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Properties

Label 16.8.68017932572...4864.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 3^{8}\cdot 17^{6}$
Root discriminant $20.05$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_4:D_4$ (as 16T265)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 56, 228, 344, 226, -24, -230, -272, -169, -8, 88, 72, 19, -12, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 12*x^13 + 19*x^12 + 72*x^11 + 88*x^10 - 8*x^9 - 169*x^8 - 272*x^7 - 230*x^6 - 24*x^5 + 226*x^4 + 344*x^3 + 228*x^2 + 56*x + 4)
 
gp: K = bnfinit(x^16 - 10*x^14 - 12*x^13 + 19*x^12 + 72*x^11 + 88*x^10 - 8*x^9 - 169*x^8 - 272*x^7 - 230*x^6 - 24*x^5 + 226*x^4 + 344*x^3 + 228*x^2 + 56*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 12 x^{13} + 19 x^{12} + 72 x^{11} + 88 x^{10} - 8 x^{9} - 169 x^{8} - 272 x^{7} - 230 x^{6} - 24 x^{5} + 226 x^{4} + 344 x^{3} + 228 x^{2} + 56 x + 4 \)

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:  \(680179325726688804864=2^{32}\cdot 3^{8}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.05$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{14} a^{13} + \frac{1}{14} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{1}{14} a^{9} + \frac{5}{14} a^{8} + \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{2} a^{5} - \frac{3}{14} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{14} a^{14} - \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{5}{14} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{14} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{4153501954} a^{15} - \frac{39037805}{2076750977} a^{14} + \frac{67588891}{4153501954} a^{13} + \frac{6967165}{133983934} a^{12} - \frac{1069322847}{4153501954} a^{11} + \frac{1035230788}{2076750977} a^{10} + \frac{23008845}{593357422} a^{9} + \frac{2045796003}{4153501954} a^{8} + \frac{1809551}{218605366} a^{7} + \frac{372915429}{2076750977} a^{6} - \frac{1033266651}{4153501954} a^{5} - \frac{1757697495}{4153501954} a^{4} - \frac{660443690}{2076750977} a^{3} + \frac{659869251}{2076750977} a^{2} - \frac{498635467}{2076750977} a - \frac{124593339}{2076750977}$

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:  \( 38499.520375 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:D_4$ (as 16T265):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2\times D_4:D_4$
Character table for $C_2\times D_4:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.4.321978368.1, 8.4.26080247808.2, 8.8.1534132224.1

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$