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Properties

Label 16.0.18244960011...2161.2
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 $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![549, -897, 2182, -2103, 2785, -1815, 663, 228, 234, -156, 243, -48, -3, 6, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 + 6*x^13 - 3*x^12 - 48*x^11 + 243*x^10 - 156*x^9 + 234*x^8 + 228*x^7 + 663*x^6 - 1815*x^5 + 2785*x^4 - 2103*x^3 + 2182*x^2 - 897*x + 549)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 + 6*x^13 - 3*x^12 - 48*x^11 + 243*x^10 - 156*x^9 + 234*x^8 + 228*x^7 + 663*x^6 - 1815*x^5 + 2785*x^4 - 2103*x^3 + 2182*x^2 - 897*x + 549, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} + 6 x^{13} - 3 x^{12} - 48 x^{11} + 243 x^{10} - 156 x^{9} + 234 x^{8} + 228 x^{7} + 663 x^{6} - 1815 x^{5} + 2785 x^{4} - 2103 x^{3} + 2182 x^{2} - 897 x + 549 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{9} - \frac{1}{18} a^{7} + \frac{1}{3} a^{6} - \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{7}{18} a^{3} - \frac{1}{6} a^{2} - \frac{5}{18} a - \frac{1}{6}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{10} - \frac{1}{12} a^{9} - \frac{1}{36} a^{8} + \frac{1}{12} a^{7} - \frac{13}{36} a^{6} + \frac{1}{3} a^{5} + \frac{7}{36} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{1}{12} a - \frac{1}{4}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{5}{36} a^{7} + \frac{1}{3} a^{6} - \frac{5}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{9} a^{3} + \frac{5}{12} a^{2} + \frac{7}{36} a - \frac{1}{3}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{11} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} + \frac{1}{36} a^{7} - \frac{5}{18} a^{6} + \frac{1}{9} a^{5} - \frac{5}{12} a^{4} - \frac{7}{36} a^{3} + \frac{11}{36} a^{2} - \frac{7}{36} a - \frac{5}{12}$, $\frac{1}{2969227921828417596} a^{15} - \frac{5171616656739203}{494871320304736266} a^{14} + \frac{14319301060286999}{2969227921828417596} a^{13} - \frac{6465637983723404}{742306980457104399} a^{12} - \frac{15161106564550687}{2969227921828417596} a^{11} + \frac{23595521646840089}{1484613960914208798} a^{10} + \frac{6656931689866692}{82478553384122711} a^{9} - \frac{29069833141573363}{2969227921828417596} a^{8} - \frac{151373186177550241}{1484613960914208798} a^{7} - \frac{831199488650903113}{2969227921828417596} a^{6} - \frac{56276687476202687}{494871320304736266} a^{5} - \frac{279813323892649237}{1484613960914208798} a^{4} - \frac{426751135901684707}{989742640609472532} a^{3} - \frac{37985416303863119}{742306980457104399} a^{2} - \frac{379761201772400065}{2969227921828417596} a + \frac{339101748996331429}{989742640609472532}$

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:  \( 290030.205662 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.2197.1, 4.2.3887.1, 4.2.50531.1, 8.4.103903004413.1, 8.0.103903004413.1, 8.0.2553381961.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$