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Properties

Label 16.0.87393197611...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}$
Root discriminant $55.76$
Ramified primes $2, 5, 11, 19$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group 16T868

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13781, 31410, 140326, 170956, 259628, 37678, 166514, -43862, 56291, -21298, 11058, -3414, 1142, -236, 56, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 56*x^14 - 236*x^13 + 1142*x^12 - 3414*x^11 + 11058*x^10 - 21298*x^9 + 56291*x^8 - 43862*x^7 + 166514*x^6 + 37678*x^5 + 259628*x^4 + 170956*x^3 + 140326*x^2 + 31410*x + 13781)
 
gp: K = bnfinit(x^16 - 6*x^15 + 56*x^14 - 236*x^13 + 1142*x^12 - 3414*x^11 + 11058*x^10 - 21298*x^9 + 56291*x^8 - 43862*x^7 + 166514*x^6 + 37678*x^5 + 259628*x^4 + 170956*x^3 + 140326*x^2 + 31410*x + 13781, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 56 x^{14} - 236 x^{13} + 1142 x^{12} - 3414 x^{11} + 11058 x^{10} - 21298 x^{9} + 56291 x^{8} - 43862 x^{7} + 166514 x^{6} + 37678 x^{5} + 259628 x^{4} + 170956 x^{3} + 140326 x^{2} + 31410 x + 13781 \)

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:  \(8739319761101494681600000000=2^{28}\cdot 5^{8}\cdot 11^{6}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.76$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{1547451103003474189895094211541147524325} a^{15} - \frac{7367017484959933080265092720439193182}{309490220600694837979018842308229504865} a^{14} - \frac{413249793340117189958519755530945463884}{1547451103003474189895094211541147524325} a^{13} + \frac{133489044919384779832310898740212141219}{309490220600694837979018842308229504865} a^{12} - \frac{597264158648127772448731926005192978548}{1547451103003474189895094211541147524325} a^{11} - \frac{49305660510242490458005202185448603182}{1547451103003474189895094211541147524325} a^{10} - \frac{2898943394070012037841906111648115957}{11295263525572804305803607383512025725} a^{9} - \frac{14759601734578675624170317652701295238}{81444794894919694205004958502165659175} a^{8} + \frac{403518876314057590576287228500579055489}{1547451103003474189895094211541147524325} a^{7} + \frac{638935088925365749398067930273233121377}{1547451103003474189895094211541147524325} a^{6} - \frac{342359524252712166708087621541826006519}{1547451103003474189895094211541147524325} a^{5} + \frac{328022692134061509915525647079186247694}{1547451103003474189895094211541147524325} a^{4} - \frac{489457178843607465344061139629398005963}{1547451103003474189895094211541147524325} a^{3} - \frac{511047901466001159260252804505337122127}{1547451103003474189895094211541147524325} a^{2} + \frac{499516435010952448154461691954726802339}{1547451103003474189895094211541147524325} a - \frac{408567116914474000774878742760631784081}{1547451103003474189895094211541147524325}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( 1509026.65444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T868:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n868 are not computed
Character table for t16n868 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.4400.1, 4.0.83600.1, 4.4.7600.1, 8.0.111823360000.8

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
5.4.2.1$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}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$