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Properties

Label 16.0.19696584134...4729.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 29^{8}$
Root discriminant $50.80$
Ramified primes $13, 29$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16641, 0, 86536, 0, 29096, 0, -3961, 0, -684, 0, 459, 0, 80, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 8*x^14 + 80*x^12 + 459*x^10 - 684*x^8 - 3961*x^6 + 29096*x^4 + 86536*x^2 + 16641)
 
gp: K = bnfinit(x^16 + 8*x^14 + 80*x^12 + 459*x^10 - 684*x^8 - 3961*x^6 + 29096*x^4 + 86536*x^2 + 16641, 1)
 

Normalized defining polynomial

\( x^{16} + 8 x^{14} + 80 x^{12} + 459 x^{10} - 684 x^{8} - 3961 x^{6} + 29096 x^{4} + 86536 x^{2} + 16641 \)

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:  \(1969658413423416139858084729=13^{14}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.80$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{18} a^{10} - \frac{1}{18} a^{8} - \frac{7}{18} a^{6} - \frac{1}{2} a^{5} + \frac{4}{9} a^{4} - \frac{1}{2} a^{3} + \frac{2}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{9} - \frac{1}{18} a^{7} - \frac{1}{2} a^{6} - \frac{2}{9} a^{5} - \frac{1}{2} a^{4} - \frac{4}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{54} a^{12} + \frac{1}{54} a^{8} - \frac{4}{27} a^{6} - \frac{4}{9} a^{4} - \frac{1}{2} a^{3} - \frac{5}{54} a^{2} - \frac{1}{3}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{9} + \frac{1}{54} a^{7} - \frac{1}{2} a^{6} - \frac{5}{18} a^{5} - \frac{1}{2} a^{4} - \frac{23}{54} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{9973812439548} a^{14} - \frac{53189629241}{9973812439548} a^{12} + \frac{224498393839}{9973812439548} a^{10} - \frac{86316709474}{2493453109887} a^{8} - \frac{507944731501}{4986906219774} a^{6} - \frac{1}{2} a^{5} - \frac{1330464546929}{9973812439548} a^{4} - \frac{2051343019595}{9973812439548} a^{2} - \frac{1}{2} a + \frac{175864207807}{1108201382172}$, $\frac{1}{857747869801128} a^{15} - \frac{1}{19947624879096} a^{14} + \frac{7519519815601}{857747869801128} a^{13} - \frac{131510601121}{19947624879096} a^{12} + \frac{14077015670989}{857747869801128} a^{11} + \frac{329602297247}{19947624879096} a^{10} - \frac{5399465035235}{107218483725141} a^{9} - \frac{49191760444}{2493453109887} a^{8} + \frac{2204854225445}{107218483725141} a^{7} - \frac{173151691463}{2493453109887} a^{6} - \frac{165344269108385}{857747869801128} a^{5} + \frac{222263164757}{19947624879096} a^{4} - \frac{217227111391325}{857747869801128} a^{3} - \frac{4782565503799}{19947624879096} a^{2} - \frac{6842744545949}{95305318866792} a - \frac{914665129255}{2216402764344}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 2825523.54935 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.63713.1, 4.0.1847677.1, 4.0.4901.1, 8.4.44380833852277.1, 8.4.52771502797.1, 8.0.3413910296329.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.2$x^{8} - 52$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$