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Properties

Label 16.0.42286532693...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{4}\cdot 7^{4}$
Root discriminant $16.85$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $C_2\times Q_8:C_2^2$ (as 16T69)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![433, -1020, 388, 1596, -1824, -92, 1082, -624, 169, 60, -170, 96, -8, -8, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 8*x^13 - 8*x^12 + 96*x^11 - 170*x^10 + 60*x^9 + 169*x^8 - 624*x^7 + 1082*x^6 - 92*x^5 - 1824*x^4 + 1596*x^3 + 388*x^2 - 1020*x + 433)
 
gp: K = bnfinit(x^16 - 4*x^15 + 6*x^14 - 8*x^13 - 8*x^12 + 96*x^11 - 170*x^10 + 60*x^9 + 169*x^8 - 624*x^7 + 1082*x^6 - 92*x^5 - 1824*x^4 + 1596*x^3 + 388*x^2 - 1020*x + 433, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 6 x^{14} - 8 x^{13} - 8 x^{12} + 96 x^{11} - 170 x^{10} + 60 x^{9} + 169 x^{8} - 624 x^{7} + 1082 x^{6} - 92 x^{5} - 1824 x^{4} + 1596 x^{3} + 388 x^{2} - 1020 x + 433 \)

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:  \(42286532693852160000=2^{32}\cdot 3^{8}\cdot 5^{4}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.85$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{4}{9} a^{7} - \frac{1}{6} a^{5} + \frac{1}{18} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} + \frac{7}{18} a + \frac{7}{18}$, $\frac{1}{18} a^{10} - \frac{4}{9} a^{7} - \frac{1}{6} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{5}{18} a^{2} - \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{8} - \frac{1}{6} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{5} + \frac{5}{18} a^{4} + \frac{5}{18} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{1}{2}$, $\frac{1}{54} a^{12} - \frac{1}{54} a^{9} + \frac{1}{27} a^{8} - \frac{11}{27} a^{7} + \frac{1}{27} a^{6} + \frac{5}{54} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{27} a^{2} - \frac{11}{54} a - \frac{17}{54}$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{10} - \frac{1}{54} a^{9} - \frac{1}{54} a^{8} + \frac{13}{27} a^{7} + \frac{23}{54} a^{6} - \frac{7}{18} a^{5} + \frac{1}{18} a^{4} - \frac{5}{27} a^{3} - \frac{5}{54} a^{2} - \frac{10}{27} a + \frac{5}{18}$, $\frac{1}{2430} a^{14} + \frac{8}{1215} a^{13} + \frac{2}{1215} a^{12} + \frac{5}{243} a^{11} - \frac{13}{1215} a^{10} - \frac{1}{45} a^{9} + \frac{17}{405} a^{8} + \frac{53}{405} a^{7} - \frac{107}{243} a^{6} + \frac{113}{243} a^{5} + \frac{511}{1215} a^{4} - \frac{4}{135} a^{3} + \frac{157}{2430} a^{2} + \frac{11}{243} a + \frac{317}{1215}$, $\frac{1}{479417130} a^{15} + \frac{7688}{239708565} a^{14} - \frac{330448}{239708565} a^{13} - \frac{251566}{47941713} a^{12} - \frac{3804928}{239708565} a^{11} + \frac{2194186}{79902855} a^{10} - \frac{1294408}{79902855} a^{9} - \frac{8387699}{159805710} a^{8} - \frac{20344562}{47941713} a^{7} + \frac{7200659}{47941713} a^{6} + \frac{16716406}{239708565} a^{5} - \frac{38775719}{159805710} a^{4} + \frac{4198117}{479417130} a^{3} + \frac{19374983}{47941713} a^{2} + \frac{69136892}{239708565} a + \frac{3472957}{31961142}$

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:  \( -\frac{492253}{239708565} a^{15} + \frac{1298039}{95883426} a^{14} - \frac{6201349}{239708565} a^{13} + \frac{15245999}{479417130} a^{12} - \frac{4568257}{239708565} a^{11} - \frac{42439969}{159805710} a^{10} + \frac{116911213}{159805710} a^{9} - \frac{39603391}{79902855} a^{8} - \frac{39686281}{239708565} a^{7} + \frac{149716663}{95883426} a^{6} - \frac{2169935437}{479417130} a^{5} + \frac{230938889}{79902855} a^{4} + \frac{1058998283}{239708565} a^{3} - \frac{1312038434}{239708565} a^{2} + \frac{38516581}{479417130} a + \frac{244914083}{159805710} \) (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:  \( 31622.0541047 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times Q_8:C_2^2$ (as 16T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 34 conjugacy class representatives for $C_2\times Q_8:C_2^2$
Character table for $C_2\times Q_8:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{24})\), 8.0.6502809600.1, 8.0.6502809600.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$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$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$