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Properties

Label 16.0.45086848686...0000.2
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![4, 0, 144, -464, 1036, -1592, 2000, -2024, 1766, -1280, 808, -416, 186, -64, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^14 - 64*x^13 + 186*x^12 - 416*x^11 + 808*x^10 - 1280*x^9 + 1766*x^8 - 2024*x^7 + 2000*x^6 - 1592*x^5 + 1036*x^4 - 464*x^3 + 144*x^2 + 4)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^14 - 64*x^13 + 186*x^12 - 416*x^11 + 808*x^10 - 1280*x^9 + 1766*x^8 - 2024*x^7 + 2000*x^6 - 1592*x^5 + 1036*x^4 - 464*x^3 + 144*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{14} - 64 x^{13} + 186 x^{12} - 416 x^{11} + 808 x^{10} - 1280 x^{9} + 1766 x^{8} - 2024 x^{7} + 2000 x^{6} - 1592 x^{5} + 1036 x^{4} - 464 x^{3} + 144 x^{2} + 4 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{154} a^{14} - \frac{4}{77} a^{13} + \frac{5}{22} a^{12} + \frac{9}{154} a^{11} + \frac{17}{154} a^{10} - \frac{3}{22} a^{9} - \frac{13}{154} a^{8} - \frac{12}{77} a^{7} + \frac{2}{77} a^{6} + \frac{31}{77} a^{5} - \frac{5}{77} a^{4} + \frac{6}{77} a^{3} - \frac{37}{77} a^{2} - \frac{32}{77} a - \frac{18}{77}$, $\frac{1}{1299546710} a^{15} + \frac{1028743}{1299546710} a^{14} + \frac{94950788}{649773355} a^{13} + \frac{233183043}{1299546710} a^{12} - \frac{161058063}{1299546710} a^{11} + \frac{11303504}{59070305} a^{10} - \frac{180776721}{1299546710} a^{9} - \frac{31928111}{185649530} a^{8} + \frac{301874896}{649773355} a^{7} - \frac{27615246}{129954671} a^{6} + \frac{41055467}{129954671} a^{5} + \frac{93783079}{649773355} a^{4} + \frac{240859691}{649773355} a^{3} - \frac{11934897}{129954671} a^{2} - \frac{128753788}{649773355} a + \frac{280148234}{649773355}$

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:  \( \frac{27279473}{649773355} a^{15} - \frac{76801091}{649773355} a^{14} + \frac{921541451}{1299546710} a^{13} - \frac{1265309226}{649773355} a^{12} + \frac{538234758}{92824765} a^{11} - \frac{15483357807}{1299546710} a^{10} + \frac{15011272322}{649773355} a^{9} - \frac{44433443657}{1299546710} a^{8} + \frac{30248722566}{649773355} a^{7} - \frac{920139665}{18564953} a^{6} + \frac{6148411090}{129954671} a^{5} - \frac{22085791536}{649773355} a^{4} + \frac{13261138656}{649773355} a^{3} - \frac{847189822}{129954671} a^{2} + \frac{1061705892}{649773355} a + \frac{550616219}{649773355} \) (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:  \( 5843.72447927 \)
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{24})\), 8.0.2123366400.7 x2, 8.4.2123366400.1 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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$