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Properties

Label 16.4.19738226851...0000.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{32}\cdot 5^{8}\cdot 7^{6}$
Root discriminant $18.55$
Ramified primes $2, 5, 7$
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, -4, 8, 0, -25, -48, -4, 96, 66, -52, -138, -88, -3, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 3*x^12 - 88*x^11 - 138*x^10 - 52*x^9 + 66*x^8 + 96*x^7 - 4*x^6 - 48*x^5 - 25*x^4 + 8*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^14 - 3*x^12 - 88*x^11 - 138*x^10 - 52*x^9 + 66*x^8 + 96*x^7 - 4*x^6 - 48*x^5 - 25*x^4 + 8*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 3 x^{12} - 88 x^{11} - 138 x^{10} - 52 x^{9} + 66 x^{8} + 96 x^{7} - 4 x^{6} - 48 x^{5} - 25 x^{4} + 8 x^{2} - 4 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(197382268518400000000=2^{32}\cdot 5^{8}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.55$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{46} a^{13} - \frac{11}{46} a^{12} + \frac{3}{23} a^{11} + \frac{1}{23} a^{10} - \frac{7}{23} a^{9} + \frac{4}{23} a^{8} + \frac{2}{23} a^{7} + \frac{4}{23} a^{6} + \frac{8}{23} a^{5} - \frac{1}{23} a^{4} - \frac{5}{23} a^{3} - \frac{7}{23} a^{2} - \frac{15}{46} a - \frac{17}{46}$, $\frac{1}{92} a^{14} - \frac{1}{4} a^{12} - \frac{6}{23} a^{11} - \frac{19}{46} a^{10} - \frac{2}{23} a^{9} - \frac{10}{23} a^{7} - \frac{17}{46} a^{6} + \frac{9}{23} a^{5} + \frac{7}{46} a^{4} - \frac{8}{23} a^{3} - \frac{31}{92} a^{2} - \frac{11}{23} a + \frac{43}{92}$, $\frac{1}{2314115468} a^{15} - \frac{3312735}{2314115468} a^{14} + \frac{23641809}{2314115468} a^{13} + \frac{26681951}{2314115468} a^{12} + \frac{187729447}{1157057734} a^{11} - \frac{3394369}{1157057734} a^{10} + \frac{103504780}{578528867} a^{9} - \frac{60194587}{578528867} a^{8} - \frac{96915103}{1157057734} a^{7} + \frac{4912359}{50306858} a^{6} - \frac{24297403}{50306858} a^{5} - \frac{492679415}{1157057734} a^{4} - \frac{943801507}{2314115468} a^{3} + \frac{807847777}{2314115468} a^{2} + \frac{644666123}{2314115468} a - \frac{57618443}{2314115468}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8439.59757225 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.2.11200.2, 4.2.448.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.2.14049280000.1, 8.2.22478848.1, 8.4.125440000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
5Data 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}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$