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Properties

Label 16.4.18000574394...1681.1
Degree $16$
Signature $[4, 6]$
Discriminant $41^{14}\cdot 83^{4}$
Root discriminant $77.80$
Ramified primes $41, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23369, -322426, 902493, -1123766, 661683, 2310, -279342, 192420, -40093, -22288, 18244, -4798, -66, 286, -32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 32*x^14 + 286*x^13 - 66*x^12 - 4798*x^11 + 18244*x^10 - 22288*x^9 - 40093*x^8 + 192420*x^7 - 279342*x^6 + 2310*x^5 + 661683*x^4 - 1123766*x^3 + 902493*x^2 - 322426*x - 23369)
 
gp: K = bnfinit(x^16 - 6*x^15 - 32*x^14 + 286*x^13 - 66*x^12 - 4798*x^11 + 18244*x^10 - 22288*x^9 - 40093*x^8 + 192420*x^7 - 279342*x^6 + 2310*x^5 + 661683*x^4 - 1123766*x^3 + 902493*x^2 - 322426*x - 23369, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 32 x^{14} + 286 x^{13} - 66 x^{12} - 4798 x^{11} + 18244 x^{10} - 22288 x^{9} - 40093 x^{8} + 192420 x^{7} - 279342 x^{6} + 2310 x^{5} + 661683 x^{4} - 1123766 x^{3} + 902493 x^{2} - 322426 x - 23369 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1800057439498232157527332231681=41^{14}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.80$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{587896878122286342771875510325989238538} a^{15} + \frac{33967544795041077009507902442329867457}{293948439061143171385937755162994619269} a^{14} + \frac{145470423518334678038189678546952610713}{587896878122286342771875510325989238538} a^{13} + \frac{120482135864641642374880600592920406711}{587896878122286342771875510325989238538} a^{12} + \frac{8129199692615943721542981930787622969}{293948439061143171385937755162994619269} a^{11} - \frac{5020167210618590731325893636668137347}{293948439061143171385937755162994619269} a^{10} + \frac{9822556265430763705772418425188744522}{293948439061143171385937755162994619269} a^{9} - \frac{117829304951465463497232448560021881023}{587896878122286342771875510325989238538} a^{8} + \frac{283020247410391747851865162673247338439}{587896878122286342771875510325989238538} a^{7} + \frac{77201956062163618631373577765509455917}{293948439061143171385937755162994619269} a^{6} + \frac{16376589880696772008535281682592708621}{587896878122286342771875510325989238538} a^{5} + \frac{45989398141347394512303926531283394289}{587896878122286342771875510325989238538} a^{4} - \frac{143069489332875205001379618319559916317}{587896878122286342771875510325989238538} a^{3} - \frac{268796305475162712513257765457658032785}{587896878122286342771875510325989238538} a^{2} - \frac{187263694356887283900518929630683138465}{587896878122286342771875510325989238538} a - \frac{181623792302423527372268105832196784525}{587896878122286342771875510325989238538}$

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:  $9$
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:  \( 715150414.327 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.2.394258652003.1, 8.4.1341662192766209.3, 8.6.16164604732123.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
41Data not computed
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$