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Properties

Label 22.22.2440279632...3024.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{71}\cdot 3^{21}\cdot 337^{8}\cdot 653\cdot 46723\cdot 310501^{8}\cdot 2253079$
Root discriminant $93{,}789.96$
Ramified primes $2, 3, 337, 653, 46723, 310501, 2253079$
Class number Not computed
Class group Not computed
Galois group 22T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26396816028057984, 0, 108887749553059407, 0, -24604405843720824, 0, 2444260894986117, 0, -139151595045408, 0, 5030770408512, 0, -120874641864, 0, 1957294872, 0, -21111552, 0, 145259, 0, -576, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 576*x^20 + 145259*x^18 - 21111552*x^16 + 1957294872*x^14 - 120874641864*x^12 + 5030770408512*x^10 - 139151595045408*x^8 + 2444260894986117*x^6 - 24604405843720824*x^4 + 108887749553059407*x^2 - 26396816028057984)
 
gp: K = bnfinit(x^22 - 576*x^20 + 145259*x^18 - 21111552*x^16 + 1957294872*x^14 - 120874641864*x^12 + 5030770408512*x^10 - 139151595045408*x^8 + 2444260894986117*x^6 - 24604405843720824*x^4 + 108887749553059407*x^2 - 26396816028057984, 1)
 

Normalized defining polynomial

\( x^{22} - 576 x^{20} + 145259 x^{18} - 21111552 x^{16} + 1957294872 x^{14} - 120874641864 x^{12} + 5030770408512 x^{10} - 139151595045408 x^{8} + 2444260894986117 x^{6} - 24604405843720824 x^{4} + 108887749553059407 x^{2} - 26396816028057984 \)

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:  \(24402796323422549444668858510960358552762329591784234782163089041212069572638422789466222769600101521528193024=2^{71}\cdot 3^{21}\cdot 337^{8}\cdot 653\cdot 46723\cdot 310501^{8}\cdot 2253079\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93{,}789.96$
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:  $2, 3, 337, 653, 46723, 310501, 2253079$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491805}{57887320607168480035725746532474501690526662399435613406516} a^{18} + \frac{2309673043835677186437772172189178620982827969187124636359}{14471830151792120008931436633118625422631665599858903351629} a^{16} + \frac{972408849458959380261985691931419558909714252274992879950}{14471830151792120008931436633118625422631665599858903351629} a^{14} + \frac{3828459901465854407083373639528292298093106624567666318060}{14471830151792120008931436633118625422631665599858903351629} a^{12} + \frac{3235940742112591300230203168691733975909036606436698288477}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{6969347845033027283122915325106161642316099685921550449030}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{1683454240084727152190446729609907751011138665286808381533}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{23406594799410675185712678366656526469474928879699153560519}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{10931181295379298123048569637512221939407919342710667800461}{57887320607168480035725746532474501690526662399435613406516} a^{2} - \frac{1423410185825622095563838850634910175814548013453702125692}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{115774641214336960071451493064949003381053324798871226813032} a^{21} + \frac{1440173476862044625438378554874387873293511702887773107478}{14471830151792120008931436633118625422631665599858903351629} a^{19} - \frac{5233137976449411263180347944361910938700353723110404806193}{115774641214336960071451493064949003381053324798871226813032} a^{17} + \frac{486204424729479690130992845965709779454857126137496439975}{14471830151792120008931436633118625422631665599858903351629} a^{15} + \frac{1914229950732927203541686819764146149046553312283833159030}{14471830151792120008931436633118625422631665599858903351629} a^{13} - \frac{5617944704839764354350616732213445723361314496711102531576}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{3484673922516513641561457662553080821158049842960775224515}{14471830151792120008931436633118625422631665599858903351629} a^{9} - \frac{6394187955853696428370494951754358835810263467286047485048}{14471830151792120008931436633118625422631665599858903351629} a^{7} + \frac{34480725807757804850013068165817975221051733519736459845997}{115774641214336960071451493064949003381053324798871226813032} a^{5} + \frac{442581107051602735735358374450800435402968282143529443896}{14471830151792120008931436633118625422631665599858903351629} a^{3} - \frac{20165470895094608391186792035658266125889857653673711854397}{115774641214336960071451493064949003381053324798871226813032} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 16220160
The 104 conjugacy class representatives for t22n44 are not computed
Character table for t22n44 is not computed

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

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 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 R R $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
337Data not computed
653Data not computed
46723Data not computed
310501Data not computed
2253079Data not computed