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Properties

Label 22.14.6005478695...6784.1
Degree $22$
Signature $[14, 4]$
Discriminant $2^{71}\cdot 3^{20}\cdot 337^{8}\cdot 947\cdot 310501^{8}\cdot 53591959$
Root discriminant $64{,}285.91$
Ramified primes $2, 3, 337, 947, 310501, 53591959$
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![-32887027192104, 0, 4008220307087, 0, 4672121576416, 0, -556843607163, 0, -33761599008, 0, 5691932352, 0, -66023304, 0, -10363368, 0, 269688, 0, 2859, 0, -136, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 136*x^20 + 2859*x^18 + 269688*x^16 - 10363368*x^14 - 66023304*x^12 + 5691932352*x^10 - 33761599008*x^8 - 556843607163*x^6 + 4672121576416*x^4 + 4008220307087*x^2 - 32887027192104)
 
gp: K = bnfinit(x^22 - 136*x^20 + 2859*x^18 + 269688*x^16 - 10363368*x^14 - 66023304*x^12 + 5691932352*x^10 - 33761599008*x^8 - 556843607163*x^6 + 4672121576416*x^4 + 4008220307087*x^2 - 32887027192104, 1)
 

Normalized defining polynomial

\( x^{22} - 136 x^{20} + 2859 x^{18} + 269688 x^{16} - 10363368 x^{14} - 66023304 x^{12} + 5691932352 x^{10} - 33761599008 x^{8} - 556843607163 x^{6} + 4672121576416 x^{4} + 4008220307087 x^{2} - 32887027192104 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6005478695920935077275180178728170344905283779505833696386712583652456011831687867181255322542385643126784=2^{71}\cdot 3^{20}\cdot 337^{8}\cdot 947\cdot 310501^{8}\cdot 53591959\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64{,}285.91$
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:  $2, 3, 337, 947, 310501, 53591959$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{3}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491405}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{2737194544524833140993105902792583037168669141457279278769}{14471830151792120008931436633118625422631665599858903351629} a^{16} + \frac{448010863468600608733251996452712734757586789849524172045}{1315620922890192728084676057556238674784696872714445759239} a^{14} + \frac{3640387740344206143639524182262773313786356565007975397991}{14471830151792120008931436633118625422631665599858903351629} a^{12} + \frac{6636518046319182006055249163725422074414943557514932705598}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{4202901319594642126930619201822855742277940242245688826046}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{877980835922178739125904347982375677210722872205491182553}{14471830151792120008931436633118625422631665599858903351629} a^{6} + \frac{7624824584200078298315306400558358002958294580349176864833}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{26237684833586530354514463222437439180316890934918410021673}{57887320607168480035725746532474501690526662399435613406516} a^{2} - \frac{3510112041698694215882322931925786995678446606960950735414}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{520985885464516320321531718792270515214739961594920520658644} a^{21} - \frac{89781423247648483059013027992835274972073565575910138601179}{520985885464516320321531718792270515214739961594920520658644} a^{19} - \frac{912398181508277713664368634264194345722889713819093092923}{43415490455376360026794309899355876267894996799576710054887} a^{17} - \frac{1604824276030723434535150744590747321460400233669419621637}{3946862768670578184254028172668716024354090618143337277717} a^{15} + \frac{10861349347976148720500799149500008053016562588241927367083}{43415490455376360026794309899355876267894996799576710054887} a^{13} - \frac{12259657469685726006913020245210151397826684414020592449763}{43415490455376360026794309899355876267894996799576710054887} a^{11} - \frac{4356954348420190877762632299725891169513006284147891247649}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{19588433814363552924950550293485625789245795090547034863023}{43415490455376360026794309899355876267894996799576710054887} a^{7} + \frac{7279127243485395370449005881448095521498328553309421141261}{57887320607168480035725746532474501690526662399435613406516} a^{5} + \frac{118480616684334669734799903108748815045999765063670623494617}{520985885464516320321531718792270515214739961594920520658644} a^{3} - \frac{61397432648867174251608069464400288686205109006396564141930}{130246471366129080080382929698067628803684990398730130164661} a$

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:  $17$
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 $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ $22$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
947Data not computed
310501Data not computed
53591959Data not computed