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Properties

Label 16.0.23807523437...5216.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 7^{4}\cdot 137^{4}$
Root discriminant $44.52$
Ramified primes $2, 7, 137$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group 16T969

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83521, 0, 5780, 0, 11642, 0, -1320, 0, 499, 0, -216, 0, 26, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 26*x^12 - 216*x^10 + 499*x^8 - 1320*x^6 + 11642*x^4 + 5780*x^2 + 83521)
 
gp: K = bnfinit(x^16 - 4*x^14 + 26*x^12 - 216*x^10 + 499*x^8 - 1320*x^6 + 11642*x^4 + 5780*x^2 + 83521, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 26 x^{12} - 216 x^{10} + 499 x^{8} - 1320 x^{6} + 11642 x^{4} + 5780 x^{2} + 83521 \)

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:  \(238075234378744256807305216=2^{48}\cdot 7^{4}\cdot 137^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.52$
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:  $2, 7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{24} a^{10} - \frac{1}{8} a^{9} - \frac{1}{24} a^{8} - \frac{1}{4} a^{7} + \frac{5}{24} a^{6} + \frac{1}{8} a^{5} + \frac{5}{24} a^{4} + \frac{1}{4} a^{3} + \frac{11}{24} a^{2} - \frac{1}{8} a - \frac{11}{24}$, $\frac{1}{24} a^{11} + \frac{1}{12} a^{9} - \frac{1}{8} a^{8} - \frac{1}{24} a^{7} - \frac{1}{4} a^{6} + \frac{1}{12} a^{5} + \frac{1}{8} a^{4} + \frac{5}{24} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a - \frac{1}{8}$, $\frac{1}{72} a^{12} - \frac{1}{8} a^{9} - \frac{5}{72} a^{8} - \frac{1}{4} a^{7} + \frac{2}{9} a^{6} + \frac{1}{8} a^{5} - \frac{11}{72} a^{4} + \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{8} a + \frac{2}{9}$, $\frac{1}{1224} a^{13} - \frac{7}{408} a^{11} - \frac{55}{612} a^{9} - \frac{233}{1224} a^{7} - \frac{31}{612} a^{5} - \frac{151}{408} a^{3} - \frac{275}{1224} a$, $\frac{1}{49492369008} a^{14} - \frac{1}{2448} a^{13} - \frac{7641883}{6186546126} a^{12} + \frac{7}{816} a^{11} + \frac{166096129}{49492369008} a^{10} + \frac{55}{1224} a^{9} - \frac{48453667}{49492369008} a^{8} + \frac{233}{2448} a^{7} + \frac{10880308913}{49492369008} a^{6} - \frac{275}{1224} a^{5} - \frac{4312077551}{49492369008} a^{4} - \frac{53}{816} a^{3} + \frac{4665675121}{12373092252} a^{2} + \frac{887}{2448} a + \frac{5262649}{171253872}$, $\frac{1}{841370273136} a^{15} + \frac{282561943}{841370273136} a^{13} - \frac{1}{144} a^{12} - \frac{216248723}{210342568284} a^{11} - \frac{1}{48} a^{10} - \frac{99720585697}{841370273136} a^{9} + \frac{1}{18} a^{8} + \frac{5899274543}{210342568284} a^{7} + \frac{5}{144} a^{6} - \frac{75113660993}{841370273136} a^{5} + \frac{2}{9} a^{4} - \frac{291695696837}{841370273136} a^{3} - \frac{3}{16} a^{2} - \frac{54897485}{181957239} a + \frac{17}{144}$

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

Class group and class number

$C_{8}$, 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:  $7$
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:  \( 453515.357319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T969:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n969
Character table for t16n969 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7168.1, 8.4.112625451008.2, 8.4.112625451008.1, 8.0.964355424256.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
137.4.2.1$x^{4} + 1507 x^{2} + 675684$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$