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Properties

Label 16.4.24471271802...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{12}\cdot 5^{14}\cdot 9929^{3}$
Root discriminant $38.62$
Ramified primes $2, 5, 9929$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1871

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-228404, 55160, 131479, -36975, 15777, 35190, -5833, -5325, 820, -335, -67, -90, -43, 55, -4, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 4*x^14 + 55*x^13 - 43*x^12 - 90*x^11 - 67*x^10 - 335*x^9 + 820*x^8 - 5325*x^7 - 5833*x^6 + 35190*x^5 + 15777*x^4 - 36975*x^3 + 131479*x^2 + 55160*x - 228404)
 
gp: K = bnfinit(x^16 - 5*x^15 - 4*x^14 + 55*x^13 - 43*x^12 - 90*x^11 - 67*x^10 - 335*x^9 + 820*x^8 - 5325*x^7 - 5833*x^6 + 35190*x^5 + 15777*x^4 - 36975*x^3 + 131479*x^2 + 55160*x - 228404, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 4 x^{14} + 55 x^{13} - 43 x^{12} - 90 x^{11} - 67 x^{10} - 335 x^{9} + 820 x^{8} - 5325 x^{7} - 5833 x^{6} + 35190 x^{5} + 15777 x^{4} - 36975 x^{3} + 131479 x^{2} + 55160 x - 228404 \)

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:  \(24471271802225000000000000=2^{12}\cdot 5^{14}\cdot 9929^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.62$
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:  $2, 5, 9929$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{3}{11} a^{13} + \frac{3}{11} a^{11} - \frac{4}{11} a^{10} - \frac{2}{11} a^{9} - \frac{1}{11} a^{8} - \frac{5}{11} a^{7} - \frac{3}{11} a^{6} - \frac{2}{11} a^{5} - \frac{4}{11} a^{4} - \frac{5}{11} a^{3} - \frac{3}{11} a^{2} - \frac{5}{11} a$, $\frac{1}{1070682760446329415235976183209244279384} a^{15} - \frac{21351177016918288617942537194427723415}{1070682760446329415235976183209244279384} a^{14} - \frac{62933142612172216427238672839384554811}{535341380223164707617988091604622139692} a^{13} + \frac{57246009461666190983802168180035077195}{1070682760446329415235976183209244279384} a^{12} + \frac{131145491916767010433156394066642869815}{1070682760446329415235976183209244279384} a^{11} - \frac{56446441886371608447140862230648373152}{133835345055791176904497022901155534923} a^{10} + \frac{517440430565799761785676355728223840957}{1070682760446329415235976183209244279384} a^{9} - \frac{396759013497087868338106755754589309129}{1070682760446329415235976183209244279384} a^{8} - \frac{224475372199411440627950547309145413173}{535341380223164707617988091604622139692} a^{7} + \frac{423951525907879719299555222369285051711}{1070682760446329415235976183209244279384} a^{6} - \frac{94973400856673084301298032776622420847}{1070682760446329415235976183209244279384} a^{5} - \frac{61999986126923528301047826076153143003}{267670690111582353808994045802311069846} a^{4} - \frac{156226473845886083848952067978171863255}{1070682760446329415235976183209244279384} a^{3} + \frac{439398237173498986085465855839125667143}{1070682760446329415235976183209244279384} a^{2} - \frac{395095064068543025122696850604813085343}{1070682760446329415235976183209244279384} a + \frac{7762665767174547155254644748164663577}{48667398202105882510726190145874739972}$

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:  \( 4430959.66595 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1871:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 104 conjugacy class representatives for t16n1871 are not computed
Character table for t16n1871 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.155140625.1

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 $16$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.10$x^{12} - 6 x^{10} + 23 x^{8} - 28 x^{6} - 9 x^{4} - 30 x^{2} - 15$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
5Data not computed
9929Data not computed