↑

Properties

Label 16.0.15795415826...7616.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 11^{6}$
Root discriminant $15.85$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -24, 66, -132, 229, -300, 354, -348, 294, -216, 138, -72, 34, -12, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^13 + 34*x^12 - 72*x^11 + 138*x^10 - 216*x^9 + 294*x^8 - 348*x^7 + 354*x^6 - 300*x^5 + 229*x^4 - 132*x^3 + 66*x^2 - 24*x + 4)
 
gp: K = bnfinit(x^16 - 12*x^13 + 34*x^12 - 72*x^11 + 138*x^10 - 216*x^9 + 294*x^8 - 348*x^7 + 354*x^6 - 300*x^5 + 229*x^4 - 132*x^3 + 66*x^2 - 24*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{13} + 34 x^{12} - 72 x^{11} + 138 x^{10} - 216 x^{9} + 294 x^{8} - 348 x^{7} + 354 x^{6} - 300 x^{5} + 229 x^{4} - 132 x^{3} + 66 x^{2} - 24 x + 4 \)

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:  \(15795415826212847616=2^{24}\cdot 3^{12}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.85$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{330660712} a^{15} + \frac{31857673}{330660712} a^{14} + \frac{21892357}{330660712} a^{13} + \frac{56933489}{330660712} a^{12} + \frac{66357855}{330660712} a^{11} - \frac{1904729}{330660712} a^{10} - \frac{25054935}{330660712} a^{9} + \frac{51582045}{330660712} a^{8} + \frac{43176603}{330660712} a^{7} - \frac{137724393}{330660712} a^{6} + \frac{139790933}{330660712} a^{5} - \frac{27979727}{330660712} a^{4} + \frac{15909155}{165330356} a^{3} - \frac{63135085}{165330356} a^{2} + \frac{25819217}{82665178} a - \frac{7074601}{82665178}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{62525}{103916} a^{15} - \frac{8965}{51958} a^{14} + \frac{15789}{103916} a^{13} + \frac{187395}{25979} a^{12} - \frac{1926675}{103916} a^{11} + \frac{926033}{25979} a^{10} - \frac{6778539}{103916} a^{9} + \frac{5030839}{51958} a^{8} - \frac{12850719}{103916} a^{7} + \frac{7003745}{51958} a^{6} - \frac{12954459}{103916} a^{5} + \frac{4753151}{51958} a^{4} - \frac{3282793}{51958} a^{3} + \frac{1436277}{51958} a^{2} - \frac{311955}{25979} a + \frac{86409}{25979} \) (order $6$)
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:  \( 6419.52605547 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), 4.0.2112.1, 4.0.2112.2, \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.0.40144896.1

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 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 ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.2$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}$
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$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$