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Properties

Label 16.4.16640273717...3921.1
Degree $16$
Signature $[4, 6]$
Discriminant $13^{4}\cdot 17^{12}$
Root discriminant $15.90$
Ramified primes $13, 17$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 0, -30, -47, 16, 60, 27, -16, -6, -9, -4, 4, 1, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^13 + 4*x^12 - 4*x^11 - 9*x^10 - 6*x^9 - 16*x^8 + 27*x^7 + 60*x^6 + 16*x^5 - 47*x^4 - 30*x^3 + 4*x + 1)
 
gp: K = bnfinit(x^16 - x^15 + x^13 + 4*x^12 - 4*x^11 - 9*x^10 - 6*x^9 - 16*x^8 + 27*x^7 + 60*x^6 + 16*x^5 - 47*x^4 - 30*x^3 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + x^{13} + 4 x^{12} - 4 x^{11} - 9 x^{10} - 6 x^{9} - 16 x^{8} + 27 x^{7} + 60 x^{6} + 16 x^{5} - 47 x^{4} - 30 x^{3} + 4 x + 1 \)

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:  \(16640273717519203921=13^{4}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.90$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} - \frac{5}{17} a^{11} - \frac{1}{17} a^{10} - \frac{15}{34} a^{9} + \frac{2}{17} a^{8} + \frac{4}{17} a^{7} - \frac{13}{34} a^{6} - \frac{1}{17} a^{5} - \frac{2}{17} a^{4} + \frac{9}{34} a^{3} - \frac{1}{17} a^{2} - \frac{3}{17} a - \frac{1}{34}$, $\frac{1}{34} a^{13} - \frac{1}{34} a^{10} - \frac{5}{17} a^{9} + \frac{7}{17} a^{8} - \frac{1}{34} a^{7} + \frac{2}{17} a^{6} + \frac{5}{17} a^{5} + \frac{3}{34} a^{4} - \frac{7}{17} a^{3} + \frac{4}{17} a^{2} + \frac{7}{34} a - \frac{5}{17}$, $\frac{1}{34} a^{14} - \frac{1}{34} a^{11} - \frac{5}{17} a^{10} + \frac{7}{17} a^{9} - \frac{1}{34} a^{8} + \frac{2}{17} a^{7} + \frac{5}{17} a^{6} + \frac{3}{34} a^{5} - \frac{7}{17} a^{4} + \frac{4}{17} a^{3} + \frac{7}{34} a^{2} - \frac{5}{17} a$, $\frac{1}{3183182} a^{15} - \frac{16599}{3183182} a^{14} - \frac{1311}{187246} a^{13} + \frac{7577}{1591591} a^{12} - \frac{710071}{3183182} a^{11} + \frac{23623}{187246} a^{10} + \frac{315521}{1591591} a^{9} - \frac{336489}{3183182} a^{8} + \frac{1275063}{3183182} a^{7} - \frac{335929}{1591591} a^{6} - \frac{89881}{187246} a^{5} - \frac{78585}{3183182} a^{4} - \frac{328607}{1591591} a^{3} + \frac{1452065}{3183182} a^{2} + \frac{1211119}{3183182} a + \frac{858271}{3183182}$

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:  \( 1759.6652294 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

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

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.313788397.1 x2, 8.4.4079249161.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$