↑

Properties

Label 16.8.15712755557...5489.5
Degree $16$
Signature $[8, 4]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![505613, -263472, -3939369, 8919920, -6133505, -1067350, 3353338, -1177018, -316281, 339490, -56878, -21932, 5302, 616, -130, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 130*x^14 + 616*x^13 + 5302*x^12 - 21932*x^11 - 56878*x^10 + 339490*x^9 - 316281*x^8 - 1177018*x^7 + 3353338*x^6 - 1067350*x^5 - 6133505*x^4 + 8919920*x^3 - 3939369*x^2 - 263472*x + 505613)
 
gp: K = bnfinit(x^16 - 6*x^15 - 130*x^14 + 616*x^13 + 5302*x^12 - 21932*x^11 - 56878*x^10 + 339490*x^9 - 316281*x^8 - 1177018*x^7 + 3353338*x^6 - 1067350*x^5 - 6133505*x^4 + 8919920*x^3 - 3939369*x^2 - 263472*x + 505613, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 130 x^{14} + 616 x^{13} + 5302 x^{12} - 21932 x^{11} - 56878 x^{10} + 339490 x^{9} - 316281 x^{8} - 1177018 x^{7} + 3353338 x^{6} - 1067350 x^{5} - 6133505 x^{4} + 8919920 x^{3} - 3939369 x^{2} - 263472 x + 505613 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{82} a^{8} + \frac{19}{41} a^{7} - \frac{4}{41} a^{6} + \frac{19}{41} a^{5} + \frac{27}{82} a^{4} + \frac{19}{41} a^{3} + \frac{33}{82} a^{2} - \frac{3}{82} a + \frac{1}{82}$, $\frac{1}{82} a^{9} + \frac{12}{41} a^{7} + \frac{7}{41} a^{6} - \frac{23}{82} a^{5} - \frac{2}{41} a^{4} - \frac{17}{82} a^{3} - \frac{27}{82} a^{2} + \frac{33}{82} a - \frac{19}{41}$, $\frac{1}{82} a^{10} + \frac{2}{41} a^{7} + \frac{5}{82} a^{6} - \frac{7}{41} a^{5} - \frac{9}{82} a^{4} - \frac{37}{82} a^{3} - \frac{21}{82} a^{2} + \frac{17}{41} a - \frac{12}{41}$, $\frac{1}{82} a^{11} + \frac{17}{82} a^{7} + \frac{9}{41} a^{6} + \frac{3}{82} a^{5} + \frac{19}{82} a^{4} - \frac{9}{82} a^{3} - \frac{8}{41} a^{2} - \frac{6}{41} a - \frac{2}{41}$, $\frac{1}{82} a^{12} + \frac{14}{41} a^{7} - \frac{25}{82} a^{6} + \frac{29}{82} a^{5} + \frac{12}{41} a^{4} - \frac{3}{41} a^{3} + \frac{1}{82} a^{2} - \frac{35}{82} a - \frac{17}{82}$, $\frac{1}{82} a^{13} - \frac{23}{82} a^{7} + \frac{7}{82} a^{6} + \frac{13}{41} a^{5} - \frac{12}{41} a^{4} + \frac{3}{82} a^{3} + \frac{25}{82} a^{2} - \frac{15}{82} a - \frac{14}{41}$, $\frac{1}{82} a^{14} - \frac{21}{82} a^{7} + \frac{3}{41} a^{6} + \frac{15}{41} a^{5} - \frac{16}{41} a^{4} - \frac{3}{82} a^{3} + \frac{3}{41} a^{2} - \frac{15}{82} a + \frac{23}{82}$, $\frac{1}{134754020331361509827040462063824766986} a^{15} + \frac{339104932535714599157241397271536873}{67377010165680754913520231031912383493} a^{14} - \frac{53023652559461715046287486886006083}{134754020331361509827040462063824766986} a^{13} - \frac{341661696164980278073204880241620794}{67377010165680754913520231031912383493} a^{12} + \frac{102974041849096942016213427678776722}{67377010165680754913520231031912383493} a^{11} - \frac{1815934589109074655627749630604089}{629691683791408924425422719924414799} a^{10} + \frac{544883152281199268167899608339794157}{134754020331361509827040462063824766986} a^{9} - \frac{232805676328798590526170544750558043}{67377010165680754913520231031912383493} a^{8} - \frac{17587612404718754637312755640542371795}{134754020331361509827040462063824766986} a^{7} + \frac{56802152245896742418279971832251681119}{134754020331361509827040462063824766986} a^{6} + \frac{65885980797248860420235036108474444359}{134754020331361509827040462063824766986} a^{5} - \frac{30098561055480653370428361062257410136}{67377010165680754913520231031912383493} a^{4} - \frac{33629788918914348720956081724935939420}{67377010165680754913520231031912383493} a^{3} - \frac{14605120114406003140809947287162137661}{67377010165680754913520231031912383493} a^{2} - \frac{14142591279310808688738207631125857669}{67377010165680754913520231031912383493} a - \frac{47357419155749619348061893625109882573}{134754020331361509827040462063824766986}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 662816664813 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41Data not computed