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Properties

Label 16.8.15712755557...5489.4
Degree $16$
Signature $[8, 4]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1009009, -27302620, 56984839, -37693454, 3162395, 8817560, -5313570, 1175012, 93887, -138518, 48486, -12212, 3256, -722, 78, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 78*x^14 - 722*x^13 + 3256*x^12 - 12212*x^11 + 48486*x^10 - 138518*x^9 + 93887*x^8 + 1175012*x^7 - 5313570*x^6 + 8817560*x^5 + 3162395*x^4 - 37693454*x^3 + 56984839*x^2 - 27302620*x - 1009009)
 
gp: K = bnfinit(x^16 - 6*x^15 + 78*x^14 - 722*x^13 + 3256*x^12 - 12212*x^11 + 48486*x^10 - 138518*x^9 + 93887*x^8 + 1175012*x^7 - 5313570*x^6 + 8817560*x^5 + 3162395*x^4 - 37693454*x^3 + 56984839*x^2 - 27302620*x - 1009009, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 78 x^{14} - 722 x^{13} + 3256 x^{12} - 12212 x^{11} + 48486 x^{10} - 138518 x^{9} + 93887 x^{8} + 1175012 x^{7} - 5313570 x^{6} + 8817560 x^{5} + 3162395 x^{4} - 37693454 x^{3} + 56984839 x^{2} - 27302620 x - 1009009 \)

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:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1506087225458763075719049480871357637570405217049066} a^{15} + \frac{253061460271101715199054399612770263833788532392849}{1506087225458763075719049480871357637570405217049066} a^{14} - \frac{321164586727161643972499400466898431563244715630981}{1506087225458763075719049480871357637570405217049066} a^{13} + \frac{727625828564123446840572018612775786508596097203}{3478261490666889320367319817254867523257286875402} a^{12} + \frac{141096559934198091612738292691099882491199732279159}{753043612729381537859524740435678818785202608524533} a^{11} - \frac{343963951879421303538822799689671196924089556649575}{1506087225458763075719049480871357637570405217049066} a^{10} - \frac{296836961192334039747664048152879295248093557502027}{1506087225458763075719049480871357637570405217049066} a^{9} - \frac{318198721260447692092427133132957771242257043983027}{1506087225458763075719049480871357637570405217049066} a^{8} - \frac{179336681575170217118613577316960146570254240984725}{753043612729381537859524740435678818785202608524533} a^{7} - \frac{527625536190029621053426764234129558231847717136483}{1506087225458763075719049480871357637570405217049066} a^{6} - \frac{266433594988238905197469369859292203321378594740686}{753043612729381537859524740435678818785202608524533} a^{5} + \frac{134640118167482865921228489102824305180188963819630}{753043612729381537859524740435678818785202608524533} a^{4} - \frac{281156099435388073328198887935851694968021111982998}{753043612729381537859524740435678818785202608524533} a^{3} - \frac{322342085887062111594704549727042964304710898701521}{1506087225458763075719049480871357637570405217049066} a^{2} - \frac{589360971360548441301794722324509655080713607720435}{1506087225458763075719049480871357637570405217049066} a - \frac{123648787238714766424119232363769922175901413351150}{753043612729381537859524740435678818785202608524533}$

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:  \( 652376806060 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed