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Properties

Label 16.0.14030845564...1601.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{4}\cdot 53^{12}$
Root discriminant $37.30$
Ramified primes $13, 53$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2197, 2236, 637, -6361, 19273, -23713, 19783, -11494, 4913, -1233, -45, 245, -97, 13, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 + 13*x^13 - 97*x^12 + 245*x^11 - 45*x^10 - 1233*x^9 + 4913*x^8 - 11494*x^7 + 19783*x^6 - 23713*x^5 + 19273*x^4 - 6361*x^3 + 637*x^2 + 2236*x + 2197)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 + 13*x^13 - 97*x^12 + 245*x^11 - 45*x^10 - 1233*x^9 + 4913*x^8 - 11494*x^7 + 19783*x^6 - 23713*x^5 + 19273*x^4 - 6361*x^3 + 637*x^2 + 2236*x + 2197, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} + 13 x^{13} - 97 x^{12} + 245 x^{11} - 45 x^{10} - 1233 x^{9} + 4913 x^{8} - 11494 x^{7} + 19783 x^{6} - 23713 x^{5} + 19273 x^{4} - 6361 x^{3} + 637 x^{2} + 2236 x + 2197 \)

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:  \(14030845564476355702701601=13^{4}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.30$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{11} - \frac{1}{13} a^{10} - \frac{6}{13} a^{9} - \frac{4}{13} a^{8} - \frac{2}{13} a^{7} + \frac{3}{13} a^{6} + \frac{1}{13} a^{5} - \frac{2}{13} a^{4} + \frac{6}{13} a^{3} + \frac{6}{13} a^{2}$, $\frac{1}{13} a^{13} - \frac{4}{13} a^{11} - \frac{2}{13} a^{10} - \frac{6}{13} a^{9} + \frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{2}{13} a^{6} - \frac{6}{13} a^{5} + \frac{1}{13} a^{4} - \frac{5}{13} a^{3} + \frac{2}{13} a^{2}$, $\frac{1}{507} a^{14} - \frac{1}{169} a^{13} - \frac{16}{507} a^{12} + \frac{61}{169} a^{11} + \frac{103}{507} a^{10} - \frac{4}{13} a^{9} + \frac{82}{507} a^{8} - \frac{215}{507} a^{7} - \frac{230}{507} a^{6} - \frac{227}{507} a^{5} + \frac{66}{169} a^{4} - \frac{27}{169} a^{3} + \frac{19}{39} a^{2} + \frac{4}{13} a - \frac{1}{3}$, $\frac{1}{20119848290547738034839069} a^{15} - \frac{13280455519053962804146}{20119848290547738034839069} a^{14} - \frac{630076859474128998194011}{20119848290547738034839069} a^{13} - \frac{39232370664210335545184}{20119848290547738034839069} a^{12} + \frac{6402514101455955110461150}{20119848290547738034839069} a^{11} - \frac{7180324060824761501432953}{20119848290547738034839069} a^{10} + \frac{6724725976093698817198699}{20119848290547738034839069} a^{9} + \frac{803180889097583559749105}{6706616096849246011613023} a^{8} - \frac{1300530861313787240744624}{6706616096849246011613023} a^{7} - \frac{1053883993634713337308453}{6706616096849246011613023} a^{6} - \frac{3041578655486061385355083}{20119848290547738034839069} a^{5} + \frac{1697188865378182851832734}{6706616096849246011613023} a^{4} - \frac{3394418524372765129061684}{20119848290547738034839069} a^{3} + \frac{322391758978121521283732}{1547680637734441387295313} a^{2} - \frac{226648296616520048882722}{1547680637734441387295313} a + \frac{2111453424886469250403}{9157873596061783356777}$

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

Class group and class number

$C_{4}$, 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:  \( 259606.919408 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), 4.4.36517.1, 4.0.148877.1, 4.0.1935401.1, 8.4.70675038317.1 x2, 8.0.288136694677.1 x2, 8.0.3745777030801.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$$0$Trivial$[\ ]$
$\Q_{13}$$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$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$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$$[\ ]_{2}$
$53$53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.3.2$x^{4} - 212$$4$$1$$3$$C_4$$[\ ]_{4}$