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Properties

Label 16.0.16707962422...4921.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{2}\cdot 73^{14}$
Root discriminant $67.06$
Ramified primes $37, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87616, 0, 462636, 0, 530997, 0, 210227, 0, 26122, 0, 919, 0, 90, 0, 23, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 23*x^14 + 90*x^12 + 919*x^10 + 26122*x^8 + 210227*x^6 + 530997*x^4 + 462636*x^2 + 87616)
 
gp: K = bnfinit(x^16 + 23*x^14 + 90*x^12 + 919*x^10 + 26122*x^8 + 210227*x^6 + 530997*x^4 + 462636*x^2 + 87616, 1)
 

Normalized defining polynomial

\( x^{16} + 23 x^{14} + 90 x^{12} + 919 x^{10} + 26122 x^{8} + 210227 x^{6} + 530997 x^{4} + 462636 x^{2} + 87616 \)

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:  \(167079624220413079894289014921=37^{2}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.06$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{9} + \frac{3}{32} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} + \frac{1}{32} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{96} a^{12} + \frac{1}{48} a^{10} - \frac{1}{96} a^{8} + \frac{1}{48} a^{6} + \frac{17}{96} a^{4} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{3552} a^{13} + \frac{13}{1776} a^{11} + \frac{131}{3552} a^{9} + \frac{175}{1776} a^{7} - \frac{1}{8} a^{6} + \frac{125}{3552} a^{5} - \frac{1}{8} a^{4} + \frac{43}{296} a^{3} + \frac{3}{8} a^{2} - \frac{55}{444} a$, $\frac{1}{9283788621074112} a^{14} - \frac{1}{7104} a^{13} + \frac{35610472750157}{9283788621074112} a^{12} - \frac{13}{3552} a^{11} - \frac{514609460085643}{9283788621074112} a^{10} + \frac{313}{7104} a^{9} + \frac{313261783069367}{9283788621074112} a^{8} + \frac{47}{3552} a^{7} + \frac{92770147528031}{9283788621074112} a^{6} + \frac{1207}{7104} a^{5} + \frac{524050956899037}{3094596207024704} a^{4} - \frac{117}{592} a^{3} - \frac{777171846527675}{2320947155268528} a^{2} - \frac{7}{111} a + \frac{570862391012}{1306839614453}$, $\frac{1}{9283788621074112} a^{15} + \frac{162901579963}{4641894310537056} a^{13} - \frac{1}{192} a^{12} + \frac{18581102611181}{9283788621074112} a^{11} - \frac{1}{96} a^{10} + \frac{166432188643081}{4641894310537056} a^{9} + \frac{1}{192} a^{8} + \frac{798463539332651}{9283788621074112} a^{7} + \frac{11}{96} a^{6} - \frac{43123639401103}{773649051756176} a^{5} - \frac{41}{192} a^{4} - \frac{648703840239421}{4641894310537056} a^{3} + \frac{7}{16} a^{2} - \frac{54494306331911}{386824525878088} a + \frac{1}{6}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 56076129.2975 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.4.408753745206589.1, 8.0.11047398519097.1, 8.4.5599366372693.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$