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Properties

Label 22.4.38041046614...0144.1
Degree $22$
Signature $[4, 9]$
Discriminant $-\,2^{70}\cdot 3^{21}\cdot 337^{8}\cdot 2707\cdot 310501^{8}\cdot 79172602891$
Root discriminant $95{,}701.90$
Ramified primes $2, 3, 337, 2707, 310501, 79172602891$
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![10287371329244976, 0, 5159145077989967, 0, 538828326720056, 0, 9697659463557, 0, -1392387544608, 0, -86702793408, 0, -1320559944, 0, 40935192, 0, 2066928, 0, 36459, 0, 304, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 304*x^20 + 36459*x^18 + 2066928*x^16 + 40935192*x^14 - 1320559944*x^12 - 86702793408*x^10 - 1392387544608*x^8 + 9697659463557*x^6 + 538828326720056*x^4 + 5159145077989967*x^2 + 10287371329244976)
 
gp: K = bnfinit(x^22 + 304*x^20 + 36459*x^18 + 2066928*x^16 + 40935192*x^14 - 1320559944*x^12 - 86702793408*x^10 - 1392387544608*x^8 + 9697659463557*x^6 + 538828326720056*x^4 + 5159145077989967*x^2 + 10287371329244976, 1)
 

Normalized defining polynomial

\( x^{22} + 304 x^{20} + 36459 x^{18} + 2066928 x^{16} + 40935192 x^{14} - 1320559944 x^{12} - 86702793408 x^{10} - 1392387544608 x^{8} + 9697659463557 x^{6} + 538828326720056 x^{4} + 5159145077989967 x^{2} + 10287371329244976 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-38041046614734606844466182382827953894774429871771088765931049023543756062067815864066521884753399513448710144=-\,2^{70}\cdot 3^{21}\cdot 337^{8}\cdot 2707\cdot 310501^{8}\cdot 79172602891\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95{,}701.90$
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:  $2, 3, 337, 2707, 310501, 79172602891$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491005}{57887320607168480035725746532474501690526662399435613406516} a^{18} + \frac{6687768018906776540507452655344280727311499347757220193732}{14471830151792120008931436633118625422631665599858903351629} a^{16} + \frac{259348720412270225378040512031951909393001302543563854899}{14471830151792120008931436633118625422631665599858903351629} a^{14} + \frac{4926708575230916258963405970902161355561452742060275632185}{14471830151792120008931436633118625422631665599858903351629} a^{12} + \frac{1707406206816729396464984654912807160423024357417047705056}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{236020680242715256632000724604190626289830189876792789884}{14471830151792120008931436633118625422631665599858903351629} a^{8} - \frac{636886923424700149398166879481462298513843233990211525355}{1315620922890192728084676057556238674784696872714445759239} a^{6} + \frac{1275122432515599557092138488238814797037673329027135720201}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{43919234986384296921281479927394133804962358347060052409}{57887320607168480035725746532474501690526662399435613406516} a^{2} + \frac{102919139453553653244575791761816156696606812119844028859}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{21} - \frac{2950442336895763005424408194123522436283571976756718491005}{57887320607168480035725746532474501690526662399435613406516} a^{19} - \frac{1096294113978566927916531322430063968008666904344462964165}{28943660303584240017862873266237250845263331199717806703258} a^{17} + \frac{259348720412270225378040512031951909393001302543563854899}{14471830151792120008931436633118625422631665599858903351629} a^{15} + \frac{4926708575230916258963405970902161355561452742060275632185}{14471830151792120008931436633118625422631665599858903351629} a^{13} + \frac{1707406206816729396464984654912807160423024357417047705056}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{236020680242715256632000724604190626289830189876792789884}{14471830151792120008931436633118625422631665599858903351629} a^{9} - \frac{636886923424700149398166879481462298513843233990211525355}{1315620922890192728084676057556238674784696872714445759239} a^{7} + \frac{1275122432515599557092138488238814797037673329027135720201}{57887320607168480035725746532474501690526662399435613406516} a^{5} - \frac{43919234986384296921281479927394133804962358347060052409}{57887320607168480035725746532474501690526662399435613406516} a^{3} - \frac{14265991872885012702442285049594993109238451975619215293911}{28943660303584240017862873266237250845263331199717806703258} a$

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:  $12$
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 $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$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}$
337Data not computed
2707Data not computed
310501Data not computed
79172602891Data not computed