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Properties

Label 16.4.16695593025...8809.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{6}\cdot 73^{12}$
Root discriminant $37.71$
Ramified primes $3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-164, -1016, 1645, 1698, -861, -1012, -522, -302, -626, -602, -252, -44, 6, -2, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 5*x^14 - 2*x^13 + 6*x^12 - 44*x^11 - 252*x^10 - 602*x^9 - 626*x^8 - 302*x^7 - 522*x^6 - 1012*x^5 - 861*x^4 + 1698*x^3 + 1645*x^2 - 1016*x - 164)
 
gp: K = bnfinit(x^16 + 5*x^14 - 2*x^13 + 6*x^12 - 44*x^11 - 252*x^10 - 602*x^9 - 626*x^8 - 302*x^7 - 522*x^6 - 1012*x^5 - 861*x^4 + 1698*x^3 + 1645*x^2 - 1016*x - 164, 1)
 

Normalized defining polynomial

\( x^{16} + 5 x^{14} - 2 x^{13} + 6 x^{12} - 44 x^{11} - 252 x^{10} - 602 x^{9} - 626 x^{8} - 302 x^{7} - 522 x^{6} - 1012 x^{5} - 861 x^{4} + 1698 x^{3} + 1645 x^{2} - 1016 x - 164 \)

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:  \(16695593025891398600698809=3^{6}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.71$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{5}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} + \frac{3}{16} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{32} a^{14} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} + \frac{1}{8} a^{6} - \frac{3}{8} a^{3} - \frac{3}{32} a^{2} + \frac{3}{8}$, $\frac{1}{4299727636153356224} a^{15} - \frac{51831831738953307}{4299727636153356224} a^{14} + \frac{50780815689988763}{2149863818076678112} a^{13} + \frac{7665949252374929}{537465954519169528} a^{12} - \frac{101933261212281779}{2149863818076678112} a^{11} - \frac{152150499386030213}{2149863818076678112} a^{10} - \frac{3103071726763309}{2149863818076678112} a^{9} + \frac{45183973576561045}{537465954519169528} a^{8} + \frac{256310820584981427}{2149863818076678112} a^{7} - \frac{160382335725901599}{1074931909038339056} a^{6} - \frac{384970250835341593}{2149863818076678112} a^{5} - \frac{29931298298509687}{2149863818076678112} a^{4} - \frac{1953677024493592615}{4299727636153356224} a^{3} + \frac{1430169915411292555}{4299727636153356224} a^{2} + \frac{419828422600901055}{1074931909038339056} a - \frac{189457296293957019}{1074931909038339056}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 113922542.516 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.2.15987.1, 4.4.389017.1, 4.2.1167051.1, 8.4.1362008036601.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73Data not computed