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Properties

Label 16.0.42511207857...0304.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 3^{8}\cdot 17^{6}$
Root discriminant $16.86$
Ramified primes $2, 3, 17$
Class number $1$
Class group Trivial
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![1, 2, 3, 6, 7, 8, 4, 0, -2, 0, 4, -8, 7, -6, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 7*x^12 - 8*x^11 + 4*x^10 - 2*x^8 + 4*x^6 + 8*x^5 + 7*x^4 + 6*x^3 + 3*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 + 3*x^14 - 6*x^13 + 7*x^12 - 8*x^11 + 4*x^10 - 2*x^8 + 4*x^6 + 8*x^5 + 7*x^4 + 6*x^3 + 3*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 3 x^{14} - 6 x^{13} + 7 x^{12} - 8 x^{11} + 4 x^{10} - 2 x^{8} + 4 x^{6} + 8 x^{5} + 7 x^{4} + 6 x^{3} + 3 x^{2} + 2 x + 1 \)

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:  \(42511207857918050304=2^{28}\cdot 3^{8}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.86$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{8} a^{13} - \frac{3}{16} a^{12} - \frac{3}{32} a^{11} - \frac{1}{64} a^{10} + \frac{3}{8} a^{9} + \frac{19}{64} a^{8} + \frac{3}{32} a^{7} + \frac{13}{64} a^{6} + \frac{3}{8} a^{5} - \frac{31}{64} a^{4} - \frac{3}{32} a^{3} - \frac{5}{16} a^{2} + \frac{3}{8} a - \frac{1}{64}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{1}{32} a^{13} - \frac{13}{64} a^{12} - \frac{11}{128} a^{11} + \frac{17}{128} a^{10} + \frac{27}{128} a^{9} + \frac{11}{128} a^{8} + \frac{23}{128} a^{7} - \frac{13}{128} a^{6} + \frac{41}{128} a^{5} + \frac{33}{128} a^{4} - \frac{15}{64} a^{3} + \frac{3}{32} a^{2} + \frac{7}{128} a + \frac{57}{128}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{115}{128} a^{15} + \frac{281}{128} a^{14} - \frac{121}{32} a^{13} + \frac{467}{64} a^{12} - \frac{1267}{128} a^{11} + \frac{1591}{128} a^{10} - \frac{1297}{128} a^{9} + \frac{737}{128} a^{8} - \frac{241}{128} a^{7} + \frac{197}{128} a^{6} - \frac{603}{128} a^{5} - \frac{621}{128} a^{4} - \frac{309}{64} a^{3} - \frac{119}{32} a^{2} - \frac{277}{128} a - \frac{129}{128} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10751.2019537 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), 4.4.9792.1, 4.0.1088.2, \(\Q(\sqrt{2}, \sqrt{-3})\), 8.0.6520061952.1, 8.0.80494592.1, 8.0.95883264.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[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}$