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Properties

Label 16.4.82017942110...9584.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{66}\cdot 17^{4}\cdot 191^{4}$
Root discriminant $131.71$
Ramified primes $2, 17, 191$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24137569, 0, 94604728, 0, 32570300, 0, -523736, 0, -728694, 0, -57096, 0, -876, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 - 876*x^12 - 57096*x^10 - 728694*x^8 - 523736*x^6 + 32570300*x^4 + 94604728*x^2 + 24137569)
 
gp: K = bnfinit(x^16 + 40*x^14 - 876*x^12 - 57096*x^10 - 728694*x^8 - 523736*x^6 + 32570300*x^4 + 94604728*x^2 + 24137569, 1)
 

Normalized defining polynomial

\( x^{16} + 40 x^{14} - 876 x^{12} - 57096 x^{10} - 728694 x^{8} - 523736 x^{6} + 32570300 x^{4} + 94604728 x^{2} + 24137569 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8201794211074936440642419329859584=2^{66}\cdot 17^{4}\cdot 191^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.71$
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:  $2, 17, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{102} a^{8} + \frac{25}{51} a^{6} - \frac{1}{51} a^{4} + \frac{2}{51} a^{2} + \frac{1}{6}$, $\frac{1}{204} a^{9} - \frac{1}{204} a^{8} + \frac{25}{102} a^{7} - \frac{25}{102} a^{6} - \frac{1}{102} a^{5} + \frac{1}{102} a^{4} - \frac{49}{102} a^{3} + \frac{49}{102} a^{2} - \frac{5}{12} a + \frac{5}{12}$, $\frac{1}{204} a^{10} - \frac{1}{204} a^{8} + \frac{25}{51} a^{6} + \frac{1}{51} a^{4} + \frac{1}{12} a^{2} + \frac{1}{4}$, $\frac{1}{204} a^{11} - \frac{1}{204} a^{8} - \frac{9}{34} a^{7} - \frac{25}{102} a^{6} + \frac{1}{102} a^{5} + \frac{1}{102} a^{4} - \frac{27}{68} a^{3} + \frac{49}{102} a^{2} - \frac{1}{6} a + \frac{5}{12}$, $\frac{1}{3468} a^{12} + \frac{1}{578} a^{10} - \frac{3}{1156} a^{8} - \frac{232}{867} a^{6} - \frac{257}{1156} a^{4} + \frac{11}{102} a^{2} + \frac{1}{4}$, $\frac{1}{3468} a^{13} + \frac{1}{578} a^{11} + \frac{2}{867} a^{9} - \frac{1}{204} a^{8} - \frac{13}{578} a^{7} - \frac{25}{102} a^{6} - \frac{805}{3468} a^{5} + \frac{1}{102} a^{4} - \frac{19}{51} a^{3} + \frac{49}{102} a^{2} - \frac{1}{6} a + \frac{5}{12}$, $\frac{1}{1052713953699307699692} a^{14} + \frac{5903936964522355}{263178488424826924923} a^{12} - \frac{82969689024311445}{175452325616551283282} a^{10} - \frac{168378163750104647}{350904651233102566564} a^{8} - \frac{294734837092978966163}{1052713953699307699692} a^{6} - \frac{919502863842860211}{5160362518133861273} a^{4} + \frac{207549512322603483}{607101472721630738} a^{2} + \frac{59972141713472083}{214271108019399084}$, $\frac{1}{17896137212888230894764} a^{15} + \frac{5903936964522355}{4474034303222057723691} a^{13} + \frac{25303994456523437695}{17896137212888230894764} a^{11} - \frac{13153473540959810153}{8948068606444115447382} a^{9} - \frac{872695439123971428739}{17896137212888230894764} a^{7} - \frac{63165105127330839064}{263178488424826924923} a^{5} + \frac{28261312610048188739}{61924350217606335276} a^{3} + \frac{378176621388259553}{1821304418164892214} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  $9$
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:  \( 4721707978.18 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.195584.1, \(\Q(\zeta_{16})^+\), 4.4.391168.1, 8.8.2448198467584.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
191Data not computed