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Properties

Label 16.8.88114204274...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 13^{8}\cdot 101^{5}\cdot 199^{4}$
Root discriminant $362.30$
Ramified primes $2, 5, 13, 101, 199$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1581

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1068656, 27381472, -5593072, -159744048, -104734960, -241115296, 102497032, 1905944, -6108960, 52168, 247268, -11968, -4490, 1064, -118, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 118*x^14 + 1064*x^13 - 4490*x^12 - 11968*x^11 + 247268*x^10 + 52168*x^9 - 6108960*x^8 + 1905944*x^7 + 102497032*x^6 - 241115296*x^5 - 104734960*x^4 - 159744048*x^3 - 5593072*x^2 + 27381472*x + 1068656)
 
gp: K = bnfinit(x^16 - 4*x^15 - 118*x^14 + 1064*x^13 - 4490*x^12 - 11968*x^11 + 247268*x^10 + 52168*x^9 - 6108960*x^8 + 1905944*x^7 + 102497032*x^6 - 241115296*x^5 - 104734960*x^4 - 159744048*x^3 - 5593072*x^2 + 27381472*x + 1068656, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 118 x^{14} + 1064 x^{13} - 4490 x^{12} - 11968 x^{11} + 247268 x^{10} + 52168 x^{9} - 6108960 x^{8} + 1905944 x^{7} + 102497032 x^{6} - 241115296 x^{5} - 104734960 x^{4} - 159744048 x^{3} - 5593072 x^{2} + 27381472 x + 1068656 \)

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:  \(88114204274383121870561308973465600000000=2^{24}\cdot 5^{8}\cdot 13^{8}\cdot 101^{5}\cdot 199^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $362.30$
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:  $2, 5, 13, 101, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{40} a^{12} - \frac{1}{20} a^{11} - \frac{1}{10} a^{10} - \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{760} a^{13} - \frac{1}{380} a^{12} - \frac{47}{380} a^{11} - \frac{17}{380} a^{10} - \frac{43}{380} a^{9} - \frac{1}{20} a^{8} - \frac{21}{190} a^{7} - \frac{23}{95} a^{6} - \frac{8}{95} a^{5} - \frac{7}{190} a^{4} + \frac{42}{95} a^{3} + \frac{2}{5} a^{2} + \frac{8}{95} a + \frac{9}{19}$, $\frac{1}{5320} a^{14} - \frac{3}{5320} a^{13} + \frac{41}{5320} a^{12} - \frac{99}{1330} a^{11} - \frac{51}{1330} a^{10} + \frac{233}{2660} a^{9} + \frac{243}{2660} a^{8} - \frac{3}{665} a^{7} - \frac{4}{665} a^{6} + \frac{33}{665} a^{5} + \frac{53}{1330} a^{4} + \frac{243}{665} a^{3} - \frac{239}{665} a^{2} - \frac{229}{665} a + \frac{69}{665}$, $\frac{1}{6673128152553263701019367885821781073840802625722277435800} a^{15} - \frac{557550036919291523411995432256897123532400177894259}{47665201089666169292995484898727007670291447326587695970} a^{14} - \frac{527900692152753096818854820997811886684275820298353136}{834141019069157962627420985727722634230100328215284679475} a^{13} - \frac{2556471879372270896111817094511116626021930218273374801}{238326005448330846464977424493635038351457236632938479850} a^{12} - \frac{152967073768245377682311836118772073904597713016664730851}{3336564076276631850509683942910890536920401312861138717900} a^{11} - \frac{79943793805746028409271166976400839088009257422662262387}{834141019069157962627420985727722634230100328215284679475} a^{10} + \frac{110723563448261732407884760759961222187246708800279275771}{1668282038138315925254841971455445268460200656430569358950} a^{9} + \frac{269028371213535676083446830131532894852473638627015878587}{3336564076276631850509683942910890536920401312861138717900} a^{8} - \frac{204959913296303875144467069043186145060169591282629749821}{1668282038138315925254841971455445268460200656430569358950} a^{7} - \frac{18468439148302337416193321627238427792612926457174464869}{238326005448330846464977424493635038351457236632938479850} a^{6} + \frac{20593432899787121798515538986068744100989455976043062988}{834141019069157962627420985727722634230100328215284679475} a^{5} + \frac{31832462163917460362352231805242882943561710404727351197}{166828203813831592525484197145544526846020065643056935895} a^{4} - \frac{7379150680275574543320019914194568361991490967062883858}{23832600544833084646497742449363503835145723663293847985} a^{3} + \frac{11590387340587882645890996248763444951649752005425420609}{834141019069157962627420985727722634230100328215284679475} a^{2} + \frac{276518182424978211735124750783598761440615597833452391462}{834141019069157962627420985727722634230100328215284679475} a - \frac{140293361241377094857124596705999771022165358881842602258}{834141019069157962627420985727722634230100328215284679475}$

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:  \( 119486682937000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1581:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1581 are not computed
Character table for t16n1581 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.426725.1, 4.4.1358692400.1, 4.4.79600.1, 8.8.1846045037817760000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.8.4.1$x^{8} + 237606 x^{4} - 7880599 x^{2} + 14114152809$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$