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Properties

Label 16.0.19987505807...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 139^{4}\cdot 181^{6}$
Root discriminant $381.33$
Ramified primes $2, 5, 29, 139, 181$
Class number $3269215744$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 25540748]$ (GRH)
Galois group 16T868

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10285190654933041, 0, 2611054203786213, 0, 205796436670039, 0, 6431469174712, 0, 82609767019, 0, 493166662, 0, 1449212, 0, 2000, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2000*x^14 + 1449212*x^12 + 493166662*x^10 + 82609767019*x^8 + 6431469174712*x^6 + 205796436670039*x^4 + 2611054203786213*x^2 + 10285190654933041)
 
gp: K = bnfinit(x^16 + 2000*x^14 + 1449212*x^12 + 493166662*x^10 + 82609767019*x^8 + 6431469174712*x^6 + 205796436670039*x^4 + 2611054203786213*x^2 + 10285190654933041, 1)
 

Normalized defining polynomial

\( x^{16} + 2000 x^{14} + 1449212 x^{12} + 493166662 x^{10} + 82609767019 x^{8} + 6431469174712 x^{6} + 205796436670039 x^{4} + 2611054203786213 x^{2} + 10285190654933041 \)

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:  \(199875058073008840373089541044249600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 139^{4}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $381.33$
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:  $2, 5, 29, 139, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{139} a^{10} + \frac{54}{139} a^{8} - \frac{2}{139} a^{6} - \frac{56}{139} a^{4} - \frac{50}{139} a^{2}$, $\frac{1}{139} a^{11} + \frac{54}{139} a^{9} - \frac{2}{139} a^{7} - \frac{56}{139} a^{5} - \frac{50}{139} a^{3}$, $\frac{1}{400620935} a^{12} - \frac{852572}{400620935} a^{10} - \frac{26981485}{80124187} a^{8} + \frac{98177034}{400620935} a^{6} - \frac{189665689}{400620935} a^{4} - \frac{22934}{99385} a^{2} - \frac{126}{715}$, $\frac{1}{400620935} a^{13} - \frac{852572}{400620935} a^{11} - \frac{26981485}{80124187} a^{9} + \frac{98177034}{400620935} a^{7} - \frac{189665689}{400620935} a^{5} - \frac{22934}{99385} a^{3} - \frac{126}{715} a$, $\frac{1}{2851861840677166499576489143466228656353982865105} a^{14} + \frac{1739008422562255704396398900526264740518}{2851861840677166499576489143466228656353982865105} a^{12} + \frac{6564868231090822403952486397068767095488124}{19668012694325286203975787196318818319682640449} a^{10} - \frac{592773458114137071897586363651723645459899209111}{2851861840677166499576489143466228656353982865105} a^{8} - \frac{30362880041414101311372670501461922952924517736}{98340063471626431019878935981594091598413202245} a^{6} - \frac{9808785526118390584070769920109046132534070111}{20516991659547960428607835564505242132043042195} a^{4} + \frac{2469770858117371827123366777241425865542889}{5089801949776224368297652087448584006956845} a^{2} + \frac{305702029914723410428364788965205207}{3678280282693866549326394738516550381}$, $\frac{1}{2851861840677166499576489143466228656353982865105} a^{15} + \frac{1739008422562255704396398900526264740518}{2851861840677166499576489143466228656353982865105} a^{13} + \frac{6564868231090822403952486397068767095488124}{19668012694325286203975787196318818319682640449} a^{11} - \frac{592773458114137071897586363651723645459899209111}{2851861840677166499576489143466228656353982865105} a^{9} - \frac{30362880041414101311372670501461922952924517736}{98340063471626431019878935981594091598413202245} a^{7} - \frac{9808785526118390584070769920109046132534070111}{20516991659547960428607835564505242132043042195} a^{5} + \frac{2469770858117371827123366777241425865542889}{5089801949776224368297652087448584006956845} a^{3} + \frac{305702029914723410428364788965205207}{3678280282693866549326394738516550381} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{25540748}$, which has order $3269215744$ (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:  \( 29196.3261178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T868:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n868 are not computed
Character table for t16n868 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.131225.1, 4.4.4525.1, 4.4.725.1, 8.8.17220000625.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.10$x^{8} + 2 x^{6} + 8 x^{3} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$139$139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
$181$181.4.2.2$x^{4} - 181 x^{2} + 589698$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
181.4.0.1$x^{4} - x + 54$$1$$4$$0$$C_4$$[\ ]^{4}$
181.8.4.1$x^{8} + 3538188 x^{4} - 5929741 x^{2} + 3129693580836$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$