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Properties

Label 16.0.47665230952...0736.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 13^{8}$
Root discriminant $46.49$
Ramified primes $2, 3, 13$
Class number $48$ (GRH)
Class group $[2, 2, 12]$ (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![14278, -42872, 74220, -45184, 28872, -40584, 40276, -17008, 3636, -2780, 2260, -912, 174, 4, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^13 + 174*x^12 - 912*x^11 + 2260*x^10 - 2780*x^9 + 3636*x^8 - 17008*x^7 + 40276*x^6 - 40584*x^5 + 28872*x^4 - 45184*x^3 + 74220*x^2 - 42872*x + 14278)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^13 + 174*x^12 - 912*x^11 + 2260*x^10 - 2780*x^9 + 3636*x^8 - 17008*x^7 + 40276*x^6 - 40584*x^5 + 28872*x^4 - 45184*x^3 + 74220*x^2 - 42872*x + 14278, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{13} + 174 x^{12} - 912 x^{11} + 2260 x^{10} - 2780 x^{9} + 3636 x^{8} - 17008 x^{7} + 40276 x^{6} - 40584 x^{5} + 28872 x^{4} - 45184 x^{3} + 74220 x^{2} - 42872 x + 14278 \)

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:  \(476652309522958914196340736=2^{40}\cdot 3^{12}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.49$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{5}{13} a^{11} + \frac{1}{13} a^{10} + \frac{3}{13} a^{8} + \frac{4}{13} a^{7} - \frac{1}{13} a^{6} - \frac{4}{13} a^{5} + \frac{2}{13} a^{4} - \frac{3}{13} a^{3} - \frac{6}{13} a^{2} - \frac{3}{13} a - \frac{1}{13}$, $\frac{1}{13} a^{13} + \frac{2}{13} a^{11} + \frac{5}{13} a^{10} + \frac{3}{13} a^{9} + \frac{6}{13} a^{8} + \frac{6}{13} a^{7} + \frac{4}{13} a^{6} - \frac{5}{13} a^{5} - \frac{6}{13} a^{4} + \frac{5}{13} a^{3} + \frac{6}{13} a^{2} - \frac{3}{13} a - \frac{5}{13}$, $\frac{1}{143} a^{14} + \frac{1}{143} a^{13} - \frac{2}{143} a^{12} + \frac{53}{143} a^{11} - \frac{9}{143} a^{10} - \frac{30}{143} a^{9} + \frac{3}{11} a^{8} - \frac{32}{143} a^{7} + \frac{68}{143} a^{6} - \frac{21}{143} a^{5} - \frac{48}{143} a^{4} + \frac{23}{143} a^{3} - \frac{25}{143} a^{2} + \frac{4}{143} a - \frac{6}{13}$, $\frac{1}{36503515023884296566014144367636863} a^{15} + \frac{127535590651708335849469277346276}{36503515023884296566014144367636863} a^{14} - \frac{642662358676047804363292728796516}{36503515023884296566014144367636863} a^{13} - \frac{410361786511217768060583734612565}{36503515023884296566014144367636863} a^{12} - \frac{3667677201574320507729227941890712}{36503515023884296566014144367636863} a^{11} - \frac{653514866823980450555757370948933}{36503515023884296566014144367636863} a^{10} - \frac{5526220979377682184005590204113191}{36503515023884296566014144367636863} a^{9} - \frac{3607354811647768332286524852411252}{36503515023884296566014144367636863} a^{8} - \frac{4108670002435345489881395863952716}{36503515023884296566014144367636863} a^{7} + \frac{17338915461784068988250538345505054}{36503515023884296566014144367636863} a^{6} + \frac{8707879399512012785922453361930299}{36503515023884296566014144367636863} a^{5} + \frac{18144442094093213355128421615687942}{36503515023884296566014144367636863} a^{4} + \frac{17843267271328202845458870087491520}{36503515023884296566014144367636863} a^{3} + \frac{2467143931128189482941545875490177}{36503515023884296566014144367636863} a^{2} - \frac{5644358449393176475312451436915633}{36503515023884296566014144367636863} a + \frac{526238824538989158603660269819097}{3318501365807663324183104033421533}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{12}$, which has order $48$ (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:  \( 242252.826442 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{78}) \), \(\Q(\sqrt{26}) \), 4.4.29952.1, 4.4.7488.1, \(\Q(\sqrt{3}, \sqrt{26})\), 8.0.419853238272.1, 8.0.104963309568.1, 8.8.606454677504.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$