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Properties

Label 16.0.35087248320...6729.1
Degree $16$
Signature $[0, 8]$
Discriminant $67^{4}\cdot 89^{15}$
Root discriminant $192.34$
Ramified primes $67, 89$
Class number $113$ (GRH)
Class group $[113]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4052037632, -5388767872, 4169879872, -1414547264, 155453472, 23232976, 6522992, -5666394, 555525, 160413, -6878, -11049, 1382, 155, -30, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 30*x^14 + 155*x^13 + 1382*x^12 - 11049*x^11 - 6878*x^10 + 160413*x^9 + 555525*x^8 - 5666394*x^7 + 6522992*x^6 + 23232976*x^5 + 155453472*x^4 - 1414547264*x^3 + 4169879872*x^2 - 5388767872*x + 4052037632)
 
gp: K = bnfinit(x^16 - 5*x^15 - 30*x^14 + 155*x^13 + 1382*x^12 - 11049*x^11 - 6878*x^10 + 160413*x^9 + 555525*x^8 - 5666394*x^7 + 6522992*x^6 + 23232976*x^5 + 155453472*x^4 - 1414547264*x^3 + 4169879872*x^2 - 5388767872*x + 4052037632, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 30 x^{14} + 155 x^{13} + 1382 x^{12} - 11049 x^{11} - 6878 x^{10} + 160413 x^{9} + 555525 x^{8} - 5666394 x^{7} + 6522992 x^{6} + 23232976 x^{5} + 155453472 x^{4} - 1414547264 x^{3} + 4169879872 x^{2} - 5388767872 x + 4052037632 \)

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:  \(3508724832059361738820170787833596729=67^{4}\cdot 89^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.34$
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:  $67, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} + \frac{7}{32} a^{5} - \frac{1}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{8} - \frac{3}{32} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} + \frac{3}{32} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{12} - \frac{1}{64} a^{10} - \frac{7}{128} a^{9} + \frac{3}{128} a^{8} - \frac{7}{64} a^{7} + \frac{1}{16} a^{6} + \frac{13}{128} a^{5} + \frac{15}{64} a^{4} - \frac{1}{16} a^{3} + \frac{7}{16} a^{2} + \frac{3}{8} a$, $\frac{1}{256} a^{13} - \frac{1}{128} a^{11} + \frac{1}{256} a^{10} - \frac{5}{256} a^{9} - \frac{3}{128} a^{8} - \frac{1}{32} a^{7} + \frac{5}{256} a^{6} + \frac{11}{128} a^{5} - \frac{1}{16} a^{4} - \frac{5}{32} a^{3} - \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{57088} a^{14} + \frac{41}{28544} a^{13} - \frac{55}{14272} a^{12} - \frac{59}{57088} a^{11} + \frac{145}{57088} a^{10} + \frac{531}{28544} a^{9} + \frac{1699}{28544} a^{8} + \frac{4393}{57088} a^{7} + \frac{29}{892} a^{6} - \frac{587}{28544} a^{5} - \frac{1875}{14272} a^{4} - \frac{509}{3568} a^{3} + \frac{1427}{3568} a^{2} + \frac{335}{1784} a + \frac{65}{223}$, $\frac{1}{102518925930058704927700837565106718470972390904832} a^{15} - \frac{33475564347579385069510844979924037206907641}{9319902357278064084336439778646065315542944627712} a^{14} - \frac{44722202261535140009549448509567514146324616353}{25629731482514676231925209391276679617743097726208} a^{13} + \frac{221652814250871748622376226137014071633760317531}{102518925930058704927700837565106718470972390904832} a^{12} + \frac{162059161198312315684771338120159112410947261303}{25629731482514676231925209391276679617743097726208} a^{11} + \frac{1169467018041141949251396040759293548798506676623}{102518925930058704927700837565106718470972390904832} a^{10} - \frac{394876495190503941526349022008861928231739902879}{12814865741257338115962604695638339808871548863104} a^{9} + \frac{1843541120818169823701022590946798511284215066485}{102518925930058704927700837565106718470972390904832} a^{8} - \frac{12387459835515261538507656388085686730191669632225}{102518925930058704927700837565106718470972390904832} a^{7} + \frac{2126755612372043035136949815269675907904736225249}{25629731482514676231925209391276679617743097726208} a^{6} + \frac{552367872483772126298234633405456875184297907533}{3203716435314334528990651173909584952217887215776} a^{5} - \frac{513369766396877852064406339813273203545403138891}{6407432870628669057981302347819169904435774431552} a^{4} + \frac{192617067621223355466339232469881546857479812791}{3203716435314334528990651173909584952217887215776} a^{3} - \frac{514538477526692332268202215528836242347031346965}{1601858217657167264495325586954792476108943607888} a^{2} - \frac{490183601022807004475369497404643846393677527825}{1601858217657167264495325586954792476108943607888} a + \frac{26665945306573727795490882552658205986047334853}{100116138603572954030957849184674529756808975493}$

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

Class group and class number

$C_{113}$, which has order $113$ (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:  \( 1890544540260 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.0.44231334895529.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
67Data not computed
89Data not computed