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Properties

Label 16.16.3657406027...5664.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{32}\cdot 41^{4}\cdot 569^{3}\cdot 1279^{2}$
Root discriminant $81.32$
Ramified primes $2, 41, 569, 1279$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1038527, 249908, -6145068, 2012224, 5524811, -1509840, -2184304, 384176, 449959, -41020, -49502, 1912, 2911, -32, -86, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 86*x^14 - 32*x^13 + 2911*x^12 + 1912*x^11 - 49502*x^10 - 41020*x^9 + 449959*x^8 + 384176*x^7 - 2184304*x^6 - 1509840*x^5 + 5524811*x^4 + 2012224*x^3 - 6145068*x^2 + 249908*x + 1038527)
 
gp: K = bnfinit(x^16 - 86*x^14 - 32*x^13 + 2911*x^12 + 1912*x^11 - 49502*x^10 - 41020*x^9 + 449959*x^8 + 384176*x^7 - 2184304*x^6 - 1509840*x^5 + 5524811*x^4 + 2012224*x^3 - 6145068*x^2 + 249908*x + 1038527, 1)
 

Normalized defining polynomial

\( x^{16} - 86 x^{14} - 32 x^{13} + 2911 x^{12} + 1912 x^{11} - 49502 x^{10} - 41020 x^{9} + 449959 x^{8} + 384176 x^{7} - 2184304 x^{6} - 1509840 x^{5} + 5524811 x^{4} + 2012224 x^{3} - 6145068 x^{2} + 249908 x + 1038527 \)

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:  \(3657406027372345478303568625664=2^{32}\cdot 41^{4}\cdot 569^{3}\cdot 1279^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.32$
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:  $2, 41, 569, 1279$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7151656739} a^{14} + \frac{2885331129}{7151656739} a^{13} - \frac{224256566}{7151656739} a^{12} + \frac{1180046414}{7151656739} a^{11} - \frac{3474648722}{7151656739} a^{10} - \frac{917879942}{7151656739} a^{9} + \frac{1573547034}{7151656739} a^{8} + \frac{2407489806}{7151656739} a^{7} + \frac{3454206697}{7151656739} a^{6} + \frac{1754805121}{7151656739} a^{5} - \frac{1192843189}{7151656739} a^{4} + \frac{1940502817}{7151656739} a^{3} - \frac{801371044}{7151656739} a^{2} + \frac{1914518054}{7151656739} a - \frac{1778283577}{7151656739}$, $\frac{1}{110196395012295587202660065941} a^{15} - \frac{279567795119783980}{110196395012295587202660065941} a^{14} - \frac{20227252270578049453990046180}{110196395012295587202660065941} a^{13} + \frac{3165447495009350104859125653}{110196395012295587202660065941} a^{12} + \frac{10721741046187107385387440648}{110196395012295587202660065941} a^{11} - \frac{52969582414761687508331828792}{110196395012295587202660065941} a^{10} + \frac{6557049011499045716630876065}{110196395012295587202660065941} a^{9} - \frac{22328892290332444800227790956}{110196395012295587202660065941} a^{8} + \frac{10716470315140719036509971174}{110196395012295587202660065941} a^{7} + \frac{564421232021692564375802999}{1867735508682976054282373999} a^{6} + \frac{29457493286728170206810300721}{110196395012295587202660065941} a^{5} - \frac{6101046992591663069489749052}{110196395012295587202660065941} a^{4} - \frac{8199690787280617874088298735}{110196395012295587202660065941} a^{3} - \frac{43204550846387249409040248975}{110196395012295587202660065941} a^{2} - \frac{19653041813631462144236708356}{110196395012295587202660065941} a + \frac{14140519109762227790920426260}{110196395012295587202660065941}$

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:  \( 13198340676.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 148 conjugacy class representatives for t16n1781 are not computed
Character table for t16n1781 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.3917778944.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
569Data not computed
1279Data not computed