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Properties

Label 22.16.2775844325...5568.1
Degree $22$
Signature $[16, 3]$
Discriminant $-\,2^{70}\cdot 3^{21}\cdot 67\cdot 337^{8}\cdot 2851\cdot 310501^{8}\cdot 818717$
Root discriminant $68{,}918.55$
Ramified primes $2, 3, 67, 337, 2851, 310501, 818717$
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![46916659556700, 0, -44093388187521, 0, 4468447202724, 0, 871502095749, 0, -135890353824, 0, 2957044992, 0, 321816792, 0, -16217640, 0, 58284, 0, 9179, 0, -180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 180*x^20 + 9179*x^18 + 58284*x^16 - 16217640*x^14 + 321816792*x^12 + 2957044992*x^10 - 135890353824*x^8 + 871502095749*x^6 + 4468447202724*x^4 - 44093388187521*x^2 + 46916659556700)
 
gp: K = bnfinit(x^22 - 180*x^20 + 9179*x^18 + 58284*x^16 - 16217640*x^14 + 321816792*x^12 + 2957044992*x^10 - 135890353824*x^8 + 871502095749*x^6 + 4468447202724*x^4 - 44093388187521*x^2 + 46916659556700, 1)
 

Normalized defining polynomial

\( x^{22} - 180 x^{20} + 9179 x^{18} + 58284 x^{16} - 16217640 x^{14} + 321816792 x^{12} + 2957044992 x^{10} - 135890353824 x^{8} + 871502095749 x^{6} + 4468447202724 x^{4} - 44093388187521 x^{2} + 46916659556700 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-27758443257594324914983363876184067664865764900320098627846512840261521316613154967560690069271834754285568=-\,2^{70}\cdot 3^{21}\cdot 67\cdot 337^{8}\cdot 2851\cdot 310501^{8}\cdot 818717\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68{,}918.55$
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:  $2, 3, 67, 337, 2851, 310501, 818717$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{3}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491445}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{5126873816047206110036305421918131955879852550364619559202}{14471830151792120008931436633118625422631665599858903351629} a^{16} + \frac{506741901178154632153423458283651167424811358765017136330}{14471830151792120008931436633118625422631665599858903351629} a^{14} - \frac{145040689786150828367516235588371351693221295079537311841}{14471830151792120008931436633118625422631665599858903351629} a^{12} - \frac{4787953406577966332088879542160203575293637223043944268149}{14471830151792120008931436633118625422631665599858903351629} a^{10} - \frac{7131752603066048449001924920926642599891215098792184156038}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{636321335543028714604923376153280555080229802599412965950}{1315620922890192728084676057556238674784696872714445759239} a^{6} + \frac{28218717654460129733782636551063535763487933288648649657013}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{24926796245418423596073142729721447569001241096227095454325}{57887320607168480035725746532474501690526662399435613406516} a^{2} + \frac{4722177501952087235395429331766508040500370402455503237510}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{289436603035842400178628732662372508452633311997178067032580} a^{21} - \frac{590088467379152601084881638824704487256714395351343698289}{57887320607168480035725746532474501690526662399435613406516} a^{19} - \frac{24725577783886532229004047476954889334391370700588142470033}{144718301517921200089314366331186254226316655998589033516290} a^{17} - \frac{13965088250613965376778013174834974255206854241093886215299}{72359150758960600044657183165593127113158327999294516758145} a^{15} - \frac{2923374168315654167459790573741399354864977378987688132694}{14471830151792120008931436633118625422631665599858903351629} a^{13} - \frac{33731613710162206349951752808397454420556968422761750971407}{72359150758960600044657183165593127113158327999294516758145} a^{11} - \frac{21603582754858168457933361554045268022522880698651087507667}{72359150758960600044657183165593127113158327999294516758145} a^{9} + \frac{1951942258433221442689599433709519229864926675313858725189}{6578104614450963640423380287781193373923484363572228796195} a^{7} + \frac{86106038261628609769508383083538037454014595688084263063529}{289436603035842400178628732662372508452633311997178067032580} a^{5} - \frac{82814116852586903631798889262195949259527903495662708860841}{289436603035842400178628732662372508452633311997178067032580} a^{3} + \frac{52859845459280534497585168562888892348895737604487716529907}{144718301517921200089314366331186254226316655998589033516290} a$

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:  $18$
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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
67Data not computed
337Data not computed
2851Data not computed
310501Data not computed
818717Data not computed