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Properties

Label 16.16.5051539403...5536.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 41^{6}\cdot 311^{2}$
Root discriminant $46.66$
Ramified primes $2, 41, 311$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1429

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![96721, 0, -295450, 0, 336689, 0, -190700, 0, 59091, 0, -10300, 0, 999, 0, -50, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 50*x^14 + 999*x^12 - 10300*x^10 + 59091*x^8 - 190700*x^6 + 336689*x^4 - 295450*x^2 + 96721)
 
gp: K = bnfinit(x^16 - 50*x^14 + 999*x^12 - 10300*x^10 + 59091*x^8 - 190700*x^6 + 336689*x^4 - 295450*x^2 + 96721, 1)
 

Normalized defining polynomial

\( x^{16} - 50 x^{14} + 999 x^{12} - 10300 x^{10} + 59091 x^{8} - 190700 x^{6} + 336689 x^{4} - 295450 x^{2} + 96721 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(505153940312306728919105536=2^{40}\cdot 41^{6}\cdot 311^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.66$
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:  $2, 41, 311$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{25} a^{8} + \frac{2}{25} a^{4} + \frac{1}{25}$, $\frac{1}{25} a^{9} + \frac{2}{25} a^{5} + \frac{1}{25} a$, $\frac{1}{25} a^{10} + \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} + \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{125} a^{12} - \frac{2}{125} a^{8} - \frac{7}{125} a^{4} - \frac{4}{125}$, $\frac{1}{125} a^{13} - \frac{2}{125} a^{9} - \frac{7}{125} a^{5} - \frac{4}{125} a$, $\frac{1}{20331625} a^{14} + \frac{3722}{4066325} a^{12} - \frac{325862}{20331625} a^{10} + \frac{37437}{4066325} a^{8} - \frac{1017902}{20331625} a^{6} - \frac{250242}{4066325} a^{4} + \frac{1433586}{20331625} a^{2} - \frac{2812}{13075}$, $\frac{1}{20331625} a^{15} + \frac{3722}{4066325} a^{13} - \frac{325862}{20331625} a^{11} + \frac{37437}{4066325} a^{9} - \frac{1017902}{20331625} a^{7} - \frac{250242}{4066325} a^{5} + \frac{1433586}{20331625} a^{3} - \frac{2812}{13075} a$

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:  $15$
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:  \( 117031113.469 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1429:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 77 conjugacy class representatives for t16n1429 are not computed
Character table for t16n1429 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
311Data not computed