↑

Properties

Label 16.8.81464396628...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 761^{6}$
Root discriminant $131.65$
Ramified primes $2, 5, 761$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T869

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125995759, 298136964, 352409812, 209915832, 71862392, 4897716, -7009552, -2599968, -395743, 32604, 39236, 6120, -126, -168, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 - 168*x^13 - 126*x^12 + 6120*x^11 + 39236*x^10 + 32604*x^9 - 395743*x^8 - 2599968*x^7 - 7009552*x^6 + 4897716*x^5 + 71862392*x^4 + 209915832*x^3 + 352409812*x^2 + 298136964*x + 125995759)
 
gp: K = bnfinit(x^16 - 48*x^14 - 168*x^13 - 126*x^12 + 6120*x^11 + 39236*x^10 + 32604*x^9 - 395743*x^8 - 2599968*x^7 - 7009552*x^6 + 4897716*x^5 + 71862392*x^4 + 209915832*x^3 + 352409812*x^2 + 298136964*x + 125995759, 1)
 

Normalized defining polynomial

\( x^{16} - 48 x^{14} - 168 x^{13} - 126 x^{12} + 6120 x^{11} + 39236 x^{10} + 32604 x^{9} - 395743 x^{8} - 2599968 x^{7} - 7009552 x^{6} + 4897716 x^{5} + 71862392 x^{4} + 209915832 x^{3} + 352409812 x^{2} + 298136964 x + 125995759 \)

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:  \(8146439662885973461565440000000000=2^{32}\cdot 5^{10}\cdot 761^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.65$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{1522} a^{12} - \frac{73}{761} a^{11} - \frac{43}{1522} a^{10} - \frac{187}{761} a^{9} + \frac{9}{1522} a^{8} + \frac{176}{761} a^{7} - \frac{661}{1522} a^{6} - \frac{369}{761} a^{5} - \frac{271}{1522} a^{4} + \frac{258}{761} a^{3} + \frac{653}{1522} a^{2} - \frac{259}{761} a + \frac{66}{761}$, $\frac{1}{1522} a^{13} - \frac{51}{1522} a^{11} + \frac{197}{1522} a^{10} + \frac{197}{1522} a^{9} + \frac{72}{761} a^{8} + \frac{505}{1522} a^{7} - \frac{597}{1522} a^{6} + \frac{43}{1522} a^{5} + \frac{261}{761} a^{4} - \frac{111}{1522} a^{3} - \frac{305}{1522} a^{2} + \frac{302}{761} a - \frac{257}{761}$, $\frac{1}{10654} a^{14} + \frac{1}{10654} a^{13} - \frac{1}{10654} a^{12} - \frac{305}{10654} a^{11} - \frac{995}{10654} a^{10} + \frac{1427}{10654} a^{9} + \frac{2621}{10654} a^{8} + \frac{3049}{10654} a^{7} + \frac{641}{10654} a^{6} + \frac{4759}{10654} a^{5} - \frac{4007}{10654} a^{4} + \frac{271}{10654} a^{3} - \frac{310}{761} a^{2} - \frac{2251}{5327} a - \frac{1}{5327}$, $\frac{1}{389945064141077609097797889847871663009038846} a^{15} + \frac{1892787161415974471270639016522624889477}{55706437734439658442542555692553094715576978} a^{14} - \frac{106914357547989864846337346486340832462441}{389945064141077609097797889847871663009038846} a^{13} + \frac{26460325782391397031672235023103724578614}{194972532070538804548898944923935831504519423} a^{12} - \frac{50314002883480559085520538135510192093996627}{389945064141077609097797889847871663009038846} a^{11} - \frac{5754236249937334231243062116367013904846136}{27853218867219829221271277846276547357788489} a^{10} + \frac{19915290906513291603679565316419252289681765}{389945064141077609097797889847871663009038846} a^{9} - \frac{83382313863323430772057158055002474564571665}{389945064141077609097797889847871663009038846} a^{8} + \frac{16757935683531647713568821938104755778949967}{55706437734439658442542555692553094715576978} a^{7} + \frac{80401381638237065441703075409428749375823365}{194972532070538804548898944923935831504519423} a^{6} - \frac{184409875851035674476692599026705873501231873}{389945064141077609097797889847871663009038846} a^{5} + \frac{94491334326746925287607682288570468922836181}{389945064141077609097797889847871663009038846} a^{4} - \frac{78381303108466407862418191843754788893865374}{194972532070538804548898944923935831504519423} a^{3} + \frac{111647967635790888922431312394396332596158923}{389945064141077609097797889847871663009038846} a^{2} + \frac{6183723327765370644850387809234259300616071}{194972532070538804548898944923935831504519423} a - \frac{106749190669198731611013184790736505099493935}{389945064141077609097797889847871663009038846}$

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:  \( 21894684672.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n869
Character table for t16n869 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.1217600.3, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.48704.1, 8.8.1482549760000.4

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
761Data not computed