↑

Properties

Label 16.8.17555500852...1584.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{8}\cdot 41^{2}\cdot 47^{2}$
Root discriminant $50.44$
Ramified primes $2, 3, 41, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T797

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21598, 114928, 245072, 286640, 171092, 736, -80800, -58864, -10806, 8280, 5488, 632, -468, -80, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 80*x^13 - 468*x^12 + 632*x^11 + 5488*x^10 + 8280*x^9 - 10806*x^8 - 58864*x^7 - 80800*x^6 + 736*x^5 + 171092*x^4 + 286640*x^3 + 245072*x^2 + 114928*x + 21598)
 
gp: K = bnfinit(x^16 - 8*x^14 - 80*x^13 - 468*x^12 + 632*x^11 + 5488*x^10 + 8280*x^9 - 10806*x^8 - 58864*x^7 - 80800*x^6 + 736*x^5 + 171092*x^4 + 286640*x^3 + 245072*x^2 + 114928*x + 21598, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 80 x^{13} - 468 x^{12} + 632 x^{11} + 5488 x^{10} + 8280 x^{9} - 10806 x^{8} - 58864 x^{7} - 80800 x^{6} + 736 x^{5} + 171092 x^{4} + 286640 x^{3} + 245072 x^{2} + 114928 x + 21598 \)

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:  \(1755550085243509039467331584=2^{56}\cdot 3^{8}\cdot 41^{2}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.44$
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:  $2, 3, 41, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13489} a^{14} + \frac{557}{13489} a^{13} - \frac{970}{13489} a^{12} + \frac{1802}{13489} a^{11} - \frac{4090}{13489} a^{10} + \frac{728}{1927} a^{9} + \frac{1753}{13489} a^{8} - \frac{2544}{13489} a^{7} - \frac{1747}{13489} a^{6} + \frac{4128}{13489} a^{5} + \frac{4283}{13489} a^{4} - \frac{189}{1927} a^{3} - \frac{459}{13489} a^{2} - \frac{2084}{13489} a - \frac{1125}{13489}$, $\frac{1}{8408110600846307107591116007537} a^{15} + \frac{223217611164072955867461746}{8408110600846307107591116007537} a^{14} - \frac{4112711654105498969453021695163}{8408110600846307107591116007537} a^{13} - \frac{3599983012764421242059648893869}{8408110600846307107591116007537} a^{12} + \frac{2494593450274151058342160852785}{8408110600846307107591116007537} a^{11} + \frac{88606724106267756977051988500}{178895970230772491650874808671} a^{10} + \frac{1546406499897796393704167421979}{8408110600846307107591116007537} a^{9} + \frac{916271445734406246891775321892}{8408110600846307107591116007537} a^{8} + \frac{2908657424460993918509398566914}{8408110600846307107591116007537} a^{7} + \frac{1802025317982429244347221143960}{8408110600846307107591116007537} a^{6} + \frac{4089260170594615775590634007329}{8408110600846307107591116007537} a^{5} - \frac{2535045644894851381398900089733}{8408110600846307107591116007537} a^{4} - \frac{1048601313719820051415847892553}{8408110600846307107591116007537} a^{3} - \frac{71625228899061705265079189752}{1201158657263758158227302286791} a^{2} - \frac{3653740335244908657115927741129}{8408110600846307107591116007537} a + \frac{2961467283897027793930220620919}{8408110600846307107591116007537}$

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:  \( 32634577.3738 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n797 are not computed
Character table for t16n797 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.18432.1, \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\)

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$