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Properties

Label 16.0.18244960011...2161.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 23^{8}$
Root discriminant $32.83$
Ramified primes $13, 23$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5057, 20098, 30602, 19292, 2900, -6360, -5851, -2752, -214, 472, 347, 68, 34, -30, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 30*x^13 + 34*x^12 + 68*x^11 + 347*x^10 + 472*x^9 - 214*x^8 - 2752*x^7 - 5851*x^6 - 6360*x^5 + 2900*x^4 + 19292*x^3 + 30602*x^2 + 20098*x + 5057)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 30*x^13 + 34*x^12 + 68*x^11 + 347*x^10 + 472*x^9 - 214*x^8 - 2752*x^7 - 5851*x^6 - 6360*x^5 + 2900*x^4 + 19292*x^3 + 30602*x^2 + 20098*x + 5057, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 30 x^{13} + 34 x^{12} + 68 x^{11} + 347 x^{10} + 472 x^{9} - 214 x^{8} - 2752 x^{7} - 5851 x^{6} - 6360 x^{5} + 2900 x^{4} + 19292 x^{3} + 30602 x^{2} + 20098 x + 5057 \)

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:  \(1824496001102094673202161=13^{12}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.83$
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:  $13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{24} a^{14} + \frac{1}{24} a^{12} - \frac{1}{12} a^{11} - \frac{1}{24} a^{10} - \frac{1}{12} a^{9} + \frac{1}{12} a^{8} - \frac{1}{4} a^{7} + \frac{1}{3} a^{6} - \frac{5}{12} a^{5} + \frac{1}{24} a^{4} + \frac{1}{12} a^{3} - \frac{1}{8} a^{2} - \frac{5}{12} a + \frac{7}{24}$, $\frac{1}{1627842856670189927227056} a^{15} - \frac{10649575679915543233863}{542614285556729975742352} a^{14} - \frac{44918501065513747519307}{1627842856670189927227056} a^{13} + \frac{22769741953296152697471}{542614285556729975742352} a^{12} - \frac{40755364916168426458709}{542614285556729975742352} a^{11} + \frac{43942938133520812231169}{542614285556729975742352} a^{10} + \frac{25747172257830132353681}{135653571389182493935588} a^{9} + \frac{73610965568084612676521}{406960714167547481806764} a^{8} + \frac{188157012613782975371291}{813921428335094963613528} a^{7} - \frac{21656624030651332559645}{271307142778364987871176} a^{6} + \frac{228293728957026188080233}{542614285556729975742352} a^{5} - \frac{586154142514658571009995}{1627842856670189927227056} a^{4} + \frac{170588087749847099254843}{1627842856670189927227056} a^{3} + \frac{137750133021333276732545}{1627842856670189927227056} a^{2} - \frac{366378607471615917171179}{1627842856670189927227056} a + \frac{176820808676184092473573}{1627842856670189927227056}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
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:  \( 186917.564969 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.2.3887.1, 8.2.58727785103.1, 8.0.103903004413.1, 8.2.4517521931.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$