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Properties

Label 22.12.2237479871...8160.1
Degree $22$
Signature $[12, 5]$
Discriminant $-\,2^{71}\cdot 3^{21}\cdot 5\cdot 113\cdot 337^{8}\cdot 310501^{8}\cdot 11155561$
Root discriminant $61{,}464.65$
Ramified primes $2, 3, 5, 113, 337, 310501, 11155561$
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![151269407160, 0, -6084443980476, 0, 2196660144654, 0, 351858173934, 0, -33068943924, 0, -4324351728, 0, 25606542, 0, 8932590, 0, 31824, 0, -5446, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 15*x^20 - 5446*x^18 + 31824*x^16 + 8932590*x^14 + 25606542*x^12 - 4324351728*x^10 - 33068943924*x^8 + 351858173934*x^6 + 2196660144654*x^4 - 6084443980476*x^2 + 151269407160)
 
gp: K = bnfinit(x^22 - 15*x^20 - 5446*x^18 + 31824*x^16 + 8932590*x^14 + 25606542*x^12 - 4324351728*x^10 - 33068943924*x^8 + 351858173934*x^6 + 2196660144654*x^4 - 6084443980476*x^2 + 151269407160, 1)
 

Normalized defining polynomial

\( x^{22} - 15 x^{20} - 5446 x^{18} + 31824 x^{16} + 8932590 x^{14} + 25606542 x^{12} - 4324351728 x^{10} - 33068943924 x^{8} + 351858173934 x^{6} + 2196660144654 x^{4} - 6084443980476 x^{2} + 151269407160 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2237479871188499870613354915145778127677091451853959466606041823244675621995663555455152169522515054428160=-\,2^{71}\cdot 3^{21}\cdot 5\cdot 113\cdot 337^{8}\cdot 310501^{8}\cdot 11155561\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61{,}464.65$
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:  $2, 3, 5, 113, 337, 310501, 11155561$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{16}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491295}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{6802983247468734961180051083513272444057495133783090363073}{28943660303584240017862873266237250845263331199717806703258} a^{16} - \frac{425557197544732772075104745731868155907536334685750004690}{1315620922890192728084676057556238674784696872714445759239} a^{14} + \frac{4680923876592183645174394895491612475191851785022117984217}{28943660303584240017862873266237250845263331199717806703258} a^{12} + \frac{8662949786774333504986279301768253514329976689682665718719}{28943660303584240017862873266237250845263331199717806703258} a^{10} + \frac{7117784073178314594266630510011921114387750478976407924199}{14471830151792120008931436633118625422631665599858903351629} a^{8} - \frac{1664647229313214397188210012526689926626322350557904617836}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{9464108547766711189440050502515589419672609500930908153453}{28943660303584240017862873266237250845263331199717806703258} a^{4} - \frac{4883220340921872324178741201419862028547813404519471807901}{28943660303584240017862873266237250845263331199717806703258} a^{2} + \frac{811459757641502885736540144075227198835528418060766323699}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{21} - \frac{2950442336895763005424408194123522436283571976756718491295}{57887320607168480035725746532474501690526662399435613406516} a^{19} - \frac{6802983247468734961180051083513272444057495133783090363073}{28943660303584240017862873266237250845263331199717806703258} a^{17} - \frac{425557197544732772075104745731868155907536334685750004690}{1315620922890192728084676057556238674784696872714445759239} a^{15} + \frac{4680923876592183645174394895491612475191851785022117984217}{28943660303584240017862873266237250845263331199717806703258} a^{13} + \frac{8662949786774333504986279301768253514329976689682665718719}{28943660303584240017862873266237250845263331199717806703258} a^{11} + \frac{7117784073178314594266630510011921114387750478976407924199}{14471830151792120008931436633118625422631665599858903351629} a^{9} - \frac{1664647229313214397188210012526689926626322350557904617836}{14471830151792120008931436633118625422631665599858903351629} a^{7} - \frac{9464108547766711189440050502515589419672609500930908153453}{28943660303584240017862873266237250845263331199717806703258} a^{5} - \frac{4883220340921872324178741201419862028547813404519471807901}{28943660303584240017862873266237250845263331199717806703258} a^{3} + \frac{811459757641502885736540144075227198835528418060766323699}{14471830151792120008931436633118625422631665599858903351629} a$

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:  $16$
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 R $22$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ $22$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
113Data not computed
337Data not computed
310501Data not computed
11155561Data not computed