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Properties

Label 16.8.32625338328...0073.3
Degree $16$
Signature $[8, 4]$
Discriminant $61^{6}\cdot 97^{15}$
Root discriminant $340.48$
Ramified primes $61, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![183952147391, 243748531790, 75820476938, -15167591488, -7952268139, 1312007365, 506360379, -116261286, -15226884, 2756222, 345102, -108182, 8775, 1519, -235, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 235*x^14 + 1519*x^13 + 8775*x^12 - 108182*x^11 + 345102*x^10 + 2756222*x^9 - 15226884*x^8 - 116261286*x^7 + 506360379*x^6 + 1312007365*x^5 - 7952268139*x^4 - 15167591488*x^3 + 75820476938*x^2 + 243748531790*x + 183952147391)
 
gp: K = bnfinit(x^16 - 4*x^15 - 235*x^14 + 1519*x^13 + 8775*x^12 - 108182*x^11 + 345102*x^10 + 2756222*x^9 - 15226884*x^8 - 116261286*x^7 + 506360379*x^6 + 1312007365*x^5 - 7952268139*x^4 - 15167591488*x^3 + 75820476938*x^2 + 243748531790*x + 183952147391, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 235 x^{14} + 1519 x^{13} + 8775 x^{12} - 108182 x^{11} + 345102 x^{10} + 2756222 x^{9} - 15226884 x^{8} - 116261286 x^{7} + 506360379 x^{6} + 1312007365 x^{5} - 7952268139 x^{4} - 15167591488 x^{3} + 75820476938 x^{2} + 243748531790 x + 183952147391 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(32625338328875805606560925727460997390073=61^{6}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $340.48$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{6} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{5}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{16}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{7}{32} a^{9} - \frac{3}{8} a^{8} + \frac{15}{32} a^{7} + \frac{1}{4} a^{6} + \frac{11}{32} a^{5} - \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{7}{16} a^{2} + \frac{5}{32} a + \frac{1}{32}$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{12} - \frac{1}{64} a^{11} - \frac{7}{64} a^{10} + \frac{13}{64} a^{9} + \frac{3}{64} a^{8} + \frac{23}{64} a^{7} - \frac{13}{64} a^{6} - \frac{3}{64} a^{5} - \frac{3}{32} a^{4} + \frac{13}{32} a^{3} - \frac{9}{64} a^{2} + \frac{3}{32} a - \frac{31}{64}$, $\frac{1}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a^{15} + \frac{536293680036601299994762539541011152812959600847945034045278196551862661199}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a^{14} - \frac{356017756319160073248446883402124887737748644518130657498560004241446909777}{58394379409225965716929198154722207767501910427698592271773461831543277937472} a^{13} - \frac{1326058874793725556127251349918973210564302057402301203154667365725529066351}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a^{12} - \frac{878164153781653052765683744464682244546490654637239141458076995508914266691}{58394379409225965716929198154722207767501910427698592271773461831543277937472} a^{11} + \frac{1885782462582168860840036320756827082502311792045323594033681070935386818297}{29197189704612982858464599077361103883750955213849296135886730915771638968736} a^{10} - \frac{12643409981670093695054857624065290136436129007825384223288360112136340000565}{58394379409225965716929198154722207767501910427698592271773461831543277937472} a^{9} - \frac{480800601222464234063099972777226877772737188281553885254741959392542973211}{29197189704612982858464599077361103883750955213849296135886730915771638968736} a^{8} - \frac{7190032097203445281419572538911731642345236710517978152736921910980125339401}{29197189704612982858464599077361103883750955213849296135886730915771638968736} a^{7} - \frac{26776363760352354723023274311064707072708983214735287232213322223254058446051}{58394379409225965716929198154722207767501910427698592271773461831543277937472} a^{6} - \frac{36868559160601118698503221055501449247391647522943455370157351214314986641827}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a^{5} + \frac{2327228154356048540680039154136647100829383476245821809200013716080393191871}{7299297426153245714616149769340275970937738803462324033971682728942909742184} a^{4} - \frac{47838069317781803127725879088116702283116458693721637578691111842471107432627}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a^{3} + \frac{17801035236300008798275636940582101035379706454451144797780717484477739243823}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a^{2} + \frac{57119898566605239751296332152809723525725072528332152160676489493571700823179}{116788758818451931433858396309444415535003820855397184543546923663086555874944} a - \frac{20582397712680354448976664164779942796562468453040276929645513263660336769073}{116788758818451931433858396309444415535003820855397184543546923663086555874944}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $11$
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:  \( 18569730490600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed