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Properties

Label 16.0.72866928486...2281.6
Degree $16$
Signature $[0, 8]$
Discriminant $31^{12}\cdot 41^{13}$
Root discriminant $268.48$
Ramified primes $31, 41$
Class number $48$ (GRH)
Class group $[4, 12]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1291248197075, 477508143788, 168484169151, 21789615908, -129649338, -2140918182, -204896052, 42000027, 14421522, 129865, 82171, 30608, 1002, 555, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 28*x^14 + 555*x^13 + 1002*x^12 + 30608*x^11 + 82171*x^10 + 129865*x^9 + 14421522*x^8 + 42000027*x^7 - 204896052*x^6 - 2140918182*x^5 - 129649338*x^4 + 21789615908*x^3 + 168484169151*x^2 + 477508143788*x + 1291248197075)
 
gp: K = bnfinit(x^16 - 4*x^15 + 28*x^14 + 555*x^13 + 1002*x^12 + 30608*x^11 + 82171*x^10 + 129865*x^9 + 14421522*x^8 + 42000027*x^7 - 204896052*x^6 - 2140918182*x^5 - 129649338*x^4 + 21789615908*x^3 + 168484169151*x^2 + 477508143788*x + 1291248197075, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 28 x^{14} + 555 x^{13} + 1002 x^{12} + 30608 x^{11} + 82171 x^{10} + 129865 x^{9} + 14421522 x^{8} + 42000027 x^{7} - 204896052 x^{6} - 2140918182 x^{5} - 129649338 x^{4} + 21789615908 x^{3} + 168484169151 x^{2} + 477508143788 x + 1291248197075 \)

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:  \(728669284860867208184771804455817142281=31^{12}\cdot 41^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $268.48$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} - \frac{2}{41} a^{7} + \frac{12}{41} a^{6} - \frac{6}{41} a^{5} + \frac{7}{41} a^{4} + \frac{15}{41} a^{3} - \frac{7}{41} a^{2} - \frac{20}{41} a + \frac{16}{41}$, $\frac{1}{41} a^{9} + \frac{8}{41} a^{7} + \frac{18}{41} a^{6} - \frac{5}{41} a^{5} - \frac{12}{41} a^{4} - \frac{18}{41} a^{3} + \frac{7}{41} a^{2} + \frac{17}{41} a - \frac{9}{41}$, $\frac{1}{41} a^{10} - \frac{7}{41} a^{7} - \frac{19}{41} a^{6} - \frac{5}{41} a^{5} + \frac{8}{41} a^{4} + \frac{10}{41} a^{3} - \frac{9}{41} a^{2} - \frac{13}{41} a - \frac{5}{41}$, $\frac{1}{41} a^{11} + \frac{8}{41} a^{7} - \frac{3}{41} a^{6} + \frac{7}{41} a^{5} + \frac{18}{41} a^{4} + \frac{14}{41} a^{3} + \frac{20}{41} a^{2} + \frac{19}{41} a - \frac{11}{41}$, $\frac{1}{8405} a^{12} - \frac{17}{1681} a^{11} - \frac{83}{8405} a^{10} - \frac{88}{8405} a^{9} - \frac{53}{8405} a^{8} - \frac{4128}{8405} a^{7} + \frac{1573}{8405} a^{6} - \frac{2759}{8405} a^{5} + \frac{2139}{8405} a^{4} - \frac{2438}{8405} a^{3} - \frac{2517}{8405} a^{2} - \frac{3242}{8405} a + \frac{717}{1681}$, $\frac{1}{42025} a^{13} - \frac{1}{42025} a^{12} - \frac{253}{42025} a^{11} - \frac{20}{1681} a^{10} - \frac{13}{8405} a^{9} + \frac{47}{8405} a^{8} + \frac{6}{1025} a^{7} - \frac{4492}{42025} a^{6} + \frac{7568}{42025} a^{5} + \frac{528}{42025} a^{4} + \frac{5686}{42025} a^{3} + \frac{1387}{8405} a^{2} + \frac{13337}{42025} a - \frac{443}{1681}$, $\frac{1}{24164375} a^{14} + \frac{3}{589375} a^{13} + \frac{148}{24164375} a^{12} + \frac{240228}{24164375} a^{11} + \frac{22537}{4832875} a^{10} + \frac{519}{193315} a^{9} - \frac{227014}{24164375} a^{8} - \frac{8930788}{24164375} a^{7} + \frac{217842}{4832875} a^{6} - \frac{192103}{4832875} a^{5} + \frac{11918708}{24164375} a^{4} + \frac{3842624}{24164375} a^{3} - \frac{9069423}{24164375} a^{2} - \frac{1759812}{24164375} a + \frac{221458}{966575}$, $\frac{1}{48135328467892571174400495356020466021541790273233827382899828125} a^{15} - \frac{394489889156131176916031383888604440478768854860593722903}{48135328467892571174400495356020466021541790273233827382899828125} a^{14} - \frac{7527886347350002722995783056720765698281834324610895902342}{1925413138715702846976019814240818640861671610929353095315993125} a^{13} + \frac{388882243331764492740186849197417361960044488805194399410026}{9627065693578514234880099071204093204308358054646765476579965625} a^{12} - \frac{452235893945326060310478268570979176643695260084847010999940868}{48135328467892571174400495356020466021541790273233827382899828125} a^{11} + \frac{83092474040132002411069972276087191581827456573818474774939763}{9627065693578514234880099071204093204308358054646765476579965625} a^{10} - \frac{2104781991223427904343114457199190455468946485890412585915852}{306594448840080071174525448127518891856954078173463868680890625} a^{9} - \frac{585923096687122205197184230733756288557830019524012960117319549}{48135328467892571174400495356020466021541790273233827382899828125} a^{8} + \frac{4485243907553954712063302682106763312611212131202001879878984698}{48135328467892571174400495356020466021541790273233827382899828125} a^{7} - \frac{424401458907256639470766033866415721119611447336710332432965629}{1925413138715702846976019814240818640861671610929353095315993125} a^{6} + \frac{10862169966272848068093876242747442119939908035797985373159328223}{48135328467892571174400495356020466021541790273233827382899828125} a^{5} - \frac{14496129625641127728331544205706582389835044884622678002688249784}{48135328467892571174400495356020466021541790273233827382899828125} a^{4} - \frac{16974220016670031769898898301653244011057973980109007787097219897}{48135328467892571174400495356020466021541790273233827382899828125} a^{3} + \frac{22223107298030028412551610837685403158431740929391480426129304811}{48135328467892571174400495356020466021541790273233827382899828125} a^{2} + \frac{4929944664531228565598823718283868168743863119893240825063742062}{48135328467892571174400495356020466021541790273233827382899828125} a + \frac{8823184325090141128997294563338968386798052326549889231995152}{27118494911488772492619997383673501983967205787737367539661875}$

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

Class group and class number

$C_{4}\times C_{12}$, which has order $48$ (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:  \( 582788719537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-31}) \), 4.0.39401.1, 8.0.106995634603721.3

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$31$31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$