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Properties

Label 16.8.70499058785...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{10}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $97.84$
Ramified primes $2, 5, 29, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![277986479, -261568276, -90353990, 503482600, -89381223, 41315996, 234858, -7558680, 244651, -104488, 43294, 20656, -1811, -268, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 268*x^13 - 1811*x^12 + 20656*x^11 + 43294*x^10 - 104488*x^9 + 244651*x^8 - 7558680*x^7 + 234858*x^6 + 41315996*x^5 - 89381223*x^4 + 503482600*x^3 - 90353990*x^2 - 261568276*x + 277986479)
 
gp: K = bnfinit(x^16 - 40*x^14 - 268*x^13 - 1811*x^12 + 20656*x^11 + 43294*x^10 - 104488*x^9 + 244651*x^8 - 7558680*x^7 + 234858*x^6 + 41315996*x^5 - 89381223*x^4 + 503482600*x^3 - 90353990*x^2 - 261568276*x + 277986479, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 268 x^{13} - 1811 x^{12} + 20656 x^{11} + 43294 x^{10} - 104488 x^{9} + 244651 x^{8} - 7558680 x^{7} + 234858 x^{6} + 41315996 x^{5} - 89381223 x^{4} + 503482600 x^{3} - 90353990 x^{2} - 261568276 x + 277986479 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(70499058785862276874240000000000=2^{32}\cdot 5^{10}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.84$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{15} + \frac{266297664644956833420064414373835014527835592074768432786232181780310176}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{14} - \frac{307128047409466879200639647852087091168259712610331543595996389883879558}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{13} + \frac{1168037597790319911328683297802468084489693987305670246157869876325763079}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{12} - \frac{752484662976341555649723010991102421619208421468565272319345854816540836}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{11} - \frac{1078469350369624058058758884590147333946068380602488491599313023461257338}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{10} - \frac{1107320256274249865763176655263793596497454410919646672148129731487105271}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{9} + \frac{615822090170065518856015918469571983936926106403656493071488781148663876}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{8} + \frac{884237965860847999431778134061179466842941453670258812671524191331227577}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{7} - \frac{106078957370224932473683551251767568201735413262747597216732608350361733}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{6} + \frac{442840570059991795145453735370834888001938466007905173230096567224773010}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{5} + \frac{413846341952281629040757047819631933203964086770017399094339178263702116}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{4} + \frac{533851773928998172388639590822470836459060507579895998889042225578792221}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{3} - \frac{1059550773845043272802421599417690127630310339089791454641217091036149874}{2415110534515281372999014072598541569469535949149086272884960622798290839} a^{2} + \frac{1189315418208668409640679005360339528945503725472391928661593925568770274}{2415110534515281372999014072598541569469535949149086272884960622798290839} a + \frac{1184685196204106829174340664656488990816085622195629955816495097676600103}{2415110534515281372999014072598541569469535949149086272884960622798290839}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 3910099620.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

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 $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.46400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.725.1, 8.8.2152960000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ 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 ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed