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Properties

Label 16.0.81272760863...5561.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{6}\cdot 17^{14}$
Root discriminant $31.22$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -988, 7724, -6153, 10325, -9844, 9317, -7004, 5181, -3145, 1800, -853, 364, -125, 37, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 37*x^14 - 125*x^13 + 364*x^12 - 853*x^11 + 1800*x^10 - 3145*x^9 + 5181*x^8 - 7004*x^7 + 9317*x^6 - 9844*x^5 + 10325*x^4 - 6153*x^3 + 7724*x^2 - 988*x + 169)
 
gp: K = bnfinit(x^16 - 7*x^15 + 37*x^14 - 125*x^13 + 364*x^12 - 853*x^11 + 1800*x^10 - 3145*x^9 + 5181*x^8 - 7004*x^7 + 9317*x^6 - 9844*x^5 + 10325*x^4 - 6153*x^3 + 7724*x^2 - 988*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 37 x^{14} - 125 x^{13} + 364 x^{12} - 853 x^{11} + 1800 x^{10} - 3145 x^{9} + 5181 x^{8} - 7004 x^{7} + 9317 x^{6} - 9844 x^{5} + 10325 x^{4} - 6153 x^{3} + 7724 x^{2} - 988 x + 169 \)

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:  \(812727608637355438705561=13^{6}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.22$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{26} a^{12} - \frac{2}{13} a^{11} - \frac{5}{26} a^{10} - \frac{1}{13} a^{9} - \frac{3}{13} a^{8} + \frac{9}{26} a^{7} + \frac{4}{13} a^{6} + \frac{11}{26} a^{5} - \frac{6}{13} a^{4} + \frac{1}{13} a^{3} + \frac{1}{26} a^{2} + \frac{5}{13} a - \frac{1}{2}$, $\frac{1}{26} a^{13} + \frac{5}{26} a^{11} + \frac{2}{13} a^{10} + \frac{6}{13} a^{9} + \frac{11}{26} a^{8} - \frac{4}{13} a^{7} - \frac{9}{26} a^{6} + \frac{3}{13} a^{5} + \frac{3}{13} a^{4} + \frac{9}{26} a^{3} - \frac{6}{13} a^{2} + \frac{1}{26} a$, $\frac{1}{26} a^{14} - \frac{1}{13} a^{11} - \frac{1}{13} a^{10} - \frac{5}{26} a^{9} - \frac{2}{13} a^{8} - \frac{1}{13} a^{7} - \frac{4}{13} a^{6} - \frac{5}{13} a^{5} - \frac{9}{26} a^{4} + \frac{2}{13} a^{3} - \frac{2}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{243128935246418343902} a^{15} - \frac{1028665145046993409}{121564467623209171951} a^{14} - \frac{2750113227193133241}{243128935246418343902} a^{13} - \frac{3912853228115243983}{243128935246418343902} a^{12} + \frac{50857821111839839303}{243128935246418343902} a^{11} - \frac{27344137693634405974}{121564467623209171951} a^{10} + \frac{20553420830478565799}{121564467623209171951} a^{9} + \frac{117551027427983103181}{243128935246418343902} a^{8} + \frac{45677031047227893565}{243128935246418343902} a^{7} + \frac{28880880206664662687}{243128935246418343902} a^{6} - \frac{44003730279870893275}{121564467623209171951} a^{5} - \frac{7202282525836050132}{121564467623209171951} a^{4} + \frac{90854349123419121295}{243128935246418343902} a^{3} + \frac{1284229116109274177}{243128935246418343902} a^{2} - \frac{70621001390377806897}{243128935246418343902} a + \frac{691686448787371019}{18702225788186026454}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 75964.7116437 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.0.69347235737.1, 8.4.53030239093.1, 8.4.5334402749.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17Data not computed