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Properties

Label 16.4.28606891443...3125.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{4}\cdot 5^{8}\cdot 149\cdot 2791^{4}$
Root discriminant $29.24$
Ramified primes $3, 5, 149, 2791$
Class number $1$
Class group Trivial
Galois group 16T1887

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 46, -40, -205, 104, 388, -720, 903, -973, 822, -416, 116, -2, -21, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 11*x^14 - 21*x^13 - 2*x^12 + 116*x^11 - 416*x^10 + 822*x^9 - 973*x^8 + 903*x^7 - 720*x^6 + 388*x^5 + 104*x^4 - 205*x^3 - 40*x^2 + 46*x + 19)
 
gp: K = bnfinit(x^16 - 4*x^15 + 11*x^14 - 21*x^13 - 2*x^12 + 116*x^11 - 416*x^10 + 822*x^9 - 973*x^8 + 903*x^7 - 720*x^6 + 388*x^5 + 104*x^4 - 205*x^3 - 40*x^2 + 46*x + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 11 x^{14} - 21 x^{13} - 2 x^{12} + 116 x^{11} - 416 x^{10} + 822 x^{9} - 973 x^{8} + 903 x^{7} - 720 x^{6} + 388 x^{5} + 104 x^{4} - 205 x^{3} - 40 x^{2} + 46 x + 19 \)

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:  \(286068914436239651953125=3^{4}\cdot 5^{8}\cdot 149\cdot 2791^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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:  $3, 5, 149, 2791$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{171} a^{14} + \frac{1}{171} a^{13} + \frac{26}{171} a^{12} + \frac{62}{171} a^{11} + \frac{55}{171} a^{10} - \frac{5}{57} a^{9} - \frac{55}{171} a^{8} - \frac{59}{171} a^{7} + \frac{7}{19} a^{6} + \frac{58}{171} a^{5} + \frac{29}{171} a^{4} + \frac{10}{57} a^{3} + \frac{31}{171} a^{2} + \frac{79}{171} a - \frac{1}{9}$, $\frac{1}{50383236893684841} a^{15} + \frac{60421916980324}{50383236893684841} a^{14} - \frac{1927357790960011}{50383236893684841} a^{13} + \frac{8053549156845185}{50383236893684841} a^{12} - \frac{1056879736016552}{50383236893684841} a^{11} + \frac{3898296936502319}{16794412297894947} a^{10} + \frac{9140005707500147}{50383236893684841} a^{9} + \frac{21123034577870065}{50383236893684841} a^{8} + \frac{7016351199769406}{16794412297894947} a^{7} + \frac{4968945227198233}{50383236893684841} a^{6} + \frac{11078110322211104}{50383236893684841} a^{5} + \frac{983828206521242}{5598137432631649} a^{4} - \frac{22367776599780152}{50383236893684841} a^{3} + \frac{24675919437314968}{50383236893684841} a^{2} - \frac{8935416653695141}{50383236893684841} a - \frac{600775072828}{883916436731313}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 323596.670178 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1887:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147456
The 148 conjugacy class representatives for t16n1887 are not computed
Character table for t16n1887 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.43816955625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
5Data not computed
$149$$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{149}$$x + 2$$1$$1$$0$Trivial$[\ ]$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
149.3.0.1$x^{3} - x + 18$$1$$3$$0$$C_3$$[\ ]^{3}$
149.6.0.1$x^{6} - x + 14$$1$$6$$0$$C_6$$[\ ]^{6}$
2791Data not computed