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Properties

Label 16.0.17876432388...1441.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 23^{12}$
Root discriminant $37.87$
Ramified primes $13, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94249, 0, 27078, 0, 17403, 0, 2853, 0, 2099, 0, -444, 0, 114, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 114*x^12 - 444*x^10 + 2099*x^8 + 2853*x^6 + 17403*x^4 + 27078*x^2 + 94249)
 
gp: K = bnfinit(x^16 - 9*x^14 + 114*x^12 - 444*x^10 + 2099*x^8 + 2853*x^6 + 17403*x^4 + 27078*x^2 + 94249, 1)
 

Normalized defining polynomial

\( x^{16} - 9 x^{14} + 114 x^{12} - 444 x^{10} + 2099 x^{8} + 2853 x^{6} + 17403 x^{4} + 27078 x^{2} + 94249 \)

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:  \(17876432388376151935981441=13^{8}\cdot 23^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.87$
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:  $13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{306} a^{12} + \frac{4}{153} a^{10} - \frac{2}{153} a^{8} - \frac{5}{34} a^{6} - \frac{1}{2} a^{5} - \frac{149}{306} a^{4} - \frac{31}{306} a^{2} - \frac{1}{2} a - \frac{26}{153}$, $\frac{1}{306} a^{13} + \frac{4}{153} a^{11} - \frac{2}{153} a^{9} - \frac{5}{34} a^{7} - \frac{1}{2} a^{6} - \frac{149}{306} a^{5} - \frac{31}{306} a^{3} - \frac{1}{2} a^{2} - \frac{26}{153} a$, $\frac{1}{28334880203544} a^{14} + \frac{585465772}{3541860025443} a^{12} - \frac{606706389365}{14167440101772} a^{10} + \frac{227694344423}{4722480033924} a^{8} + \frac{11860938916897}{28334880203544} a^{6} - \frac{1}{2} a^{5} - \frac{6063161900399}{14167440101772} a^{4} - \frac{8824128944467}{28334880203544} a^{2} - \frac{1}{2} a + \frac{203079446297}{555585886344}$, $\frac{1}{17397616444976016} a^{15} - \frac{1}{56669760407088} a^{14} - \frac{5948227934723}{4349404111244004} a^{13} + \frac{7326160129}{4722480033924} a^{12} - \frac{71286363032579}{2899602740829336} a^{11} + \frac{977096980261}{28334880203544} a^{10} + \frac{619698358068209}{8698808222488008} a^{9} - \frac{868278328717}{28334880203544} a^{8} + \frac{195365192407349}{1023389202645648} a^{7} - \frac{16027833064477}{56669760407088} a^{6} + \frac{307611369764651}{8698808222488008} a^{5} - \frac{278454285013}{9444960067848} a^{4} + \frac{2534944086621995}{5799205481658672} a^{3} - \frac{22381278338521}{56669760407088} a^{2} + \frac{832253772959371}{17397616444976016} a + \frac{13162750760749}{56669760407088}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 812391.716616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-23}) \), 4.0.6877.1, 8.0.614810677.1, 8.0.325234848133.1, 8.0.4228053025729.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$