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Properties

Label 16.0.52155530347...1509.2
Degree $16$
Signature $[0, 8]$
Discriminant $149^{9}\cdot 331^{8}$
Root discriminant $303.62$
Ramified primes $149, 331$
Class number $1200$ (GRH)
Class group $[2, 2, 300]$ (GRH)
Galois group 16T1675

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25172218864, -3217639232, 402208164, 886766664, 1881736663, 138905845, 10311242, -14810136, 4065355, -549695, 170050, -23796, 5369, -616, 99, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 99*x^14 - 616*x^13 + 5369*x^12 - 23796*x^11 + 170050*x^10 - 549695*x^9 + 4065355*x^8 - 14810136*x^7 + 10311242*x^6 + 138905845*x^5 + 1881736663*x^4 + 886766664*x^3 + 402208164*x^2 - 3217639232*x + 25172218864)
 
gp: K = bnfinit(x^16 - 6*x^15 + 99*x^14 - 616*x^13 + 5369*x^12 - 23796*x^11 + 170050*x^10 - 549695*x^9 + 4065355*x^8 - 14810136*x^7 + 10311242*x^6 + 138905845*x^5 + 1881736663*x^4 + 886766664*x^3 + 402208164*x^2 - 3217639232*x + 25172218864, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 99 x^{14} - 616 x^{13} + 5369 x^{12} - 23796 x^{11} + 170050 x^{10} - 549695 x^{9} + 4065355 x^{8} - 14810136 x^{7} + 10311242 x^{6} + 138905845 x^{5} + 1881736663 x^{4} + 886766664 x^{3} + 402208164 x^{2} - 3217639232 x + 25172218864 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5215553034727934460124227765005778951509=149^{9}\cdot 331^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $303.62$
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:  $149, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{16} a^{12} + \frac{3}{8} a^{11} + \frac{1}{16} a^{10} + \frac{3}{8} a^{9} + \frac{1}{8} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{16} a^{3} + \frac{1}{16} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{27289735963008147602857125163223178687556948192623813646697957521138792113184} a^{15} + \frac{73126774926895388947143722190333845349762856395912329346399187765683973357}{3411216995376018450357140645402897335944618524077976705837244690142349014148} a^{14} - \frac{2734592143720634531563620997576334174142691435746136063856280461150375358033}{27289735963008147602857125163223178687556948192623813646697957521138792113184} a^{13} + \frac{3243776744260688365202985928954629515329878283092720503217692156818001215261}{13644867981504073801428562581611589343778474096311906823348978760569396056592} a^{12} - \frac{9285230376030218222465376601482013010618812389505460525411869332091863555607}{27289735963008147602857125163223178687556948192623813646697957521138792113184} a^{11} + \frac{6803015028731110437865267168475693553347124154765119389835549790879273197385}{13644867981504073801428562581611589343778474096311906823348978760569396056592} a^{10} + \frac{413942189054460188531802724078597087257095986978118523112071153581023697025}{1049605229346467215494504813970122257213728776639377447949921443120722773584} a^{9} - \frac{13525785605406984337995489898422015426659721621107375090045660703113690241}{28104774421223632958658213350384324086052469817326275640265661710750558304} a^{8} + \frac{8864710094335951170694965937320271649640633413277733241075060299601342955817}{27289735963008147602857125163223178687556948192623813646697957521138792113184} a^{7} + \frac{244482462940029384249292887740557077656521537450548336349006570108693628489}{13644867981504073801428562581611589343778474096311906823348978760569396056592} a^{6} + \frac{4567145925889767228634889893532503431098123318132017362370478730008126198121}{13644867981504073801428562581611589343778474096311906823348978760569396056592} a^{5} + \frac{1142740669163906633542886555847507344980842325769744967684283884809280258425}{27289735963008147602857125163223178687556948192623813646697957521138792113184} a^{4} + \frac{5141204548429553422946396208525305513023482449078039941309090467933076453357}{27289735963008147602857125163223178687556948192623813646697957521138792113184} a^{3} - \frac{4491771853079262756584484500178290687122212759688399876757304664170776418131}{13644867981504073801428562581611589343778474096311906823348978760569396056592} a^{2} - \frac{968887899822586361951326600030402206562511111342110421534293822271461044541}{6822433990752036900714281290805794671889237048155953411674489380284698028296} a - \frac{25741907399407278563849003073824604169978807819160045756078866756311014157}{131200653668308401936813101746265282151716097079922180993740180390090346698}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{300}$, which has order $1200$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 31016149776.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{-49319}) \), 4.2.331.1, 8.0.5916393465826065121.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
149Data not computed
331Data not computed