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Properties

Label 16.8.24471271802...0000.4
Degree $16$
Signature $[8, 4]$
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 16T1872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13981, 4483, -47594, -24178, -1604, -31191, -9696, 17956, 6632, -4129, -2001, 489, 346, -13, -29, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 29*x^14 - 13*x^13 + 346*x^12 + 489*x^11 - 2001*x^10 - 4129*x^9 + 6632*x^8 + 17956*x^7 - 9696*x^6 - 31191*x^5 - 1604*x^4 - 24178*x^3 - 47594*x^2 + 4483*x + 13981)
 
gp: K = bnfinit(x^16 - 2*x^15 - 29*x^14 - 13*x^13 + 346*x^12 + 489*x^11 - 2001*x^10 - 4129*x^9 + 6632*x^8 + 17956*x^7 - 9696*x^6 - 31191*x^5 - 1604*x^4 - 24178*x^3 - 47594*x^2 + 4483*x + 13981, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 29 x^{14} - 13 x^{13} + 346 x^{12} + 489 x^{11} - 2001 x^{10} - 4129 x^{9} + 6632 x^{8} + 17956 x^{7} - 9696 x^{6} - 31191 x^{5} - 1604 x^{4} - 24178 x^{3} - 47594 x^{2} + 4483 x + 13981 \)

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:  \(24471271802225000000000000=2^{12}\cdot 5^{14}\cdot 9929^{3}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{721348577381596672441669517577118459} a^{15} + \frac{348607485288962176620948772403476559}{721348577381596672441669517577118459} a^{14} + \frac{164804081920030924611781397446897976}{721348577381596672441669517577118459} a^{13} - \frac{192718255564254155616006359484329884}{721348577381596672441669517577118459} a^{12} - \frac{57196909425565775897301979036113524}{721348577381596672441669517577118459} a^{11} + \frac{89921529768461506165617780258004480}{721348577381596672441669517577118459} a^{10} - \frac{76007723020118755019544238249266024}{721348577381596672441669517577118459} a^{9} - \frac{282565414823932868775889780939697014}{721348577381596672441669517577118459} a^{8} - \frac{321922515047855463080275480912076808}{721348577381596672441669517577118459} a^{7} + \frac{345458122187348106331253182975176434}{721348577381596672441669517577118459} a^{6} - \frac{50782844506475482219710040191683197}{721348577381596672441669517577118459} a^{5} + \frac{23467918368443954102679904018210183}{721348577381596672441669517577118459} a^{4} + \frac{125919905492512838435001236893587659}{721348577381596672441669517577118459} a^{3} - \frac{255767386463119828468640651983529780}{721348577381596672441669517577118459} a^{2} - \frac{122060951661482118642946785373032183}{721348577381596672441669517577118459} a - \frac{280435130683794610016219979016695721}{721348577381596672441669517577118459}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6096929.86661 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1872:

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 t16n1872 are not computed
Character table for t16n1872 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 $16$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.6$x^{12} - 18 x^{10} + 11 x^{8} - 52 x^{6} - x^{4} + 6 x^{2} - 11$$2$$6$$12$12T105$[2, 2, 2, 2]^{12}$
5Data not computed
9929Data not computed