↑

Properties

Label 16.0.27231268499...8081.5
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 43^{8}$
Root discriminant $44.89$
Ramified primes $13, 43$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![439569, 0, 363857, 0, 181649, 0, 57967, 0, 13789, 0, 2333, 0, 294, 0, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 + 294*x^12 + 2333*x^10 + 13789*x^8 + 57967*x^6 + 181649*x^4 + 363857*x^2 + 439569)
 
gp: K = bnfinit(x^16 + 24*x^14 + 294*x^12 + 2333*x^10 + 13789*x^8 + 57967*x^6 + 181649*x^4 + 363857*x^2 + 439569, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} + 294 x^{12} + 2333 x^{10} + 13789 x^{8} + 57967 x^{6} + 181649 x^{4} + 363857 x^{2} + 439569 \)

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:  \(272312684996154152285848081=13^{12}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.89$
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:  $13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{286} a^{12} - \frac{27}{143} a^{10} + \frac{21}{286} a^{8} - \frac{1}{2} a^{7} - \frac{36}{143} a^{6} - \frac{67}{143} a^{4} - \frac{1}{2} a^{3} + \frac{7}{22} a^{2} - \frac{1}{2} a + \frac{2}{11}$, $\frac{1}{4862} a^{13} + \frac{89}{4862} a^{11} - \frac{551}{4862} a^{9} + \frac{1215}{4862} a^{7} - \frac{1}{2} a^{6} - \frac{277}{4862} a^{5} + \frac{75}{187} a^{3} - \frac{1}{2} a^{2} - \frac{64}{187} a$, $\frac{1}{15652063323182} a^{14} - \frac{9650933828}{7826031661591} a^{12} + \frac{952398666903}{7826031661591} a^{10} + \frac{18909242585}{109454988274} a^{8} - \frac{1}{2} a^{7} - \frac{496818893169}{7826031661591} a^{6} - \frac{1898014945858}{7826031661591} a^{4} - \frac{39400334759}{1204004871014} a^{2} + \frac{28415766705}{70823815942}$, $\frac{1}{46956189969546} a^{15} + \frac{63391821}{920709607246} a^{13} + \frac{1956071782050}{7826031661591} a^{11} - \frac{4856340475361}{23478094984773} a^{9} - \frac{579270577454}{2134372271343} a^{7} - \frac{100552355732}{2134372271343} a^{5} - \frac{1}{2} a^{4} + \frac{165407533378}{1806007306521} a^{3} - \frac{1}{2} a^{2} + \frac{6616908557}{3612014613042} a$

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:  \( 1809127.9607 \) (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

\(\Q(\sqrt{13}) \), 4.2.7267.1, 4.0.2197.1, 4.2.94471.1, 8.0.1269376879693.1, 8.4.1269376879693.1, 8.0.8924769841.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$43$43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
43.4.2.1$x^{4} + 215 x^{2} + 16641$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
43.8.6.1$x^{8} - 301 x^{4} + 149769$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$