↑

Properties

Label 16.4.10995116277...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{48}\cdot 5^{8}$
Root discriminant $17.89$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_2^2$ (as 16T128)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 56, -76, 8, 194, -256, 208, -80, 15, 24, -8, -16, 6, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 6*x^12 - 16*x^11 - 8*x^10 + 24*x^9 + 15*x^8 - 80*x^7 + 208*x^6 - 256*x^5 + 194*x^4 + 8*x^3 - 76*x^2 + 56*x - 1)
 
gp: K = bnfinit(x^16 - 4*x^14 + 6*x^12 - 16*x^11 - 8*x^10 + 24*x^9 + 15*x^8 - 80*x^7 + 208*x^6 - 256*x^5 + 194*x^4 + 8*x^3 - 76*x^2 + 56*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 6 x^{12} - 16 x^{11} - 8 x^{10} + 24 x^{9} + 15 x^{8} - 80 x^{7} + 208 x^{6} - 256 x^{5} + 194 x^{4} + 8 x^{3} - 76 x^{2} + 56 x - 1 \)

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:  \(109951162777600000000=2^{48}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.89$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{49} a^{14} - \frac{3}{49} a^{13} + \frac{15}{49} a^{12} + \frac{23}{49} a^{11} - \frac{11}{49} a^{10} + \frac{2}{49} a^{9} + \frac{23}{49} a^{8} + \frac{24}{49} a^{7} - \frac{23}{49} a^{6} - \frac{16}{49} a^{5} - \frac{23}{49} a^{4} - \frac{4}{49} a^{3} - \frac{24}{49} a^{2} - \frac{9}{49} a + \frac{5}{49}$, $\frac{1}{419925347464063} a^{15} - \frac{2572392323100}{419925347464063} a^{14} - \frac{3056356319185}{59989335352009} a^{13} - \frac{16612092683748}{59989335352009} a^{12} + \frac{112465867691603}{419925347464063} a^{11} + \frac{208159856840617}{419925347464063} a^{10} + \frac{49029936420596}{419925347464063} a^{9} - \frac{129755546237885}{419925347464063} a^{8} - \frac{110078494325255}{419925347464063} a^{7} - \frac{169244512304439}{419925347464063} a^{6} + \frac{55937893356340}{419925347464063} a^{5} - \frac{231648915931}{59989335352009} a^{4} + \frac{148724647135398}{419925347464063} a^{3} + \frac{148274016234593}{419925347464063} a^{2} + \frac{64724187417899}{419925347464063} a + \frac{49518757116378}{419925347464063}$

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:  \( 5523.66491371 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T128):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.2.25600.3, 4.2.1024.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.6.655360000.1, 8.4.655360000.2, 8.2.655360000.3

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$