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Properties

Label 16.8.91603437103...1609.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{6}\cdot 17^{12}\cdot 2389^{4}$
Root discriminant $153.15$
Ramified primes $13, 17, 2389$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43264, 0, 111488, 0, -92560, 0, -37320, 0, 17321, 0, 1604, 0, -530, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 530*x^12 + 1604*x^10 + 17321*x^8 - 37320*x^6 - 92560*x^4 + 111488*x^2 + 43264)
 
gp: K = bnfinit(x^16 - 4*x^14 - 530*x^12 + 1604*x^10 + 17321*x^8 - 37320*x^6 - 92560*x^4 + 111488*x^2 + 43264, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 530 x^{12} + 1604 x^{10} + 17321 x^{8} - 37320 x^{6} - 92560 x^{4} + 111488 x^{2} + 43264 \)

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:  \(91603437103816431399094670113021609=13^{6}\cdot 17^{12}\cdot 2389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.15$
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:  $13, 17, 2389$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{448} a^{8} - \frac{1}{224} a^{6} + \frac{13}{448} a^{4} - \frac{3}{112} a^{2} + \frac{13}{28}$, $\frac{1}{448} a^{9} - \frac{1}{224} a^{7} + \frac{13}{448} a^{5} - \frac{3}{112} a^{3} + \frac{13}{28} a$, $\frac{1}{11648} a^{10} - \frac{1}{896} a^{9} + \frac{1}{2912} a^{8} + \frac{1}{448} a^{7} + \frac{561}{11648} a^{6} - \frac{13}{896} a^{5} + \frac{705}{5824} a^{4} - \frac{53}{224} a^{3} - \frac{17}{112} a^{2} - \frac{27}{56} a - \frac{11}{28}$, $\frac{1}{11648} a^{11} - \frac{9}{11648} a^{9} - \frac{141}{11648} a^{7} - \frac{215}{11648} a^{5} + \frac{11}{224} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{23296} a^{12} - \frac{1}{23296} a^{10} - \frac{1}{896} a^{9} - \frac{5}{23296} a^{8} + \frac{1}{448} a^{7} + \frac{1153}{23296} a^{6} - \frac{13}{896} a^{5} - \frac{339}{5824} a^{4} - \frac{53}{224} a^{3} + \frac{3}{112} a^{2} - \frac{27}{56} a + \frac{5}{14}$, $\frac{1}{46592} a^{13} - \frac{1}{46592} a^{12} - \frac{1}{46592} a^{11} - \frac{1}{46592} a^{10} - \frac{31}{46592} a^{9} - \frac{3}{46592} a^{8} + \frac{1205}{46592} a^{7} - \frac{25}{512} a^{6} - \frac{121}{3328} a^{5} + \frac{545}{5824} a^{4} - \frac{103}{448} a^{3} - \frac{1}{16} a^{2} + \frac{1}{16} a + \frac{1}{56}$, $\frac{1}{15421952} a^{14} + \frac{81}{7710976} a^{12} - \frac{1}{23296} a^{11} + \frac{17}{481936} a^{10} + \frac{9}{23296} a^{9} + \frac{1931}{7710976} a^{8} - \frac{587}{23296} a^{7} - \frac{390857}{15421952} a^{6} - \frac{1241}{23296} a^{5} + \frac{59711}{963872} a^{4} - \frac{81}{448} a^{3} - \frac{225}{2317} a^{2} - \frac{3}{16} a + \frac{6185}{18536}$, $\frac{1}{30843904} a^{15} + \frac{81}{15421952} a^{13} - \frac{1}{46592} a^{12} - \frac{15}{593152} a^{11} - \frac{1}{46592} a^{10} + \frac{16495}{15421952} a^{9} + \frac{7}{6656} a^{8} + \frac{725275}{30843904} a^{7} - \frac{183}{3584} a^{6} + \frac{91487}{1927744} a^{5} - \frac{197}{11648} a^{4} + \frac{5151}{37072} a^{3} + \frac{11}{224} a^{2} + \frac{12805}{37072} a + \frac{1}{4}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1227146937300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.152583041.1, 4.4.8975473.1, 8.8.23281584400807681.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
2389Data not computed