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Properties

Label 16.4.19718090210...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{8}\cdot 11^{6}\cdot 19^{8}$
Root discriminant $67.75$
Ramified primes $2, 5, 11, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T868

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6401861, -8868558, 4315806, -816582, 4669, 1009684, -649026, -101524, 170225, 19436, -15006, -2986, 409, 238, -14, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 14*x^14 + 238*x^13 + 409*x^12 - 2986*x^11 - 15006*x^10 + 19436*x^9 + 170225*x^8 - 101524*x^7 - 649026*x^6 + 1009684*x^5 + 4669*x^4 - 816582*x^3 + 4315806*x^2 - 8868558*x + 6401861)
 
gp: K = bnfinit(x^16 - 8*x^15 - 14*x^14 + 238*x^13 + 409*x^12 - 2986*x^11 - 15006*x^10 + 19436*x^9 + 170225*x^8 - 101524*x^7 - 649026*x^6 + 1009684*x^5 + 4669*x^4 - 816582*x^3 + 4315806*x^2 - 8868558*x + 6401861, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 14 x^{14} + 238 x^{13} + 409 x^{12} - 2986 x^{11} - 15006 x^{10} + 19436 x^{9} + 170225 x^{8} - 101524 x^{7} - 649026 x^{6} + 1009684 x^{5} + 4669 x^{4} - 816582 x^{3} + 4315806 x^{2} - 8868558 x + 6401861 \)

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:  \(197180902109852473753600000000=2^{24}\cdot 5^{8}\cdot 11^{6}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.75$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{18455} a^{14} - \frac{1234}{18455} a^{13} + \frac{479}{18455} a^{12} + \frac{527}{3691} a^{11} + \frac{496}{18455} a^{10} + \frac{1893}{18455} a^{9} + \frac{482}{3691} a^{8} - \frac{1411}{3691} a^{7} - \frac{329}{3691} a^{6} + \frac{8541}{18455} a^{5} - \frac{8827}{18455} a^{4} + \frac{1547}{3691} a^{3} + \frac{4543}{18455} a^{2} + \frac{484}{3691} a + \frac{5858}{18455}$, $\frac{1}{60301261307853786677177938555464055306539776423575} a^{15} - \frac{1216519844303881633101203979285567466058983837}{60301261307853786677177938555464055306539776423575} a^{14} + \frac{3868050126510316032789025035726300056524581845004}{60301261307853786677177938555464055306539776423575} a^{13} - \frac{3739398100603773018427944282465374696200240597103}{60301261307853786677177938555464055306539776423575} a^{12} - \frac{27569452981814896787389653353852096965254154072729}{60301261307853786677177938555464055306539776423575} a^{11} - \frac{4345853367570584424349316811689289314892300699292}{12060252261570757335435587711092811061307955284715} a^{10} - \frac{18930904112944946701166500987704942319634607282981}{60301261307853786677177938555464055306539776423575} a^{9} + \frac{1153438541105728500860468113657657089950675872832}{12060252261570757335435587711092811061307955284715} a^{8} - \frac{1453314619730228212560673563942336765219285812529}{12060252261570757335435587711092811061307955284715} a^{7} + \frac{19221988692439449485598932105480958998094409508311}{60301261307853786677177938555464055306539776423575} a^{6} + \frac{1122187487528773106830027122329550795727955459270}{2412050452314151467087117542218562212261591056943} a^{5} + \frac{19211009059596177954812959871056346780531590076084}{60301261307853786677177938555464055306539776423575} a^{4} - \frac{2952238564610606569416990922961925491671588505632}{60301261307853786677177938555464055306539776423575} a^{3} - \frac{16519688941566806307647393757853359905068257611094}{60301261307853786677177938555464055306539776423575} a^{2} + \frac{19828050483214457351029813880806112745254891096482}{60301261307853786677177938555464055306539776423575} a - \frac{27982035499924548484996219927049542136511034971011}{60301261307853786677177938555464055306539776423575}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 115189335.929 \) (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

\(\Q(\sqrt{5}) \), 4.4.4400.1, 4.2.475.1, 4.2.83600.1, 8.4.6988960000.9

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$