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Properties

Label 16.0.31168238303...5257.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{21}\cdot 19\cdot 199^{4}$
Root discriminant $19.09$
Ramified primes $3, 19, 199$
Class number $1$
Class group Trivial
Galois group 16T1806

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 2, -9, 1, 47, 30, 6, -24, -60, 23, 19, -3, 9, -15, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 9*x^14 - 15*x^13 + 9*x^12 - 3*x^11 + 19*x^10 + 23*x^9 - 60*x^8 - 24*x^7 + 6*x^6 + 30*x^5 + 47*x^4 + x^3 - 9*x^2 + 2*x + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 9*x^14 - 15*x^13 + 9*x^12 - 3*x^11 + 19*x^10 + 23*x^9 - 60*x^8 - 24*x^7 + 6*x^6 + 30*x^5 + 47*x^4 + x^3 - 9*x^2 + 2*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 9 x^{14} - 15 x^{13} + 9 x^{12} - 3 x^{11} + 19 x^{10} + 23 x^{9} - 60 x^{8} - 24 x^{7} + 6 x^{6} + 30 x^{5} + 47 x^{4} + x^{3} - 9 x^{2} + 2 x + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(311682383035759705257=3^{21}\cdot 19\cdot 199^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.09$
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:  $3, 19, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{9066907572} a^{15} + \frac{193209439}{9066907572} a^{14} + \frac{34257317}{4533453786} a^{13} + \frac{1306578943}{9066907572} a^{12} - \frac{29811059}{2266726893} a^{11} - \frac{2053236521}{9066907572} a^{10} - \frac{635605525}{2266726893} a^{9} + \frac{261212951}{9066907572} a^{8} - \frac{4531464871}{9066907572} a^{7} + \frac{460305293}{9066907572} a^{6} - \frac{2606625595}{9066907572} a^{5} + \frac{3632753915}{9066907572} a^{4} + \frac{1022327083}{2266726893} a^{3} - \frac{369322043}{9066907572} a^{2} + \frac{115847812}{2266726893} a + \frac{314680965}{755575631}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{28123}{1353876} a^{15} + \frac{62533}{451292} a^{14} - \frac{41928}{112823} a^{13} + \frac{309731}{451292} a^{12} - \frac{81363}{112823} a^{11} + \frac{17217}{451292} a^{10} - \frac{51556}{338469} a^{9} - \frac{44341}{451292} a^{8} + \frac{1655459}{451292} a^{7} - \frac{1145251}{451292} a^{6} - \frac{831677}{451292} a^{5} - \frac{463931}{451292} a^{4} + \frac{63875}{676938} a^{3} + \frac{1413809}{451292} a^{2} + \frac{20593}{112823} a + \frac{29983}{338469} \) (order $6$)
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:  \( 23862.633438 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1806:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24576
The 88 conjugacy class representatives for t16n1806 are not computed
Character table for t16n1806 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 8.0.2338399449.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$199$199.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
199.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
199.8.4.1$x^{8} + 237606 x^{4} - 7880599 x^{2} + 14114152809$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$