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Properties

Label 16.8.16239385543...3136.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{26}\cdot 193^{8}\cdot 257^{10}$
Root discriminant $1374.55$
Ramified primes $2, 193, 257$
Class number $96$ (GRH)
Class group $[2, 2, 2, 12]$ (GRH)
Galois group 16T813

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6668459216896, 0, 10352687231872, 0, 533135995896, 0, -9719978496, 0, -558556815, 0, 3513246, 0, 138966, 0, -862, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 862*x^14 + 138966*x^12 + 3513246*x^10 - 558556815*x^8 - 9719978496*x^6 + 533135995896*x^4 + 10352687231872*x^2 + 6668459216896)
 
gp: K = bnfinit(x^16 - 862*x^14 + 138966*x^12 + 3513246*x^10 - 558556815*x^8 - 9719978496*x^6 + 533135995896*x^4 + 10352687231872*x^2 + 6668459216896, 1)
 

Normalized defining polynomial

\( x^{16} - 862 x^{14} + 138966 x^{12} + 3513246 x^{10} - 558556815 x^{8} - 9719978496 x^{6} + 533135995896 x^{4} + 10352687231872 x^{2} + 6668459216896 \)

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:  \(162393855437479514050983820757859401106269862363136=2^{26}\cdot 193^{8}\cdot 257^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1374.55$
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:  $2, 193, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4}$, $\frac{1}{6168} a^{12} - \frac{29}{514} a^{10} + \frac{175}{1542} a^{8} + \frac{271}{1542} a^{6} - \frac{1753}{6168} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{24672} a^{13} + \frac{199}{4112} a^{11} - \frac{421}{12336} a^{9} + \frac{2855}{12336} a^{7} + \frac{9041}{24672} a^{5} - \frac{1}{4} a^{3} - \frac{1}{12} a$, $\frac{1}{900614660400475698366407441276170135333248} a^{14} - \frac{30547702017978721340870559817029236695}{450307330200237849183203720638085067666624} a^{12} - \frac{29391572333653099368545406673591102777717}{450307330200237849183203720638085067666624} a^{10} + \frac{39762273919311351916418114870557917573967}{450307330200237849183203720638085067666624} a^{8} - \frac{55224895245855526312055001734549050332485}{300204886800158566122135813758723378444416} a^{6} - \frac{5814750805608738761557054776475757240911}{56288416275029731147900465079760633458328} a^{4} + \frac{187854810705341079254924303239293231959}{438042150000231370800781829414479637808} a^{2} - \frac{9633177427525684996927968819051530101}{27377634375014460675048864338404977363}$, $\frac{1}{1131172013462997477148207746242869689978559488} a^{15} - \frac{1}{1801229320800951396732814882552340270666496} a^{14} - \frac{1052646052018518586542694828450815058247}{565586006731498738574103873121434844989279744} a^{13} + \frac{30547702017978721340870559817029236695}{900614660400475698366407441276170135333248} a^{12} + \frac{9895329420221589068864768502370274051038139}{565586006731498738574103873121434844989279744} a^{11} - \frac{83185260216406362927255523485930164138939}{900614660400475698366407441276170135333248} a^{10} + \frac{7617617143960453688699178510055617542689557}{188528668910499579524701291040478281663093248} a^{9} - \frac{152339106469370814212219045030079184490623}{900614660400475698366407441276170135333248} a^{8} + \frac{41523380785984453288668642742611599285373649}{1131172013462997477148207746242869689978559488} a^{7} + \frac{130276116945895167842588955174229894943589}{600409773600317132244271627517446756888832} a^{6} + \frac{9744967004687041555982946402837389773728685}{70698250841437342321762984140179355623659968} a^{5} + \frac{48031063011881037122482403586296232334657}{112576832550059462295800930159521266916656} a^{4} - \frac{233069590064417863872161399859971113900801}{550180940400290601725781977744586425086848} a^{3} - \frac{406875885705456764655315217946533050863}{876084300000462741601563658828959275616} a^{2} - \frac{8003648765280487769725240515280179984607}{17193154387509081303930686804518325783964} a - \frac{8872228473744387839060447759676723631}{27377634375014460675048864338404977363}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{12}$, which has order $96$ (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:  \( 35530711807900000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T813:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 38 conjugacy class representatives for t16n813
Character table for t16n813 is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), \(\Q(\sqrt{193}) \), \(\Q(\sqrt{49601}) \), 4.4.19682073608.1, 4.4.528392.1, \(\Q(\sqrt{193}, \sqrt{257})\), 8.8.387384021510730137664.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.11.15$x^{4} + 30$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.10.2$x^{4} + 2 x^{2} - 1$$4$$1$$10$$D_{4}$$[2, 3, 7/2]$
193Data not computed
257Data not computed