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Properties

Label 16.0.86162309624...0496.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{54}\cdot 3^{14}$
Root discriminant $27.13$
Ramified primes $2, 3$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times SD_{16}$ (as 16T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, 0, -144, 0, 396, 0, -576, 0, 606, 0, -312, 0, 114, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 114*x^12 - 312*x^10 + 606*x^8 - 576*x^6 + 396*x^4 - 144*x^2 + 36)
 
gp: K = bnfinit(x^16 - 12*x^14 + 114*x^12 - 312*x^10 + 606*x^8 - 576*x^6 + 396*x^4 - 144*x^2 + 36, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} + 114 x^{12} - 312 x^{10} + 606 x^{8} - 576 x^{6} + 396 x^{4} - 144 x^{2} + 36 \)

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:  \(86162309624498535530496=2^{54}\cdot 3^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.13$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8}$, $\frac{1}{6} a^{9}$, $\frac{1}{6} a^{10}$, $\frac{1}{6} a^{11}$, $\frac{1}{18} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{18} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{1641186} a^{14} - \frac{24941}{1641186} a^{12} - \frac{8093}{182354} a^{10} + \frac{18161}{547062} a^{8} - \frac{111958}{273531} a^{6} + \frac{72916}{273531} a^{4} + \frac{24317}{91177} a^{2} + \frac{37372}{91177}$, $\frac{1}{1641186} a^{15} - \frac{24941}{1641186} a^{13} - \frac{8093}{182354} a^{11} + \frac{18161}{547062} a^{9} - \frac{111958}{273531} a^{7} + \frac{72916}{273531} a^{5} + \frac{24317}{91177} a^{3} + \frac{37372}{91177} a$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{87070}{820593} a^{14} + \frac{2015239}{1641186} a^{12} - \frac{1052470}{91177} a^{10} + \frac{15235441}{547062} a^{8} - \frac{13505986}{273531} a^{6} + \frac{9016834}{273531} a^{4} - \frac{1396624}{91177} a^{2} + \frac{163003}{91177} \) (order $12$)
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:  \( 176109.391358 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times SD_{16}$ (as 16T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2\times SD_{16}$
Character table for $C_2\times SD_{16}$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.4.13824.1, 4.0.13824.1, \(\Q(\zeta_{12})\), 8.0.293534171136.1, 8.8.293534171136.1, 8.0.191102976.5

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed