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Properties

Label 16.0.58013247259...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{8}\cdot 41^{6}$
Root discriminant $72.48$
Ramified primes $5, 29, 41$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![616025, -146575, 176275, -137475, 239995, -156970, 116130, -73660, 56256, -29958, 12580, -3638, 949, -197, 45, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 45*x^14 - 197*x^13 + 949*x^12 - 3638*x^11 + 12580*x^10 - 29958*x^9 + 56256*x^8 - 73660*x^7 + 116130*x^6 - 156970*x^5 + 239995*x^4 - 137475*x^3 + 176275*x^2 - 146575*x + 616025)
 
gp: K = bnfinit(x^16 - 7*x^15 + 45*x^14 - 197*x^13 + 949*x^12 - 3638*x^11 + 12580*x^10 - 29958*x^9 + 56256*x^8 - 73660*x^7 + 116130*x^6 - 156970*x^5 + 239995*x^4 - 137475*x^3 + 176275*x^2 - 146575*x + 616025, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 45 x^{14} - 197 x^{13} + 949 x^{12} - 3638 x^{11} + 12580 x^{10} - 29958 x^{9} + 56256 x^{8} - 73660 x^{7} + 116130 x^{6} - 156970 x^{5} + 239995 x^{4} - 137475 x^{3} + 176275 x^{2} - 146575 x + 616025 \)

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:  \(580132472595479362207275390625=5^{12}\cdot 29^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.48$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{3}{20} a^{7} + \frac{1}{4} a^{6} + \frac{1}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{4} a$, $\frac{1}{120} a^{13} - \frac{1}{40} a^{12} - \frac{1}{40} a^{11} + \frac{1}{15} a^{10} + \frac{1}{20} a^{9} - \frac{17}{120} a^{8} + \frac{7}{30} a^{7} - \frac{1}{15} a^{6} - \frac{1}{120} a^{5} + \frac{11}{30} a^{4} - \frac{1}{4} a^{3} - \frac{7}{24} a^{2} + \frac{11}{24} a - \frac{1}{24}$, $\frac{1}{240} a^{14} - \frac{1}{40} a^{12} - \frac{13}{240} a^{11} - \frac{1}{8} a^{10} - \frac{11}{240} a^{9} + \frac{1}{240} a^{8} - \frac{1}{120} a^{7} + \frac{1}{48} a^{6} - \frac{7}{240} a^{5} - \frac{17}{40} a^{4} + \frac{23}{48} a^{3} - \frac{11}{24} a^{2} - \frac{5}{24} a - \frac{1}{16}$, $\frac{1}{1269576050172529178399023100338872720} a^{15} - \frac{600506507976196596135506684174177}{634788025086264589199511550169436360} a^{14} - \frac{397840392913446432861440061562401}{211596008362088196399837183389812120} a^{13} + \frac{11088831036978242871200297574378971}{1269576050172529178399023100338872720} a^{12} - \frac{32390371613305993235602764184474517}{317394012543132294599755775084718180} a^{11} + \frac{35410734096958192268405648267616709}{1269576050172529178399023100338872720} a^{10} + \frac{12027410565115736927518880295896981}{84638403344835278559934873355924848} a^{9} - \frac{1949881301761204907714300119728979}{52899002090522049099959295847453030} a^{8} + \frac{270974863345209888444720180743027593}{1269576050172529178399023100338872720} a^{7} - \frac{183676551630160657903256858196455379}{423192016724176392799674366779624240} a^{6} - \frac{2650834916248673332896629746757384}{79348503135783073649938943771179545} a^{5} + \frac{525568225901299532642197901868300403}{1269576050172529178399023100338872720} a^{4} - \frac{1604746854201069757938023775664137}{5289900209052204909995929584745303} a^{3} + \frac{5679855865679043744112732037616205}{42319201672417639279967436677962424} a^{2} + \frac{110651436362637522152968944115916521}{253915210034505835679804620067774544} a + \frac{6360447100932304977008005060726807}{42319201672417639279967436677962424}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 19191274.6552 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 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

\(\Q(\sqrt{5}) \), 4.0.29725.2, 4.0.3625.1, 4.4.5125.1, 8.0.22089390625.5

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$