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Properties

Label 16.8.79909289421...0209.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 83^{4}$
Root discriminant $36.01$
Ramified primes $17, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 6656, 6272, -26064, -27, 19675, -8914, -8624, 3872, 4303, 99, -994, -11, 87, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 + 87*x^13 - 11*x^12 - 994*x^11 + 99*x^10 + 4303*x^9 + 3872*x^8 - 8624*x^7 - 8914*x^6 + 19675*x^5 - 27*x^4 - 26064*x^3 + 6272*x^2 + 6656*x + 4096)
 
gp: K = bnfinit(x^16 - 3*x^15 - 15*x^14 + 87*x^13 - 11*x^12 - 994*x^11 + 99*x^10 + 4303*x^9 + 3872*x^8 - 8624*x^7 - 8914*x^6 + 19675*x^5 - 27*x^4 - 26064*x^3 + 6272*x^2 + 6656*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 15 x^{14} + 87 x^{13} - 11 x^{12} - 994 x^{11} + 99 x^{10} + 4303 x^{9} + 3872 x^{8} - 8624 x^{7} - 8914 x^{6} + 19675 x^{5} - 27 x^{4} - 26064 x^{3} + 6272 x^{2} + 6656 x + 4096 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7990928942138374856180209=17^{14}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.01$
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:  $17, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{104} a^{13} + \frac{9}{104} a^{12} + \frac{25}{104} a^{11} - \frac{5}{104} a^{10} + \frac{1}{104} a^{9} - \frac{15}{52} a^{8} + \frac{27}{104} a^{7} - \frac{29}{104} a^{6} + \frac{7}{26} a^{5} - \frac{3}{26} a^{4} + \frac{1}{4} a^{3} - \frac{25}{104} a^{2} + \frac{49}{104} a + \frac{5}{13}$, $\frac{1}{832} a^{14} - \frac{3}{832} a^{13} + \frac{177}{832} a^{12} - \frac{201}{832} a^{11} - \frac{43}{832} a^{10} + \frac{31}{416} a^{9} - \frac{29}{832} a^{8} - \frac{145}{832} a^{7} - \frac{11}{26} a^{6} + \frac{1}{52} a^{5} - \frac{201}{416} a^{4} - \frac{389}{832} a^{3} + \frac{37}{832} a^{2} - \frac{5}{52} a - \frac{1}{13}$, $\frac{1}{25750418986993453095251712174592} a^{15} - \frac{7124630221647389090817395619}{25750418986993453095251712174592} a^{14} - \frac{15757732277671502239899256303}{25750418986993453095251712174592} a^{13} - \frac{2347648681228927886954267205897}{25750418986993453095251712174592} a^{12} - \frac{5667192818173056063755130561067}{25750418986993453095251712174592} a^{11} + \frac{1164989226694117761188725079775}{12875209493496726547625856087296} a^{10} + \frac{5360150922320084714007321132771}{25750418986993453095251712174592} a^{9} + \frac{132320243690944411693124061797}{384334611746170941720174808576} a^{8} - \frac{19924727722357145171309651483}{201175148335886352306654001364} a^{7} - \frac{444931777539314183231130798943}{1609401186687090818453232010912} a^{6} - \frac{1147395060099151540075895467625}{12875209493496726547625856087296} a^{5} + \frac{11213174105948435988089278351387}{25750418986993453095251712174592} a^{4} - \frac{11784594343740405158478160745595}{25750418986993453095251712174592} a^{3} - \frac{224242593563381046139414705811}{1609401186687090818453232010912} a^{2} + \frac{39421336945907074229273202051}{402350296671772704613308002728} a + \frac{7156089560910768471089789637}{50293787083971588076663500341}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2930814.43896 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.23987.1, 4.2.407779.1, 8.4.2826823118297.1, \(\Q(\zeta_{17})^+\), 8.4.166283712841.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$83$83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.4.0.1$x^{4} - x + 22$$1$$4$$0$$C_4$$[\ ]^{4}$
83.8.4.1$x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$