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Properties

Label 22.18.3596864355...3104.1
Degree $22$
Signature $[18, 2]$
Discriminant $2^{73}\cdot 3^{20}\cdot 31\cdot 337^{8}\cdot 991\cdot 310501^{8}\cdot 2473607$
Root discriminant $69{,}735.05$
Ramified primes $2, 3, 31, 337, 991, 310501, 2473607$
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![-49242609059256, 0, 197759461441604, 0, -46742377664590, 0, 2609466379182, 0, 177479869188, 0, -24773125392, 0, 833458818, 0, 4911342, 0, -973488, 0, 24474, 0, -257, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 257*x^20 + 24474*x^18 - 973488*x^16 + 4911342*x^14 + 833458818*x^12 - 24773125392*x^10 + 177479869188*x^8 + 2609466379182*x^6 - 46742377664590*x^4 + 197759461441604*x^2 - 49242609059256)
 
gp: K = bnfinit(x^22 - 257*x^20 + 24474*x^18 - 973488*x^16 + 4911342*x^14 + 833458818*x^12 - 24773125392*x^10 + 177479869188*x^8 + 2609466379182*x^6 - 46742377664590*x^4 + 197759461441604*x^2 - 49242609059256, 1)
 

Normalized defining polynomial

\( x^{22} - 257 x^{20} + 24474 x^{18} - 973488 x^{16} + 4911342 x^{14} + 833458818 x^{12} - 24773125392 x^{10} + 177479869188 x^{8} + 2609466379182 x^{6} - 46742377664590 x^{4} + 197759461441604 x^{2} - 49242609059256 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(35968643551695332182677064543344800913941547158154336510998469613936975508099279042617383314590716452143104=2^{73}\cdot 3^{20}\cdot 31\cdot 337^{8}\cdot 991\cdot 310501^{8}\cdot 2473607\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69{,}735.05$
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:  $2, 3, 31, 337, 991, 310501, 2473607$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491515}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{2072897465315298801396186263828529852308590716023013373279}{14471830151792120008931436633118625422631665599858903351629} a^{16} - \frac{45934940523762542979720200406415829103826870945991606533}{1315620922890192728084676057556238674784696872714445759239} a^{14} - \frac{7750991424342622926846838761607506147523363715528612321633}{28943660303584240017862873266237250845263331199717806703258} a^{12} + \frac{14369399608388516778481253602814827984491107713778032217183}{28943660303584240017862873266237250845263331199717806703258} a^{10} + \frac{6384950929421186309646537261575526246587170168321909433914}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{6101277060851129071657329624694490024855228449302033897019}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{8999527289941553270828354357051563526970304822402894928151}{28943660303584240017862873266237250845263331199717806703258} a^{4} - \frac{1079280007262362927759679245240881203101417689549458711869}{28943660303584240017862873266237250845263331199717806703258} a^{2} + \frac{6973875443932986974756931871720393566502063305424122925545}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{520985885464516320321531718792270515214739961594920520658644} a^{21} - \frac{60837762944064243041150154726598024126810234376192331898031}{520985885464516320321531718792270515214739961594920520658644} a^{19} - \frac{5514909205702472936775874298982385091646752105293972241636}{43415490455376360026794309899355876267894996799576710054887} a^{17} + \frac{423228660788810061701651952383274281893623333922818050902}{3946862768670578184254028172668716024354090618143337277717} a^{15} - \frac{41175210879559860999432777275518836509525562838133279711555}{86830980910752720053588619798711752535789993599153420109774} a^{13} + \frac{43381346940908492283310915555921277121848144170883086343405}{86830980910752720053588619798711752535789993599153420109774} a^{11} + \frac{2317420120134811813175330432743794629913203974242312531727}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{6857702404214416360196255419271038482495631349720312416216}{43415490455376360026794309899355876267894996799576710054887} a^{7} - \frac{999947476660172585647594928561284836330033869155877214239}{28943660303584240017862873266237250845263331199717806703258} a^{5} - \frac{116853921221599322999211172310189884584154742488420685524901}{260492942732258160160765859396135257607369980797460260329322} a^{3} + \frac{21445705595725106983688368504839018989133728905283026277174}{130246471366129080080382929698067628803684990398730130164661} a$

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:  $19$
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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.9.10.1$x^{9} + 3 x^{2} + 3$$9$$1$$10$$C_3^2:Q_8$$[5/4, 5/4]_{4}^{2}$
3.9.10.1$x^{9} + 3 x^{2} + 3$$9$$1$$10$$C_3^2:Q_8$$[5/4, 5/4]_{4}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
337Data not computed
991Data not computed
310501Data not computed
2473607Data not computed