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Properties

Label 16.0.40291974979...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 17^{8}\cdot 19^{2}$
Root discriminant $39.84$
Ramified primes $2, 5, 17, 19$
Class number $32$ (GRH)
Class group $[2, 16]$ (GRH)
Galois group $C_2^6.C_2^2$ (as 16T528)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20736, 0, -26208, 0, 16021, 0, -3229, 0, 234, 0, 163, 0, 54, 0, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^14 + 54*x^12 + 163*x^10 + 234*x^8 - 3229*x^6 + 16021*x^4 - 26208*x^2 + 20736)
 
gp: K = bnfinit(x^16 - x^14 + 54*x^12 + 163*x^10 + 234*x^8 - 3229*x^6 + 16021*x^4 - 26208*x^2 + 20736, 1)
 

Normalized defining polynomial

\( x^{16} - x^{14} + 54 x^{12} + 163 x^{10} + 234 x^{8} - 3229 x^{6} + 16021 x^{4} - 26208 x^{2} + 20736 \)

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:  \(40291974979216000000000000=2^{16}\cdot 5^{12}\cdot 17^{8}\cdot 19^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.84$
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:  $2, 5, 17, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{6} a^{5} - \frac{1}{4} a^{4} + \frac{1}{3} a^{3} + \frac{1}{4} a^{2} - \frac{5}{12} a$, $\frac{1}{12} a^{8} + \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{5}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{3} a^{2} + \frac{1}{4} a$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{5} + \frac{5}{12} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{6} + \frac{5}{12} a^{2}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{7} - \frac{1}{8} a^{6} - \frac{1}{24} a^{5} + \frac{1}{4} a^{4} + \frac{1}{24} a^{3} + \frac{1}{24} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{72} a^{12} - \frac{1}{36} a^{10} - \frac{1}{24} a^{9} - \frac{1}{72} a^{8} - \frac{2}{9} a^{6} - \frac{1}{8} a^{5} - \frac{29}{72} a^{4} - \frac{1}{4} a^{3} - \frac{7}{36} a^{2} - \frac{11}{24} a - \frac{1}{2}$, $\frac{1}{72} a^{13} + \frac{1}{72} a^{11} + \frac{1}{36} a^{9} - \frac{1}{72} a^{7} - \frac{1}{4} a^{6} - \frac{7}{36} a^{5} + \frac{1}{4} a^{4} - \frac{11}{72} a^{3} - \frac{1}{4} a^{2} - \frac{11}{24} a - \frac{1}{2}$, $\frac{1}{247089161664} a^{14} - \frac{1312656049}{247089161664} a^{12} - \frac{558124621}{13727175648} a^{10} - \frac{1698625709}{247089161664} a^{8} + \frac{344479573}{13727175648} a^{6} - \frac{1}{4} a^{5} - \frac{45142346221}{247089161664} a^{4} - \frac{1}{4} a^{3} + \frac{7990057621}{247089161664} a^{2} + \frac{1}{4} a - \frac{754062049}{1715896956}$, $\frac{1}{5930139879936} a^{15} - \frac{1}{494178323328} a^{14} - \frac{1312656049}{5930139879936} a^{13} + \frac{1312656049}{494178323328} a^{12} + \frac{585806683}{329452215552} a^{11} - \frac{585806683}{27454351296} a^{10} + \frac{142436718595}{5930139879936} a^{9} - \frac{18892137763}{494178323328} a^{8} - \frac{647794345}{109817405184} a^{7} - \frac{1640068263}{9151450432} a^{6} - \frac{642274486909}{5930139879936} a^{5} - \frac{222537578915}{494178323328} a^{4} - \frac{1412772621947}{5930139879936} a^{3} + \frac{177326813627}{494178323328} a^{2} - \frac{19628928565}{41181526944} a - \frac{961834907}{3431793912}$

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

Class group and class number

$C_{2}\times C_{16}$, which has order $32$ (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:  \( 108588.634681 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^6.C_2^2$ (as 16T528):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^6.C_2^2$
Character table for $C_2^6.C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 8.0.15868990000.1, 8.0.1586899000000.3, 8.8.83521000000.1

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.16.14$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{2} + 12$$4$$2$$16$$C_2^2:C_4$$[2, 2, 3]^{2}$
5Data not computed
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$