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Properties

Label 16.0.16620815899...6144.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{10}$
Root discriminant $15.90$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_2^2$ (as 16T128)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -64, -24, 184, 28, -96, 112, -48, -44, 48, -16, -8, 24, 0, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 24*x^12 - 8*x^11 - 16*x^10 + 48*x^9 - 44*x^8 - 48*x^7 + 112*x^6 - 96*x^5 + 28*x^4 + 184*x^3 - 24*x^2 - 64*x + 19)
 
gp: K = bnfinit(x^16 - 8*x^14 + 24*x^12 - 8*x^11 - 16*x^10 + 48*x^9 - 44*x^8 - 48*x^7 + 112*x^6 - 96*x^5 + 28*x^4 + 184*x^3 - 24*x^2 - 64*x + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} + 24 x^{12} - 8 x^{11} - 16 x^{10} + 48 x^{9} - 44 x^{8} - 48 x^{7} + 112 x^{6} - 96 x^{5} + 28 x^{4} + 184 x^{3} - 24 x^{2} - 64 x + 19 \)

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:  \(16620815899787526144=2^{48}\cdot 3^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.90$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{54} a^{14} + \frac{1}{27} a^{13} + \frac{1}{54} a^{12} + \frac{1}{18} a^{11} - \frac{1}{54} a^{10} - \frac{2}{27} a^{9} - \frac{1}{27} a^{8} + \frac{4}{9} a^{7} + \frac{1}{18} a^{6} + \frac{1}{9} a^{5} - \frac{23}{54} a^{4} + \frac{5}{54} a^{3} - \frac{5}{54} a^{2} - \frac{4}{27} a - \frac{10}{27}$, $\frac{1}{1429288406334} a^{15} - \frac{12700127167}{1429288406334} a^{14} - \frac{40625732641}{714644203167} a^{13} - \frac{13118422577}{476429468778} a^{12} - \frac{37856241919}{1429288406334} a^{11} - \frac{22993229234}{714644203167} a^{10} + \frac{217759724467}{1429288406334} a^{9} + \frac{35338902065}{476429468778} a^{8} + \frac{162796233073}{476429468778} a^{7} - \frac{36605832661}{476429468778} a^{6} + \frac{2180563040}{714644203167} a^{5} + \frac{33305215685}{1429288406334} a^{4} - \frac{153255058151}{1429288406334} a^{3} - \frac{37786614331}{714644203167} a^{2} + \frac{191851137349}{1429288406334} a + \frac{79107678523}{158809822926}$

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:  \( -\frac{153368}{2548269} a^{15} - \frac{40424}{2548269} a^{14} + \frac{1195376}{2548269} a^{13} + \frac{326813}{2548269} a^{12} - \frac{3421582}{2548269} a^{11} + \frac{81014}{849423} a^{10} + \frac{1967912}{2548269} a^{9} - \frac{12935237}{5096538} a^{8} + \frac{1796060}{849423} a^{7} + \frac{2597606}{849423} a^{6} - \frac{13842464}{2548269} a^{5} + \frac{11451502}{2548269} a^{4} - \frac{3499510}{2548269} a^{3} - \frac{8558044}{849423} a^{2} - \frac{612660}{283141} a + \frac{14347453}{5096538} \) (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:  \( 6607.64912558 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.0.3072.2, 4.0.3072.1, 8.0.84934656.1, 8.0.254803968.1, 8.0.254803968.2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$