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Properties

Label 16.8.94284067507...4928.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 17^{15}\cdot 26626013^{4}$
Root discriminant $2045.98$
Ramified primes $2, 17, 26626013$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8544240815100822030896306381537, 0, -1357399571910239703957606270, 0, -1577424202471246092688679, 0, -105732588674541402, 0, 60897826532184704, 0, 1798329785850, 0, -638585467, 0, -19890, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19890*x^14 - 638585467*x^12 + 1798329785850*x^10 + 60897826532184704*x^8 - 105732588674541402*x^6 - 1577424202471246092688679*x^4 - 1357399571910239703957606270*x^2 + 8544240815100822030896306381537)
 
gp: K = bnfinit(x^16 - 19890*x^14 - 638585467*x^12 + 1798329785850*x^10 + 60897826532184704*x^8 - 105732588674541402*x^6 - 1577424202471246092688679*x^4 - 1357399571910239703957606270*x^2 + 8544240815100822030896306381537, 1)
 

Normalized defining polynomial

\( x^{16} - 19890 x^{14} - 638585467 x^{12} + 1798329785850 x^{10} + 60897826532184704 x^{8} - 105732588674541402 x^{6} - 1577424202471246092688679 x^{4} - 1357399571910239703957606270 x^{2} + 8544240815100822030896306381537 \)

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:  \(94284067507195126346045323522879461199867166580604928=2^{16}\cdot 17^{15}\cdot 26626013^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2045.98$
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:  $2, 17, 26626013$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{106504052} a^{10} - \frac{9945}{53252026} a^{8} + \frac{438845}{106504052} a^{6} - \frac{17758183}{106504052} a^{4} + \frac{23812967}{53252026} a^{2} + \frac{1}{4}$, $\frac{1}{106504052} a^{11} - \frac{9945}{53252026} a^{9} + \frac{438845}{106504052} a^{7} - \frac{17758183}{106504052} a^{5} + \frac{23812967}{53252026} a^{3} + \frac{1}{4} a$, $\frac{1}{5671556546209352} a^{12} - \frac{9945}{2835778273104676} a^{10} + \frac{354471964845351}{2835778273104676} a^{8} - \frac{177011350845811}{708944568276169} a^{6} - \frac{35703169782915}{2835778273104676} a^{4} - \frac{15188015}{106504052} a^{2} + \frac{3}{8}$, $\frac{1}{5671556546209352} a^{13} - \frac{9945}{2835778273104676} a^{11} + \frac{354471964845351}{2835778273104676} a^{9} - \frac{177011350845811}{708944568276169} a^{7} - \frac{35703169782915}{2835778273104676} a^{5} - \frac{15188015}{106504052} a^{3} + \frac{3}{8} a$, $\frac{1}{1923101510642529814403860203575394194695985823171983378622232652366812974607318264496} a^{14} - \frac{133477362874473844919636746556416354119086803207043570241631444291713}{1923101510642529814403860203575394194695985823171983378622232652366812974607318264496} a^{12} + \frac{181021825275091568946644002837738813012560060154410230726660130213237032132}{120193844415158113400241262723462137168499113948248961163889540772925810912957391531} a^{10} + \frac{9076148832996555819143169560734874137097559656121759184204229677121769047581825947}{961550755321264907201930101787697097347992911585991689311116326183406487303659132248} a^{8} + \frac{9559519246435931157401854281616316433250652317539521652246122018257981049655448907}{73965442717020377477071546291361315180614839352768591485470486629492806715666087096} a^{6} - \frac{394898009077909856621688231158268882457860563261054902160103967459391624240}{4514151045263822014968642234324085140666727457402238974490455659768731086887} a^{4} + \frac{16476086123186608296102384797295501328334702904306631797209127962977}{208663543037985944391756101735158142154116610348267669931099943376368} a^{2} + \frac{38696735471936655292276461282781995276510945620711334056392851}{101878792724010811423130805297701005704591067935236105725040368}$, $\frac{1}{1923101510642529814403860203575394194695985823171983378622232652366812974607318264496} a^{15} - \frac{133477362874473844919636746556416354119086803207043570241631444291713}{1923101510642529814403860203575394194695985823171983378622232652366812974607318264496} a^{13} + \frac{181021825275091568946644002837738813012560060154410230726660130213237032132}{120193844415158113400241262723462137168499113948248961163889540772925810912957391531} a^{11} + \frac{9076148832996555819143169560734874137097559656121759184204229677121769047581825947}{961550755321264907201930101787697097347992911585991689311116326183406487303659132248} a^{9} + \frac{9559519246435931157401854281616316433250652317539521652246122018257981049655448907}{73965442717020377477071546291361315180614839352768591485470486629492806715666087096} a^{7} - \frac{394898009077909856621688231158268882457860563261054902160103967459391624240}{4514151045263822014968642234324085140666727457402238974490455659768731086887} a^{5} + \frac{16476086123186608296102384797295501328334702904306631797209127962977}{208663543037985944391756101735158142154116610348267669931099943376368} a^{3} + \frac{38696735471936655292276461282781995276510945620711334056392851}{101878792724010811423130805297701005704591067935236105725040368} a$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 67879824549300000000 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
17Data not computed
26626013Data not computed