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Properties

Label 16.0.19324652256...6393.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{7}\cdot 31^{15}$
Root discriminant $58.60$
Ramified primes $7, 31$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14175, 30240, 34983, 22722, 16660, 1822, 10, 337, -416, 711, 545, 96, 88, -1, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 - x^13 + 88*x^12 + 96*x^11 + 545*x^10 + 711*x^9 - 416*x^8 + 337*x^7 + 10*x^6 + 1822*x^5 + 16660*x^4 + 22722*x^3 + 34983*x^2 + 30240*x + 14175)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 - x^13 + 88*x^12 + 96*x^11 + 545*x^10 + 711*x^9 - 416*x^8 + 337*x^7 + 10*x^6 + 1822*x^5 + 16660*x^4 + 22722*x^3 + 34983*x^2 + 30240*x + 14175, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} - x^{13} + 88 x^{12} + 96 x^{11} + 545 x^{10} + 711 x^{9} - 416 x^{8} + 337 x^{7} + 10 x^{6} + 1822 x^{5} + 16660 x^{4} + 22722 x^{3} + 34983 x^{2} + 30240 x + 14175 \)

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:  \(19324652256549748184809636393=7^{7}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.60$
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:  $7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{11} + \frac{2}{15} a^{10} - \frac{2}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{7}{15} a^{6} - \frac{1}{3} a^{5} - \frac{7}{15} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{135} a^{12} - \frac{2}{135} a^{11} + \frac{2}{27} a^{10} - \frac{13}{135} a^{9} + \frac{1}{9} a^{8} + \frac{14}{135} a^{7} + \frac{7}{15} a^{6} - \frac{7}{135} a^{5} - \frac{64}{135} a^{4} - \frac{8}{135} a^{3} - \frac{13}{135} a^{2} + \frac{11}{45} a - \frac{1}{3}$, $\frac{1}{945} a^{13} + \frac{1}{315} a^{12} + \frac{1}{105} a^{11} + \frac{11}{189} a^{10} + \frac{67}{945} a^{9} - \frac{20}{189} a^{8} + \frac{97}{945} a^{7} + \frac{40}{189} a^{6} - \frac{12}{35} a^{5} + \frac{239}{945} a^{4} + \frac{199}{945} a^{3} - \frac{131}{945} a^{2} - \frac{8}{45} a + \frac{1}{3}$, $\frac{1}{61425} a^{14} + \frac{1}{4725} a^{13} + \frac{41}{20475} a^{12} + \frac{197}{12285} a^{11} + \frac{9143}{61425} a^{10} + \frac{418}{6825} a^{9} + \frac{89}{2925} a^{8} - \frac{1087}{20475} a^{7} - \frac{23083}{61425} a^{6} + \frac{17831}{61425} a^{5} + \frac{8899}{20475} a^{4} + \frac{15551}{61425} a^{3} - \frac{24998}{61425} a^{2} - \frac{989}{2925} a + \frac{46}{195}$, $\frac{1}{1824896133739382947425} a^{15} - \frac{9600411283903177}{1824896133739382947425} a^{14} + \frac{605208075256788958}{1824896133739382947425} a^{13} - \frac{269489981070317498}{364979226747876589485} a^{12} + \frac{22514865847046749483}{1824896133739382947425} a^{11} + \frac{41721402166316967884}{608298711246460982475} a^{10} + \frac{276643537642922017424}{1824896133739382947425} a^{9} - \frac{6714319442353024}{4573674520650082575} a^{8} - \frac{860457737712301544}{5944287080584309275} a^{7} + \frac{88511080950266214406}{1824896133739382947425} a^{6} - \frac{758598602080052775698}{1824896133739382947425} a^{5} - \frac{83851201327266275564}{1824896133739382947425} a^{4} - \frac{711173534448722302493}{1824896133739382947425} a^{3} - \frac{73560077805163974758}{608298711246460982475} a^{2} + \frac{19819064384244323}{340783591734712035} a - \frac{40956383973722701}{128740467988668991}$

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:  \( -1 \) (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:  \( 118753618.963 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{-31}) \), 4.0.208537.1, 8.0.9436826640073.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16$ $16$ R $16$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
31Data not computed