{469}(174,·)$, $\chi_{469}(442,·)$, $\chi_{469}(330,·)$, $\chi_{469}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1073} a^{18} - \frac{312}{1073} a^{17} + \frac{83}{1073} a^{16} + \frac{530}{1073} a^{15} + \frac{2}{37} a^{14} - \frac{221}{1073} a^{13} + \frac{23}{1073} a^{12} + \frac{4}{1073} a^{11} - \frac{6}{37} a^{10} - \frac{473}{1073} a^{9} + \frac{89}{1073} a^{8} - \frac{301}{1073} a^{7} - \frac{5}{37} a^{6} - \frac{372}{1073} a^{5} + \frac{427}{1073} a^{4} + \frac{4}{1073} a^{3} - \frac{161}{1073} a^{2} + \frac{498}{1073} a + \frac{442}{1073}$, $\frac{1}{256447} a^{19} - \frac{97}{256447} a^{18} + \frac{54252}{256447} a^{17} + \frac{37689}{256447} a^{16} - \frac{53380}{256447} a^{15} - \frac{114365}{256447} a^{14} + \frac{46932}{256447} a^{13} + \frac{117614}{256447} a^{12} - \frac{13263}{256447} a^{11} - \frac{38956}{256447} a^{10} + \frac{22862}{256447} a^{9} - \frac{128167}{256447} a^{8} + \frac{31710}{256447} a^{7} + \frac{73607}{256447} a^{6} + \frac{68521}{256447} a^{5} + \frac{38159}{256447} a^{4} - \frac{22907}{256447} a^{3} + \frac{53869}{256447} a^{2} + \frac{57081}{256447} a + \frac{31723}{256447}$, $\frac{1}{256447} a^{20} - \frac{89}{256447} a^{18} + \frac{85392}{256447} a^{17} - \frac{126903}{256447} a^{16} - \frac{127674}{256447} a^{15} - \frac{60838}{256447} a^{14} - \frac{17489}{256447} a^{13} + \frac{103979}{256447} a^{12} + \frac{33487}{256447} a^{11} - \frac{40854}{256447} a^{10} + \frac{5606}{256447} a^{9} + \frac{13171}{256447} a^{8} + \frac{4954}{256447} a^{7} - \frac{124598}{256447} a^{6} + \frac{62723}{256447} a^{5} - \frac{120628}{256447} a^{4} - \frac{39815}{256447} a^{3} - \frac{49477}{256447} a^{2} + \frac{117946}{256447} a - \frac{113758}{256447}$, $\frac{1}{58067213892884303030968469749624884899803816778787790847901151} a^{21} + \frac{98867339522426793542840003867674904402916169809589463827}{58067213892884303030968469749624884899803816778787790847901151} a^{20} + \frac{33697364823932548549219652630555441471568178295086917199}{58067213892884303030968469749624884899803816778787790847901151} a^{19} - \frac{19636104557840160927117334965577676853704074930662740638385}{58067213892884303030968469749624884899803816778787790847901151} a^{18} + \frac{22499558456465447118848265308784471631997346793497744787621685}{58067213892884303030968469749624884899803816778787790847901151} a^{17} + \frac{733999909075753463800925360534173702040999128306549853631470}{2002317720444286311412705853435340858613924716509923822341419} a^{16} + \frac{6775069682813019452075042162539117781769887954544223192168563}{58067213892884303030968469749624884899803816778787790847901151} a^{15} - \frac{28480626094940829314712597925306633891048972383407910199390688}{58067213892884303030968469749624884899803816778787790847901151} a^{14} + \frac{27602868383387396102786355417438753575908648002744993519567153}{58067213892884303030968469749624884899803816778787790847901151} a^{13} - \frac{16860692779640987852259256564849784420197580085603247870517431}{58067213892884303030968469749624884899803816778787790847901151} a^{12} + \frac{16587554791170274879375673216898685120672736096864597243317383}{58067213892884303030968469749624884899803816778787790847901151} a^{11} - \frac{552307872241016035343097749458502714048444472783589754283905}{1569384159267143325161309993233104997291995048075345698591923} a^{10} + \frac{20674211356164478090420278203670239849030221927428312668064143}{58067213892884303030968469749624884899803816778787790847901151} a^{9} - \frac{14043394091444845077180643806708901207575612630165666601887860}{58067213892884303030968469749624884899803816778787790847901151} a^{8} + \frac{814250739373575792333217105053510734528885439104506117838552}{2002317720444286311412705853435340858613924716509923822341419} a^{7} - \frac{15611717893653921697694710202508439192179278389585757227554627}{58067213892884303030968469749624884899803816778787790847901151} a^{6} - \frac{6508983927167517275558120466600026811655162404162572300721396}{58067213892884303030968469749624884899803816778787790847901151} a^{5} + \frac{28877686863204584996493077499466976169099282455705638926869972}{58067213892884303030968469749624884899803816778787790847901151} a^{4} + \frac{726393014852311430954693196293394315739970662387015153990755}{2002317720444286311412705853435340858613924716509923822341419} a^{3} - \frac{13652993040355497137062019425910207591101686933948666624356948}{58067213892884303030968469749624884899803816778787790847901151} a^{2} - \frac{13041750729315378296012539860715623210960729874759895141859519}{58067213892884303030968469749624884899803816778787790847901151} a - \frac{22369625906420479300323858147947258728480670669518947389056253}{58067213892884303030968469749624884899803816778787790847901151}$

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

Class group and class number

$C_{420607}$, which has order $420607$ (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:  \( 338444542.042557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 11.11.1822837804551761449.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

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
7Data not computed
$67$67.11.10.1$x^{11} - 67$$11$$1$$10$$C_{11}$$[\ ]_{11}$
67.11.10.1$x^{11} - 67$$11$$1$$10$$C_{11}$$[\ ]_{11}$