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Properties

Label 16.4.23728052512...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 9929^{4}$
Root discriminant $33.38$
Ramified primes $5, 9929$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2389, 14673, -19806, 11070, -4943, 279, 349, 993, -1113, 1115, -314, 68, 80, -52, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 26*x^14 - 52*x^13 + 80*x^12 + 68*x^11 - 314*x^10 + 1115*x^9 - 1113*x^8 + 993*x^7 + 349*x^6 + 279*x^5 - 4943*x^4 + 11070*x^3 - 19806*x^2 + 14673*x - 2389)
 
gp: K = bnfinit(x^16 - 6*x^15 + 26*x^14 - 52*x^13 + 80*x^12 + 68*x^11 - 314*x^10 + 1115*x^9 - 1113*x^8 + 993*x^7 + 349*x^6 + 279*x^5 - 4943*x^4 + 11070*x^3 - 19806*x^2 + 14673*x - 2389, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 26 x^{14} - 52 x^{13} + 80 x^{12} + 68 x^{11} - 314 x^{10} + 1115 x^{9} - 1113 x^{8} + 993 x^{7} + 349 x^{6} + 279 x^{5} - 4943 x^{4} + 11070 x^{3} - 19806 x^{2} + 14673 x - 2389 \)

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:  \(2372805251213789306640625=5^{12}\cdot 9929^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.38$
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:  $5, 9929$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2854968594215120521850050582769} a^{15} + \frac{1107853503580241616940132530202}{2854968594215120521850050582769} a^{14} + \frac{1369402579062476311024929478327}{2854968594215120521850050582769} a^{13} - \frac{1977903302731887401791310877}{14792583389715650372280054833} a^{12} - \frac{1214946209112079323557086485453}{2854968594215120521850050582769} a^{11} + \frac{698575518272235724572130084518}{2854968594215120521850050582769} a^{10} - \frac{33792896522346253975336739459}{2854968594215120521850050582769} a^{9} - \frac{623478627589208200138342727000}{2854968594215120521850050582769} a^{8} + \frac{113429075479768162103530174372}{2854968594215120521850050582769} a^{7} + \frac{585067974230483635819931305338}{2854968594215120521850050582769} a^{6} + \frac{1195562929655209986920040993202}{2854968594215120521850050582769} a^{5} - \frac{1232191884881201142186514550242}{2854968594215120521850050582769} a^{4} - \frac{1112374982870446969804825058372}{2854968594215120521850050582769} a^{3} + \frac{733668066805152667784790361726}{2854968594215120521850050582769} a^{2} + \frac{58564397317326082336780340237}{2854968594215120521850050582769} a - \frac{395175037736867634849136229564}{2854968594215120521850050582769}$

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:  \( 734836.868092 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
9929Data not computed