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Properties

Label 16.16.2024866473...0176.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 449^{2}\cdot 1889^{2}$
Root discriminant $44.07$
Ramified primes $2, 449, 1889$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1720

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-32, -768, -1408, 4704, 9012, -7872, -15272, 5808, 10754, -2112, -3536, 328, 562, -16, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 16*x^13 + 562*x^12 + 328*x^11 - 3536*x^10 - 2112*x^9 + 10754*x^8 + 5808*x^7 - 15272*x^6 - 7872*x^5 + 9012*x^4 + 4704*x^3 - 1408*x^2 - 768*x - 32)
 
gp: K = bnfinit(x^16 - 40*x^14 - 16*x^13 + 562*x^12 + 328*x^11 - 3536*x^10 - 2112*x^9 + 10754*x^8 + 5808*x^7 - 15272*x^6 - 7872*x^5 + 9012*x^4 + 4704*x^3 - 1408*x^2 - 768*x - 32, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 16 x^{13} + 562 x^{12} + 328 x^{11} - 3536 x^{10} - 2112 x^{9} + 10754 x^{8} + 5808 x^{7} - 15272 x^{6} - 7872 x^{5} + 9012 x^{4} + 4704 x^{3} - 1408 x^{2} - 768 x - 32 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(202486647379893148425650176=2^{48}\cdot 449^{2}\cdot 1889^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.07$
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:  $2, 449, 1889$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{56} a^{14} + \frac{1}{28} a^{13} - \frac{1}{14} a^{12} + \frac{3}{14} a^{11} + \frac{5}{28} a^{10} + \frac{1}{14} a^{9} + \frac{3}{14} a^{8} - \frac{3}{28} a^{6} - \frac{1}{2} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{5}{14} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{2256010344055176464} a^{15} - \frac{187089683583813}{161143596003941176} a^{14} + \frac{12294767823228217}{564002586013794116} a^{13} - \frac{15558662333064797}{564002586013794116} a^{12} + \frac{12548803238453915}{161143596003941176} a^{11} - \frac{131012815059514007}{564002586013794116} a^{10} - \frac{29000770399010849}{141000646503448529} a^{9} - \frac{35221124670484863}{141000646503448529} a^{8} + \frac{16102871430170137}{1128005172027588232} a^{7} - \frac{153490642210247255}{564002586013794116} a^{6} - \frac{24608624713271407}{141000646503448529} a^{5} - \frac{124653995332977695}{282001293006897058} a^{4} - \frac{119134064941915795}{564002586013794116} a^{3} + \frac{84023168314949875}{282001293006897058} a^{2} + \frac{20884432365258600}{141000646503448529} a - \frac{68246010453148984}{141000646503448529}$

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:  $15$
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:  \( 512753317.509 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1720:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 116 conjugacy class representatives for t16n1720 are not computed
Character table for t16n1720 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.8.26.8$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
449Data not computed
1889Data not computed