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Properties

Label 16.8.27578424983...7808.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 337^{3}$
Root discriminant $33.69$
Ramified primes $2, 337$
Class number $1$
Class group Trivial
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 24, -164, 224, 102, -136, -416, 1088, -141, -488, 648, 144, -58, -16, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 16*x^13 - 58*x^12 + 144*x^11 + 648*x^10 - 488*x^9 - 141*x^8 + 1088*x^7 - 416*x^6 - 136*x^5 + 102*x^4 + 224*x^3 - 164*x^2 + 24*x - 1)
 
gp: K = bnfinit(x^16 - 12*x^14 - 16*x^13 - 58*x^12 + 144*x^11 + 648*x^10 - 488*x^9 - 141*x^8 + 1088*x^7 - 416*x^6 - 136*x^5 + 102*x^4 + 224*x^3 - 164*x^2 + 24*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} - 16 x^{13} - 58 x^{12} + 144 x^{11} + 648 x^{10} - 488 x^{9} - 141 x^{8} + 1088 x^{7} - 416 x^{6} - 136 x^{5} + 102 x^{4} + 224 x^{3} - 164 x^{2} + 24 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2757842498387888526327808=2^{56}\cdot 337^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.69$
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:  $2, 337$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5550241271150725161751} a^{15} + \frac{917309360141175133167}{5550241271150725161751} a^{14} - \frac{609187009759651136579}{5550241271150725161751} a^{13} - \frac{2009054769965935594560}{5550241271150725161751} a^{12} + \frac{804889824016345707548}{5550241271150725161751} a^{11} + \frac{1070954312453961541136}{5550241271150725161751} a^{10} - \frac{15333725795599089790}{118090239811717556633} a^{9} - \frac{732466681342401823479}{5550241271150725161751} a^{8} + \frac{761065866345693163132}{5550241271150725161751} a^{7} + \frac{1421707214810728359120}{5550241271150725161751} a^{6} + \frac{319946333642299848456}{5550241271150725161751} a^{5} - \frac{2035434773189129479078}{5550241271150725161751} a^{4} + \frac{1400258130301328406418}{5550241271150725161751} a^{3} - \frac{369376422460312073650}{5550241271150725161751} a^{2} - \frac{1606838364781581990451}{5550241271150725161751} a - \frac{948157856214500454966}{5550241271150725161751}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2198186.32965 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.1413480448.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ $16$ $16$

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
337Data not computed