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Properties

Label 16.4.16072565967...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 71^{2}\cdot 1901^{4}$
Root discriminant $37.62$
Ramified primes $5, 71, 1901$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1884

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46591, 59685, 109439, 79233, 77203, 41414, 22489, 10926, 2734, 1574, -17, 5, -28, -38, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 38*x^13 - 28*x^12 + 5*x^11 - 17*x^10 + 1574*x^9 + 2734*x^8 + 10926*x^7 + 22489*x^6 + 41414*x^5 + 77203*x^4 + 79233*x^3 + 109439*x^2 + 59685*x + 46591)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 38*x^13 - 28*x^12 + 5*x^11 - 17*x^10 + 1574*x^9 + 2734*x^8 + 10926*x^7 + 22489*x^6 + 41414*x^5 + 77203*x^4 + 79233*x^3 + 109439*x^2 + 59685*x + 46591, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 38 x^{13} - 28 x^{12} + 5 x^{11} - 17 x^{10} + 1574 x^{9} + 2734 x^{8} + 10926 x^{7} + 22489 x^{6} + 41414 x^{5} + 77203 x^{4} + 79233 x^{3} + 109439 x^{2} + 59685 x + 46591 \)

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:  \(16072565967377109619140625=5^{12}\cdot 71^{2}\cdot 1901^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.62$
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:  $5, 71, 1901$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3786324749228146117713115993397119} a^{15} - \frac{771309965911365527066688607053611}{3786324749228146117713115993397119} a^{14} + \frac{1265786137593975361757890634548160}{3786324749228146117713115993397119} a^{13} - \frac{1395892854909867766052793731267108}{3786324749228146117713115993397119} a^{12} + \frac{1296873132093236812003498061273156}{3786324749228146117713115993397119} a^{11} + \frac{1166867104351178097013480377973250}{3786324749228146117713115993397119} a^{10} + \frac{1379051166554466826839824638989841}{3786324749228146117713115993397119} a^{9} + \frac{1777865553682138751152016085068556}{3786324749228146117713115993397119} a^{8} + \frac{974879432697111052066733227551173}{3786324749228146117713115993397119} a^{7} - \frac{724746929765912879910726613698564}{3786324749228146117713115993397119} a^{6} + \frac{453233495792650018582679716482335}{3786324749228146117713115993397119} a^{5} - \frac{375173610092811658098890738184172}{3786324749228146117713115993397119} a^{4} - \frac{1157518972684194524125162569458835}{3786324749228146117713115993397119} a^{3} - \frac{135646377948405716467118612672209}{3786324749228146117713115993397119} a^{2} + \frac{1033065282739767518254780629686294}{3786324749228146117713115993397119} a - \frac{182737906092739925908966456777912}{3786324749228146117713115993397119}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1359649.37553 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1884:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147456
The 136 conjugacy class representatives for t16n1884 are not computed
Character table for t16n1884 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.6.84356875.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.8.0.1$x^{8} - 7 x + 13$$1$$8$$0$$C_8$$[\ ]^{8}$
1901Data not computed