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Properties

Label 16.0.22858790439...8513.5
Degree $16$
Signature $[0, 8]$
Discriminant $37^{6}\cdot 73^{15}$
Root discriminant $216.24$
Ramified primes $37, 73$
Class number $712$ (GRH)
Class group $[2, 2, 178]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28380726272, -15384971776, 3724493056, -3203742192, 477553112, -14629984, 97125874, 5391363, 48414, -478033, 47429, -18171, 3421, -611, 117, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 117*x^14 - 611*x^13 + 3421*x^12 - 18171*x^11 + 47429*x^10 - 478033*x^9 + 48414*x^8 + 5391363*x^7 + 97125874*x^6 - 14629984*x^5 + 477553112*x^4 - 3203742192*x^3 + 3724493056*x^2 - 15384971776*x + 28380726272)
 
gp: K = bnfinit(x^16 - 4*x^15 + 117*x^14 - 611*x^13 + 3421*x^12 - 18171*x^11 + 47429*x^10 - 478033*x^9 + 48414*x^8 + 5391363*x^7 + 97125874*x^6 - 14629984*x^5 + 477553112*x^4 - 3203742192*x^3 + 3724493056*x^2 - 15384971776*x + 28380726272, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 117 x^{14} - 611 x^{13} + 3421 x^{12} - 18171 x^{11} + 47429 x^{10} - 478033 x^{9} + 48414 x^{8} + 5391363 x^{7} + 97125874 x^{6} - 14629984 x^{5} + 477553112 x^{4} - 3203742192 x^{3} + 3724493056 x^{2} - 15384971776 x + 28380726272 \)

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:  \(22858790439424412670626523398015008513=37^{6}\cdot 73^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.24$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} + \frac{3}{32} a^{6} - \frac{5}{32} a^{5} - \frac{5}{32} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{5}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{3}{32} a^{6} + \frac{11}{64} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{256} a^{13} + \frac{1}{128} a^{11} - \frac{3}{256} a^{10} - \frac{13}{256} a^{9} + \frac{5}{128} a^{8} - \frac{5}{64} a^{7} - \frac{15}{256} a^{6} + \frac{11}{128} a^{5} + \frac{7}{32} a^{4} + \frac{5}{32} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{35072} a^{14} - \frac{5}{35072} a^{13} - \frac{123}{17536} a^{12} + \frac{379}{35072} a^{11} + \frac{205}{17536} a^{10} + \frac{1963}{35072} a^{9} - \frac{647}{17536} a^{8} - \frac{99}{35072} a^{7} - \frac{279}{35072} a^{6} + \frac{1437}{17536} a^{5} - \frac{373}{2192} a^{4} + \frac{401}{4384} a^{3} + \frac{313}{2192} a^{2} + \frac{203}{548} a + \frac{63}{137}$, $\frac{1}{1847757715293505452662216396946986517667836618599113352050615808} a^{15} - \frac{14277602856300044213332838723325387029638024921443985288289}{1847757715293505452662216396946986517667836618599113352050615808} a^{14} + \frac{46509465515359456971195794580028530021013633659276395069319}{28871214301461022697847131202296664338559947165611146125790872} a^{13} + \frac{2160538967331675398110197757090722258557603106422263264795771}{1847757715293505452662216396946986517667836618599113352050615808} a^{12} - \frac{9101511271195064872141528251673548074637598421056365780143951}{923878857646752726331108198473493258833918309299556676025307904} a^{11} - \frac{19200830244071426750047006346832901695008158713324869048192499}{1847757715293505452662216396946986517667836618599113352050615808} a^{10} + \frac{15123069247274675313132244813385720661408609922723919576710667}{461939428823376363165554099236746629416959154649778338012653952} a^{9} + \frac{9126035628542267533860878259813515872063536805131459643042929}{1847757715293505452662216396946986517667836618599113352050615808} a^{8} + \frac{106051012087181478310393335803494595899282003726517233835603549}{1847757715293505452662216396946986517667836618599113352050615808} a^{7} + \frac{4350366336332378183576036856066019229908480832353913817094281}{115484857205844090791388524809186657354239788662444584503163488} a^{6} - \frac{105201808317529900850187226558022583961964429037115576732596987}{461939428823376363165554099236746629416959154649778338012653952} a^{5} + \frac{9423342933638102980405359100209113780318895217406302044408767}{230969714411688181582777049618373314708479577324889169006326976} a^{4} + \frac{255000613549553077770171188190028894162411906190992131899315}{28871214301461022697847131202296664338559947165611146125790872} a^{3} + \frac{14702430602074784315496552254336629414536296019991676250483383}{57742428602922045395694262404593328677119894331222292251581744} a^{2} + \frac{120850020154270600057993552666040804815907013783976841409363}{390151544614338144565501773004008977548107394129880353051228} a + \frac{17273780929978761652366820758876463894553993217659214494705}{97537886153584536141375443251002244387026848532470088262807}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{178}$, which has order $712$ (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:  \( 793547094607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.0.11047398519097.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ $16$ $16$ $16$ $16$

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
37Data not computed
73Data not computed