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Properties

Label 16.0.14832394327...8416.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 17^{8}\cdot 89^{2}$
Root discriminant $24.31$
Ramified primes $2, 17, 89$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_2^4.C_2$ (as 16T608)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -448, 1608, -3696, 6496, -9392, 10446, -8260, 5318, -3328, 1649, -696, 300, -78, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 28*x^14 - 78*x^13 + 300*x^12 - 696*x^11 + 1649*x^10 - 3328*x^9 + 5318*x^8 - 8260*x^7 + 10446*x^6 - 9392*x^5 + 6496*x^4 - 3696*x^3 + 1608*x^2 - 448*x + 64)
 
gp: K = bnfinit(x^16 - 4*x^15 + 28*x^14 - 78*x^13 + 300*x^12 - 696*x^11 + 1649*x^10 - 3328*x^9 + 5318*x^8 - 8260*x^7 + 10446*x^6 - 9392*x^5 + 6496*x^4 - 3696*x^3 + 1608*x^2 - 448*x + 64, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 28 x^{14} - 78 x^{13} + 300 x^{12} - 696 x^{11} + 1649 x^{10} - 3328 x^{9} + 5318 x^{8} - 8260 x^{7} + 10446 x^{6} - 9392 x^{5} + 6496 x^{4} - 3696 x^{3} + 1608 x^{2} - 448 x + 64 \)

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:  \(14832394327221826748416=2^{28}\cdot 17^{8}\cdot 89^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.31$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{248} a^{13} + \frac{9}{124} a^{12} - \frac{27}{124} a^{11} - \frac{23}{124} a^{10} + \frac{15}{62} a^{9} + \frac{5}{62} a^{8} - \frac{103}{248} a^{7} - \frac{41}{124} a^{6} - \frac{3}{62} a^{5} - \frac{14}{31} a^{4} + \frac{7}{124} a^{3} - \frac{7}{62} a^{2} + \frac{5}{62} a + \frac{13}{31}$, $\frac{1}{992} a^{14} - \frac{3}{496} a^{12} - \frac{95}{496} a^{11} + \frac{9}{62} a^{10} + \frac{45}{248} a^{9} - \frac{463}{992} a^{8} + \frac{71}{248} a^{7} - \frac{99}{248} a^{6} + \frac{119}{248} a^{5} + \frac{147}{496} a^{4} + \frac{29}{62} a^{3} + \frac{69}{248} a^{2} + \frac{15}{62} a - \frac{12}{31}$, $\frac{1}{34699542364859552} a^{15} - \frac{5244832149689}{34699542364859552} a^{14} + \frac{12765937642355}{17349771182429776} a^{13} - \frac{591234320895}{81838543313348} a^{12} + \frac{350876418438571}{17349771182429776} a^{11} + \frac{2051786653944751}{8674885591214888} a^{10} - \frac{417327213929667}{34699542364859552} a^{9} + \frac{12882301272912323}{34699542364859552} a^{8} + \frac{3171968504867001}{8674885591214888} a^{7} - \frac{807837080401723}{4337442795607444} a^{6} + \frac{8233632567027101}{17349771182429776} a^{5} - \frac{154807741443643}{17349771182429776} a^{4} + \frac{3038006846106283}{8674885591214888} a^{3} + \frac{2680141920029839}{8674885591214888} a^{2} + \frac{324705373551488}{1084360698901861} a - \frac{216008727172932}{1084360698901861}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 356433.950423 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T608):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), 4.0.1088.2 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.2312.1 x2, 8.0.342102016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$