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Properties

Label 22.14.1215217932...4176.1
Degree $22$
Signature $[14, 4]$
Discriminant $2^{71}\cdot 3^{20}\cdot 11\cdot 337^{8}\cdot 4789\cdot 310501^{8}\cdot 19494793$
Root discriminant $73{,}702.85$
Ramified primes $2, 3, 11, 337, 4789, 310501, 19494793$
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![-18485391608046, 0, -19818898708405, 0, 6739137357122, 0, -54779058171, 0, -91451924736, 0, 5499758880, 0, 106875912, 0, -14130840, 0, 192426, 0, 5799, 0, -158, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 158*x^20 + 5799*x^18 + 192426*x^16 - 14130840*x^14 + 106875912*x^12 + 5499758880*x^10 - 91451924736*x^8 - 54779058171*x^6 + 6739137357122*x^4 - 19818898708405*x^2 - 18485391608046)
 
gp: K = bnfinit(x^22 - 158*x^20 + 5799*x^18 + 192426*x^16 - 14130840*x^14 + 106875912*x^12 + 5499758880*x^10 - 91451924736*x^8 - 54779058171*x^6 + 6739137357122*x^4 - 19818898708405*x^2 - 18485391608046, 1)
 

Normalized defining polynomial

\( x^{22} - 158 x^{20} + 5799 x^{18} + 192426 x^{16} - 14130840 x^{14} + 106875912 x^{12} + 5499758880 x^{10} - 91451924736 x^{8} - 54779058171 x^{6} + 6739137357122 x^{4} - 19818898708405 x^{2} - 18485391608046 \)

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:  \(121521793204922702401925547871513287519587064054154046165862083482052135980296415200485327438880728653234176=2^{71}\cdot 3^{20}\cdot 11\cdot 337^{8}\cdot 4789\cdot 310501^{8}\cdot 19494793\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73{,}702.85$
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:  $2, 3, 11, 337, 4789, 310501, 19494793$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{3}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491425}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{7864068360572039251029411324710714993048521691821898838061}{28943660303584240017862873266237250845263331199717806703258} a^{16} + \frac{394628689472948792402517949688862661350232187854982068495}{14471830151792120008931436633118625422631665599858903351629} a^{14} - \frac{6491232988242494561591497469540147509817851218635449926848}{14471830151792120008931436633118625422631665599858903351629} a^{12} + \frac{379479944499403111464025246751598608579032886983333304630}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{1712055329964228154619011588592338376355269556820087867485}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{6665730580054876086478648920975304229722129469638265792334}{14471830151792120008931436633118625422631665599858903351629} a^{6} - \frac{28268327301576765177633202998451644372891321806113100005183}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{20340117148558599765493003641515711915227634526496937886781}{57887320607168480035725746532474501690526662399435613406516} a^{2} - \frac{355297054274060653207354539590737003099606590089302394393}{2631241845780385456169352115112477349569393745428891518478}$, $\frac{1}{173661961821505440107177239597423505071579987198306840219548} a^{21} + \frac{25993217966688477012438465072113728408979759222961088211833}{173661961821505440107177239597423505071579987198306840219548} a^{19} + \frac{7026530647670733588944487313842178617404936502631969288399}{28943660303584240017862873266237250845263331199717806703258} a^{17} + \frac{4955486280421689600444651527602496027993965929237961806708}{14471830151792120008931436633118625422631665599858903351629} a^{15} + \frac{2660199054516541815779979721192825970937938127074484474927}{14471830151792120008931436633118625422631665599858903351629} a^{13} - \frac{4697450069097572299155803795455675604684210904291856682333}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{570685109988076051539670529530779458785089852273362622495}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{2221910193351625362159549640325101409907376489879421930778}{14471830151792120008931436633118625422631665599858903351629} a^{7} - \frac{28718549302915081737786316510308715354472661401849571137233}{57887320607168480035725746532474501690526662399435613406516} a^{5} + \frac{66490863762194120288095616157196040620562359072656482222993}{173661961821505440107177239597423505071579987198306840219548} a^{3} - \frac{2986538900054446109376706654703214352669000335518193912871}{7893725537341156368508056345337432048708181236286674555434} a$

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 $16{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ $22$ R ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
337Data not computed
4789Data not computed
310501Data not computed
19494793Data not computed