↑

Properties

Label 16.0.19112453149...1536.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 11^{8}$
Root discriminant $21.38$
Ramified primes $2, 3, 11$
Class number $2$
Class group $[2]$
Galois group $C_8:C_2^2$ (as 16T35)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -32, 12, 56, -42, 72, 193, -64, -60, 34, -23, 48, 9, -14, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 14*x^13 + 9*x^12 + 48*x^11 - 23*x^10 + 34*x^9 - 60*x^8 - 64*x^7 + 193*x^6 + 72*x^5 - 42*x^4 + 56*x^3 + 12*x^2 - 32*x + 64)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 14*x^13 + 9*x^12 + 48*x^11 - 23*x^10 + 34*x^9 - 60*x^8 - 64*x^7 + 193*x^6 + 72*x^5 - 42*x^4 + 56*x^3 + 12*x^2 - 32*x + 64, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 14 x^{13} + 9 x^{12} + 48 x^{11} - 23 x^{10} + 34 x^{9} - 60 x^{8} - 64 x^{7} + 193 x^{6} + 72 x^{5} - 42 x^{4} + 56 x^{3} + 12 x^{2} - 32 x + 64 \)

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:  \(1911245314971754561536=2^{24}\cdot 3^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.38$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{8} - \frac{5}{16} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{7347955664937248} a^{15} - \frac{106204589485129}{7347955664937248} a^{14} - \frac{54376887202327}{7347955664937248} a^{13} - \frac{438326101194461}{7347955664937248} a^{12} - \frac{199577625980227}{1836988916234312} a^{11} + \frac{719913433328893}{3673977832468624} a^{10} + \frac{390762625815299}{7347955664937248} a^{9} + \frac{122068898017975}{7347955664937248} a^{8} + \frac{2259940909185387}{7347955664937248} a^{7} + \frac{1937421991830687}{7347955664937248} a^{6} - \frac{347296308933929}{1836988916234312} a^{5} + \frac{1561146795044525}{3673977832468624} a^{4} - \frac{45921053500095}{229623614529289} a^{3} - \frac{17287623935985}{44804607713032} a^{2} - \frac{129513865033165}{918494458117156} a + \frac{132324013898}{229623614529289}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 95377.382067 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-66}) \), 4.2.8712.2 x2, 4.0.2112.1 x2, \(\Q(\sqrt{-2}, \sqrt{33})\), 8.2.5464723968.1 x2, 8.2.5464723968.2 x2, 8.0.4857532416.8

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 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$