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Properties

Label 16.4.23728052512...0625.2
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![32756, -90016, 103105, -65145, 19594, 4549, -9298, 8355, -4769, 1319, -168, -57, 74, -37, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 11*x^14 - 37*x^13 + 74*x^12 - 57*x^11 - 168*x^10 + 1319*x^9 - 4769*x^8 + 8355*x^7 - 9298*x^6 + 4549*x^5 + 19594*x^4 - 65145*x^3 + 103105*x^2 - 90016*x + 32756)
 
gp: K = bnfinit(x^16 - 3*x^15 + 11*x^14 - 37*x^13 + 74*x^12 - 57*x^11 - 168*x^10 + 1319*x^9 - 4769*x^8 + 8355*x^7 - 9298*x^6 + 4549*x^5 + 19594*x^4 - 65145*x^3 + 103105*x^2 - 90016*x + 32756, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 11 x^{14} - 37 x^{13} + 74 x^{12} - 57 x^{11} - 168 x^{10} + 1319 x^{9} - 4769 x^{8} + 8355 x^{7} - 9298 x^{6} + 4549 x^{5} + 19594 x^{4} - 65145 x^{3} + 103105 x^{2} - 90016 x + 32756 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5258474666304548782716916047782176} a^{15} + \frac{1289397799357288039711872112809579}{5258474666304548782716916047782176} a^{14} + \frac{600524036224263141024690992322053}{5258474666304548782716916047782176} a^{13} + \frac{2345254589958295667332631351569377}{5258474666304548782716916047782176} a^{12} - \frac{151100489473640112210162098656201}{657309333288068597839614505972772} a^{11} + \frac{2154065239804590814770481206144407}{5258474666304548782716916047782176} a^{10} - \frac{71409408736169563779352800600211}{2629237333152274391358458023891088} a^{9} + \frac{2064667215423735045561365694505267}{5258474666304548782716916047782176} a^{8} + \frac{2167631957173153161428272957369897}{5258474666304548782716916047782176} a^{7} - \frac{1538338806208032756116745207735519}{5258474666304548782716916047782176} a^{6} + \frac{386542104580102194846408715385775}{1314618666576137195679229011945544} a^{5} + \frac{1763586318016885873713766005881517}{5258474666304548782716916047782176} a^{4} - \frac{75221324084514673191397129935104}{164327333322017149459903626493193} a^{3} - \frac{1068541981815855248657441593080473}{5258474666304548782716916047782176} a^{2} + \frac{1819242687762507650954538498202627}{5258474666304548782716916047782176} a - \frac{544425598717989946882715340161259}{2629237333152274391358458023891088}$

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:  \( 1064406.05241 \) (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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
9929Data not computed