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Properties

Label 16.0.51399544780...7056.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63, -252, 576, -738, 585, -204, -39, 72, -56, 8, 41, -40, 43, -20, 11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 11*x^14 - 20*x^13 + 43*x^12 - 40*x^11 + 41*x^10 + 8*x^9 - 56*x^8 + 72*x^7 - 39*x^6 - 204*x^5 + 585*x^4 - 738*x^3 + 576*x^2 - 252*x + 63)
 
gp: K = bnfinit(x^16 - 2*x^15 + 11*x^14 - 20*x^13 + 43*x^12 - 40*x^11 + 41*x^10 + 8*x^9 - 56*x^8 + 72*x^7 - 39*x^6 - 204*x^5 + 585*x^4 - 738*x^3 + 576*x^2 - 252*x + 63, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 11 x^{14} - 20 x^{13} + 43 x^{12} - 40 x^{11} + 41 x^{10} + 8 x^{9} - 56 x^{8} + 72 x^{7} - 39 x^{6} - 204 x^{5} + 585 x^{4} - 738 x^{3} + 576 x^{2} - 252 x + 63 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{42} a^{12} - \frac{1}{21} a^{11} + \frac{1}{21} a^{10} - \frac{1}{21} a^{9} - \frac{4}{21} a^{8} - \frac{2}{21} a^{7} - \frac{13}{42} a^{6} + \frac{4}{21} a^{5} - \frac{5}{42} a^{4} + \frac{3}{14} a^{2} - \frac{1}{2}$, $\frac{1}{294} a^{13} + \frac{2}{147} a^{11} + \frac{10}{147} a^{10} + \frac{1}{49} a^{9} + \frac{8}{147} a^{8} + \frac{23}{98} a^{7} - \frac{23}{49} a^{6} + \frac{131}{294} a^{5} - \frac{38}{147} a^{4} + \frac{23}{98} a^{3} + \frac{18}{49} a^{2} + \frac{3}{14} a - \frac{3}{7}$, $\frac{1}{294} a^{14} - \frac{1}{98} a^{12} - \frac{4}{147} a^{11} - \frac{4}{147} a^{10} - \frac{2}{49} a^{9} + \frac{83}{294} a^{8} + \frac{50}{147} a^{7} - \frac{12}{49} a^{6} - \frac{1}{49} a^{5} - \frac{53}{147} a^{4} - \frac{24}{49} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{1}{2}$, $\frac{1}{11289181638} a^{15} - \frac{7236710}{5644590819} a^{14} - \frac{586162}{806370117} a^{13} + \frac{55869650}{5644590819} a^{12} - \frac{56643493}{806370117} a^{11} - \frac{267279256}{5644590819} a^{10} - \frac{325184401}{11289181638} a^{9} + \frac{507924882}{1881530273} a^{8} + \frac{4636660493}{11289181638} a^{7} + \frac{909795699}{1881530273} a^{6} + \frac{3059296825}{11289181638} a^{5} + \frac{437497520}{1881530273} a^{4} + \frac{56956805}{3763060546} a^{3} + \frac{485793152}{1881530273} a^{2} - \frac{107359431}{268790039} a - \frac{14994198}{268790039}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{8720}{129579} a^{15} - \frac{297656}{2116457} a^{14} + \frac{4523264}{6349371} a^{13} - \frac{8652101}{6349371} a^{12} + \frac{5565936}{2116457} a^{11} - \frac{15666796}{6349371} a^{10} + \frac{3833036}{2116457} a^{9} + \frac{230302}{302351} a^{8} - \frac{29240780}{6349371} a^{7} + \frac{25764430}{6349371} a^{6} - \frac{63572}{43193} a^{5} - \frac{30951087}{2116457} a^{4} + \frac{85229412}{2116457} a^{3} - \frac{94716982}{2116457} a^{2} + \frac{8005920}{302351} a - \frac{2127170}{302351} \) (order $6$)
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:  \( 5348.4217908 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_2$ (as 16T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{42}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.796594176.2, 8.4.7169347584.2 x2, 8.0.146313216.5 x2, 8.0.112021056.1 x2

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

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
$7$7.2.1.2$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$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$