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Properties

Label 16.0.49943352689...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 67^{12}$
Root discriminant $95.75$
Ramified primes $5, 67$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1335975601, 0, 2074962124, 0, 610268042, 0, 64738207, 0, 1985485, 0, -56357, 0, -2218, 0, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 36*x^14 - 2218*x^12 - 56357*x^10 + 1985485*x^8 + 64738207*x^6 + 610268042*x^4 + 2074962124*x^2 + 1335975601)
 
gp: K = bnfinit(x^16 + 36*x^14 - 2218*x^12 - 56357*x^10 + 1985485*x^8 + 64738207*x^6 + 610268042*x^4 + 2074962124*x^2 + 1335975601, 1)
 

Normalized defining polynomial

\( x^{16} + 36 x^{14} - 2218 x^{12} - 56357 x^{10} + 1985485 x^{8} + 64738207 x^{6} + 610268042 x^{4} + 2074962124 x^{2} + 1335975601 \)

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:  \(49943352689409528470220947265625=5^{14}\cdot 67^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.75$
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:  $5, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{56245981334242573657371058019602} a^{14} + \frac{1668059177040127509235332696161}{28122990667121286828685529009801} a^{12} + \frac{1704113378525521178876786759947}{28122990667121286828685529009801} a^{10} - \frac{252778188258550899801323799172}{28122990667121286828685529009801} a^{8} + \frac{11184758683662893557635211905316}{28122990667121286828685529009801} a^{6} - \frac{1}{2} a^{5} + \frac{1185879587250543793250905537158}{28122990667121286828685529009801} a^{4} - \frac{16633891095206754280177858096101}{56245981334242573657371058019602} a^{2} - \frac{1}{2} a - \frac{25046516296703081130414193608475}{56245981334242573657371058019602}$, $\frac{1}{2055846863747900309750569541674472702} a^{15} + \frac{215901867410667159111328041540938438}{1027923431873950154875284770837236351} a^{13} + \frac{25396764685789047527481909482610250}{1027923431873950154875284770837236351} a^{11} + \frac{142498688153927043259149972845063323}{2055846863747900309750569541674472702} a^{9} - \frac{6288365150751505356067923286290108}{1027923431873950154875284770837236351} a^{7} - \frac{473448176121812362673334379068925519}{2055846863747900309750569541674472702} a^{5} - \frac{1}{2} a^{4} - \frac{253179795876521030786107310004324703}{2055846863747900309750569541674472702} a^{3} - \frac{1}{2} a^{2} + \frac{326621951845131834331202869587529477}{1027923431873950154875284770837236351} a$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( \frac{4764231301504977617388626}{2384973159800348387181635199158321} a^{15} - \frac{21430851107015}{5640279078827371107842} a^{14} + \frac{241256927173153840363945783}{4769946319600696774363270398316642} a^{13} - \frac{473871850504487}{5640279078827371107842} a^{12} - \frac{11811305339064742839002135425}{2384973159800348387181635199158321} a^{11} + \frac{27457892144040270}{2820139539413685553921} a^{10} - \frac{281521304325604374123693720891}{4769946319600696774363270398316642} a^{9} + \frac{462577907338336177}{5640279078827371107842} a^{8} + \frac{10854453407838671021877862459956}{2384973159800348387181635199158321} a^{7} - \frac{25500745113749642977}{2820139539413685553921} a^{6} + \frac{379599819509318810174280666412687}{4769946319600696774363270398316642} a^{5} - \frac{695109894795923430307}{5640279078827371107842} a^{4} + \frac{1967181731693041592090019222339717}{4769946319600696774363270398316642} a^{3} - \frac{816986963759335118115}{2820139539413685553921} a^{2} + \frac{2231002237950198337451422445716216}{2384973159800348387181635199158321} a + \frac{1828656112075454365025}{2820139539413685553921} \) (order $10$)
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:  \( 371298502.259 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-67}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-335}) \), \(\Q(\sqrt{5}, \sqrt{-67})\), 4.4.561125.1, \(\Q(\zeta_{5})\), 8.0.314861265625.1, 8.4.7067061106953125.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
67Data not computed