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Properties

Label 16.0.95344240396...0321.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 43^{8}$
Root discriminant $23.64$
Ramified primes $13, 43$
Class number $2$
Class group $[2]$
Galois group $D_{8}$ (as 16T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1521, 3549, 5642, 3822, 64, -3761, -1950, 843, 552, -92, -10, -59, 27, 21, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 21*x^13 + 27*x^12 - 59*x^11 - 10*x^10 - 92*x^9 + 552*x^8 + 843*x^7 - 1950*x^6 - 3761*x^5 + 64*x^4 + 3822*x^3 + 5642*x^2 + 3549*x + 1521)
 
gp: K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 21*x^13 + 27*x^12 - 59*x^11 - 10*x^10 - 92*x^9 + 552*x^8 + 843*x^7 - 1950*x^6 - 3761*x^5 + 64*x^4 + 3822*x^3 + 5642*x^2 + 3549*x + 1521, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 5 x^{14} + 21 x^{13} + 27 x^{12} - 59 x^{11} - 10 x^{10} - 92 x^{9} + 552 x^{8} + 843 x^{7} - 1950 x^{6} - 3761 x^{5} + 64 x^{4} + 3822 x^{3} + 5642 x^{2} + 3549 x + 1521 \)

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:  \(9534424039639863880321=13^{8}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.64$
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:  $13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{39} a^{10} + \frac{2}{39} a^{9} + \frac{5}{39} a^{8} - \frac{19}{39} a^{7} - \frac{7}{39} a^{6} + \frac{5}{39} a^{5} - \frac{16}{39} a^{4} + \frac{5}{39} a^{3} + \frac{10}{39} a^{2}$, $\frac{1}{39} a^{11} + \frac{1}{39} a^{9} - \frac{1}{13} a^{8} + \frac{6}{13} a^{7} + \frac{2}{13} a^{6} - \frac{5}{13} a^{4} - \frac{1}{3} a^{3} + \frac{2}{13} a^{2} - \frac{1}{3} a$, $\frac{1}{39} a^{12} - \frac{5}{39} a^{9} + \frac{4}{13} a^{7} - \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{10}{39} a^{4} - \frac{4}{13} a^{3} + \frac{1}{13} a^{2} - \frac{1}{3} a$, $\frac{1}{117} a^{13} - \frac{1}{117} a^{10} - \frac{5}{117} a^{9} + \frac{19}{117} a^{8} - \frac{56}{117} a^{7} + \frac{43}{117} a^{6} + \frac{4}{13} a^{5} + \frac{28}{117} a^{4} + \frac{10}{117} a^{3} + \frac{14}{117} a^{2} + \frac{1}{3} a$, $\frac{1}{1287} a^{14} + \frac{4}{1287} a^{13} - \frac{2}{429} a^{12} - \frac{16}{1287} a^{11} + \frac{5}{429} a^{10} + \frac{101}{1287} a^{9} - \frac{10}{1287} a^{8} - \frac{4}{1287} a^{7} - \frac{443}{1287} a^{6} + \frac{529}{1287} a^{5} - \frac{406}{1287} a^{4} + \frac{82}{429} a^{3} - \frac{553}{1287} a^{2} + \frac{5}{33} a - \frac{3}{11}$, $\frac{1}{81132291731390885271} a^{15} + \frac{704204005648358}{2458554294890632887} a^{14} + \frac{20471286403872851}{7375662884671898661} a^{13} - \frac{567232580702922544}{81132291731390885271} a^{12} + \frac{493687226493027283}{81132291731390885271} a^{11} - \frac{97637798685958807}{27044097243796961757} a^{10} - \frac{3794107082196914512}{81132291731390885271} a^{9} + \frac{7018005838894773308}{81132291731390885271} a^{8} + \frac{5371755096169755790}{81132291731390885271} a^{7} - \frac{28004065838725071874}{81132291731390885271} a^{6} + \frac{17747082789511078792}{81132291731390885271} a^{5} - \frac{335765412864271940}{2458554294890632887} a^{4} - \frac{15097207982375955092}{81132291731390885271} a^{3} + \frac{39812909459496801065}{81132291731390885271} a^{2} + \frac{292904678212722151}{2080315172599766289} a + \frac{32532332650457641}{693438390866588763}$

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:  \( 26474.6238892 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 16T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-559}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-43}) \), \(\Q(\sqrt{13}, \sqrt{-43})\), 4.2.7267.1 x2, 4.0.24037.1 x2, 8.0.97644375361.1, 8.2.2270799427.1 x4, 8.0.7511105797.1 x4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$