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Properties

Label 16.0.32258158221...1521.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 61^{4}$
Root discriminant $19.13$
Ramified primes $13, 61$
Class number $1$
Class group Trivial
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 48, 191, 393, 561, 407, 262, 141, 28, 31, 71, -30, 44, -22, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 22*x^13 + 44*x^12 - 30*x^11 + 71*x^10 + 31*x^9 + 28*x^8 + 141*x^7 + 262*x^6 + 407*x^5 + 561*x^4 + 393*x^3 + 191*x^2 + 48*x + 9)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 22*x^13 + 44*x^12 - 30*x^11 + 71*x^10 + 31*x^9 + 28*x^8 + 141*x^7 + 262*x^6 + 407*x^5 + 561*x^4 + 393*x^3 + 191*x^2 + 48*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} - 22 x^{13} + 44 x^{12} - 30 x^{11} + 71 x^{10} + 31 x^{9} + 28 x^{8} + 141 x^{7} + 262 x^{6} + 407 x^{5} + 561 x^{4} + 393 x^{3} + 191 x^{2} + 48 x + 9 \)

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:  \(322581582210337451521=13^{12}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.13$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{6} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{7} - \frac{2}{9} a^{3}$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{27} a^{8} - \frac{2}{27} a^{7} + \frac{4}{27} a^{6} + \frac{1}{9} a^{5} - \frac{11}{27} a^{4} + \frac{8}{27} a^{3} + \frac{5}{27} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{13} + \frac{1}{27} a^{11} - \frac{1}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{9} a^{8} - \frac{1}{27} a^{7} - \frac{2}{27} a^{6} + \frac{1}{27} a^{5} - \frac{1}{9} a^{4} + \frac{7}{27} a^{3} + \frac{8}{27} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{81} a^{14} + \frac{1}{81} a^{13} - \frac{1}{81} a^{12} - \frac{4}{81} a^{11} - \frac{1}{27} a^{10} - \frac{13}{81} a^{9} - \frac{11}{81} a^{8} - \frac{2}{81} a^{7} + \frac{1}{27} a^{6} - \frac{8}{81} a^{5} + \frac{17}{81} a^{4} + \frac{29}{81} a^{3} - \frac{11}{81} a^{2} + \frac{8}{27} a + \frac{2}{9}$, $\frac{1}{3339360027} a^{15} + \frac{3844445}{1113120009} a^{14} + \frac{50973406}{3339360027} a^{13} - \frac{9679157}{1113120009} a^{12} - \frac{33747035}{3339360027} a^{11} - \frac{110112568}{3339360027} a^{10} - \frac{415163761}{3339360027} a^{9} - \frac{133425413}{1113120009} a^{8} - \frac{90082753}{3339360027} a^{7} - \frac{490132364}{3339360027} a^{6} + \frac{374634781}{3339360027} a^{5} - \frac{11371348}{41226667} a^{4} - \frac{684583696}{3339360027} a^{3} - \frac{954885115}{3339360027} a^{2} - \frac{461290871}{1113120009} a - \frac{106989338}{371040003}$

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:  \( -1 \) (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:  \( 5480.44012635 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

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, 4.4.10309.1, 4.0.134017.1, 8.0.294435349.1 x2, 8.4.1381581253.1 x2, 8.0.17960556289.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$