↑

Properties

Label 16.4.68457752166...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{40}\cdot 5^{14}\cdot 101^{2}$
Root discriminant $41.18$
Ramified primes $2, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1162

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6821, -23420, 34384, -14720, -9518, 8740, 4852, 2320, -4290, -2020, 1408, 320, -218, -20, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 20*x^13 - 218*x^12 + 320*x^11 + 1408*x^10 - 2020*x^9 - 4290*x^8 + 2320*x^7 + 4852*x^6 + 8740*x^5 - 9518*x^4 - 14720*x^3 + 34384*x^2 - 23420*x + 6821)
 
gp: K = bnfinit(x^16 - 4*x^14 - 20*x^13 - 218*x^12 + 320*x^11 + 1408*x^10 - 2020*x^9 - 4290*x^8 + 2320*x^7 + 4852*x^6 + 8740*x^5 - 9518*x^4 - 14720*x^3 + 34384*x^2 - 23420*x + 6821, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 20 x^{13} - 218 x^{12} + 320 x^{11} + 1408 x^{10} - 2020 x^{9} - 4290 x^{8} + 2320 x^{7} + 4852 x^{6} + 8740 x^{5} - 9518 x^{4} - 14720 x^{3} + 34384 x^{2} - 23420 x + 6821 \)

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:  \(68457752166400000000000000=2^{40}\cdot 5^{14}\cdot 101^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.18$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{47841108382654279970321632056894478} a^{15} - \frac{11389377395200696546245191974380017}{47841108382654279970321632056894478} a^{14} - \frac{282976254792152062685968462571873}{23920554191327139985160816028447239} a^{13} + \frac{5730297541117568304040746643239670}{23920554191327139985160816028447239} a^{12} + \frac{4844925318432495304443965316183449}{23920554191327139985160816028447239} a^{11} + \frac{1824799068369362416585175238574785}{47841108382654279970321632056894478} a^{10} - \frac{10184765304907485460802186207817601}{47841108382654279970321632056894478} a^{9} + \frac{3329592225227562168450690512368091}{47841108382654279970321632056894478} a^{8} + \frac{7775055798368317864272733451458895}{47841108382654279970321632056894478} a^{7} - \frac{9512521294320662167079380879400637}{47841108382654279970321632056894478} a^{6} + \frac{6505463556363527465130228014242212}{23920554191327139985160816028447239} a^{5} + \frac{972128716764436838887395449980239}{23920554191327139985160816028447239} a^{4} - \frac{2349605325305030109987919158083453}{23920554191327139985160816028447239} a^{3} - \frac{5512540146038048255448997155629629}{47841108382654279970321632056894478} a^{2} - \frac{4499056309391253477672448171920781}{47841108382654279970321632056894478} a - \frac{9102490440714950944487430815561929}{47841108382654279970321632056894478}$

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:  \( 3931675.02839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.5120000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
5Data not computed
$101$101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$