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Properties

Label 16.0.12812035518...7689.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 61^{12}$
Root discriminant $57.11$
Ramified primes $13, 61$
Class number $18$ (GRH)
Class group $[18]$ (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4225, 0, 34402, 0, 71813, 0, 63323, 0, 27490, 0, 6164, 0, 728, 0, 43, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 43*x^14 + 728*x^12 + 6164*x^10 + 27490*x^8 + 63323*x^6 + 71813*x^4 + 34402*x^2 + 4225)
 
gp: K = bnfinit(x^16 + 43*x^14 + 728*x^12 + 6164*x^10 + 27490*x^8 + 63323*x^6 + 71813*x^4 + 34402*x^2 + 4225, 1)
 

Normalized defining polynomial

\( x^{16} + 43 x^{14} + 728 x^{12} + 6164 x^{10} + 27490 x^{8} + 63323 x^{6} + 71813 x^{4} + 34402 x^{2} + 4225 \)

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:  \(12812035518280357547259227689=13^{6}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.11$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{7} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{3} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{2}{5} a^{6} - \frac{1}{2} a^{4} + \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{9} + \frac{1}{10} a^{7} - \frac{1}{2} a^{5} - \frac{2}{5} a^{3} - \frac{1}{2}$, $\frac{1}{2275322710} a^{14} - \frac{16293051}{1137661355} a^{12} + \frac{280711023}{1137661355} a^{10} - \frac{197006538}{1137661355} a^{8} - \frac{554103057}{2275322710} a^{6} - \frac{1}{2} a^{5} - \frac{323324947}{1137661355} a^{4} - \frac{1}{2} a^{3} - \frac{116155437}{2275322710} a^{2} + \frac{10877612}{227532271}$, $\frac{1}{29579195230} a^{15} + \frac{438771491}{14789597615} a^{13} + \frac{39095638}{1137661355} a^{11} + \frac{3929100073}{29579195230} a^{9} - \frac{2438608103}{14789597615} a^{7} - \frac{1}{2} a^{6} - \frac{444930727}{1137661355} a^{5} - \frac{1481349063}{29579195230} a^{3} - \frac{1}{2} a^{2} - \frac{9675111533}{29579195230} a - \frac{1}{2}$

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

Class group and class number

$C_{18}$, which has order $18$ (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:  \( 2309523.86989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{61}) \), 4.4.48373.1, 4.0.226981.1, 4.0.2950753.1, 8.8.113190262471117.1, 8.0.30419312677.1, 8.0.8706943267009.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$$0$Trivial$[\ ]$
$\Q_{13}$$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}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$