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Properties

Label 16.0.45086848686...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 5^{4}$
Root discriminant $14.65$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -72, 204, -168, -278, 576, -12, -644, 401, 16, 80, -164, 36, 24, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 24*x^13 + 36*x^12 - 164*x^11 + 80*x^10 + 16*x^9 + 401*x^8 - 644*x^7 - 12*x^6 + 576*x^5 - 278*x^4 - 168*x^3 + 204*x^2 - 72*x + 9)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 24*x^13 + 36*x^12 - 164*x^11 + 80*x^10 + 16*x^9 + 401*x^8 - 644*x^7 - 12*x^6 + 576*x^5 - 278*x^4 - 168*x^3 + 204*x^2 - 72*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 24 x^{13} + 36 x^{12} - 164 x^{11} + 80 x^{10} + 16 x^{9} + 401 x^{8} - 644 x^{7} - 12 x^{6} + 576 x^{5} - 278 x^{4} - 168 x^{3} + 204 x^{2} - 72 x + 9 \)

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:  \(4508684868648960000=2^{40}\cdot 3^{8}\cdot 5^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.65$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9} a^{12} - \frac{1}{3} a^{11} - \frac{2}{9} a^{10} + \frac{4}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{2}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{2}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{7857} a^{14} + \frac{82}{2619} a^{13} + \frac{136}{7857} a^{12} + \frac{460}{7857} a^{11} + \frac{371}{873} a^{10} + \frac{317}{873} a^{9} - \frac{73}{7857} a^{8} + \frac{107}{873} a^{7} + \frac{331}{2619} a^{6} + \frac{2491}{7857} a^{5} + \frac{844}{7857} a^{4} - \frac{32}{291} a^{3} - \frac{709}{2619} a^{2} - \frac{376}{873} a - \frac{71}{873}$, $\frac{1}{7857} a^{15} - \frac{143}{7857} a^{13} + \frac{178}{7857} a^{12} + \frac{59}{2619} a^{11} + \frac{37}{873} a^{10} - \frac{3511}{7857} a^{9} + \frac{1069}{2619} a^{8} - \frac{938}{2619} a^{7} + \frac{34}{7857} a^{6} + \frac{3523}{7857} a^{5} + \frac{1217}{2619} a^{4} + \frac{8}{2619} a^{3} + \frac{338}{873} a^{2} - \frac{113}{873} a - \frac{95}{291}$

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:  \( -\frac{5785}{2619} a^{15} + \frac{65642}{7857} a^{14} + \frac{29641}{2619} a^{13} - \frac{410812}{7857} a^{12} - \frac{756643}{7857} a^{11} + \frac{102037}{291} a^{10} - \frac{159038}{2619} a^{9} - \frac{843530}{7857} a^{8} - \frac{836357}{873} a^{7} + \frac{3108766}{2619} a^{6} + \frac{4241246}{7857} a^{5} - \frac{9551863}{7857} a^{4} + \frac{36718}{291} a^{3} + \frac{1311535}{2619} a^{2} - \frac{232961}{873} a + \frac{35042}{873} \) (order $24$)
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:  \( 7455.41021685 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2$ (as 16T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{24})\), 8.0.235929600.1 x2, 8.0.132710400.2 x2

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 8 siblings: data not computed
Degree 16 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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$