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Properties

Label 16.0.10366174140...5216.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{12}\cdot 13^{6}\cdot 97^{2}$
Root discriminant $42.26$
Ramified primes $2, 3, 13, 97$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T707)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9801, -14256, 17496, 26028, 8388, 2052, 2592, -3456, 226, 872, -40, -124, 100, -68, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 68*x^13 + 100*x^12 - 124*x^11 - 40*x^10 + 872*x^9 + 226*x^8 - 3456*x^7 + 2592*x^6 + 2052*x^5 + 8388*x^4 + 26028*x^3 + 17496*x^2 - 14256*x + 9801)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 68*x^13 + 100*x^12 - 124*x^11 - 40*x^10 + 872*x^9 + 226*x^8 - 3456*x^7 + 2592*x^6 + 2052*x^5 + 8388*x^4 + 26028*x^3 + 17496*x^2 - 14256*x + 9801, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 68 x^{13} + 100 x^{12} - 124 x^{11} - 40 x^{10} + 872 x^{9} + 226 x^{8} - 3456 x^{7} + 2592 x^{6} + 2052 x^{5} + 8388 x^{4} + 26028 x^{3} + 17496 x^{2} - 14256 x + 9801 \)

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:  \(103661741408596533461385216=2^{32}\cdot 3^{12}\cdot 13^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.26$
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:  $2, 3, 13, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{12} a^{10} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{5}{12} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{12} a^{3} + \frac{1}{6} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{11} - \frac{1}{36} a^{10} - \frac{1}{18} a^{9} - \frac{1}{18} a^{8} - \frac{1}{9} a^{7} - \frac{5}{18} a^{6} + \frac{7}{18} a^{5} + \frac{7}{36} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2}$, $\frac{1}{396} a^{13} - \frac{1}{198} a^{12} + \frac{1}{198} a^{11} + \frac{13}{396} a^{10} - \frac{23}{396} a^{9} - \frac{17}{198} a^{8} + \frac{26}{99} a^{7} - \frac{34}{99} a^{6} - \frac{119}{396} a^{5} - \frac{2}{11} a^{4} - \frac{31}{66} a^{3} - \frac{37}{132} a^{2} + \frac{9}{44} a$, $\frac{1}{2376} a^{14} - \frac{1}{1188} a^{13} - \frac{31}{2376} a^{12} + \frac{23}{1188} a^{11} - \frac{23}{2376} a^{10} - \frac{29}{297} a^{9} - \frac{259}{2376} a^{8} - \frac{116}{297} a^{7} - \frac{1043}{2376} a^{6} + \frac{65}{396} a^{5} + \frac{235}{792} a^{4} + \frac{1}{44} a^{3} + \frac{97}{264} a^{2} + \frac{1}{8}$, $\frac{1}{506907983503024808070168} a^{15} - \frac{118463743877570819}{2831888175994552000392} a^{14} - \frac{159455903417595850877}{168969327834341602690056} a^{13} - \frac{298429869119137600265}{168969327834341602690056} a^{12} + \frac{10629250180243203947465}{506907983503024808070168} a^{11} - \frac{1471022200429613108335}{56323109278113867563352} a^{10} - \frac{51718788696287814972209}{506907983503024808070168} a^{9} + \frac{55885887084635478817729}{506907983503024808070168} a^{8} - \frac{62570805765067728919865}{168969327834341602690056} a^{7} + \frac{237101072850915816662143}{506907983503024808070168} a^{6} - \frac{26309793468221438369395}{56323109278113867563352} a^{5} + \frac{3957024162185185568957}{56323109278113867563352} a^{4} + \frac{2730252591805942830625}{56323109278113867563352} a^{3} + \frac{28091405093019152606101}{56323109278113867563352} a^{2} - \frac{7325405954417756166963}{18774369759371289187784} a - \frac{328609409133426282019}{1706760887215571744344}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 232586.859044 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T707):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 8.4.48949235712.1, 8.0.104963309568.1, 8.4.1131271225344.5

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$