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Properties

Label 16.8.88114204274...0000.1
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 $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1581

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-129643679, 76456848, 338879286, -403444758, 126060809, -7160878, -3705988, 3591426, -1113909, 124450, -12508, 146, -1751, 360, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 30*x^14 + 360*x^13 - 1751*x^12 + 146*x^11 - 12508*x^10 + 124450*x^9 - 1113909*x^8 + 3591426*x^7 - 3705988*x^6 - 7160878*x^5 + 126060809*x^4 - 403444758*x^3 + 338879286*x^2 + 76456848*x - 129643679)
 
gp: K = bnfinit(x^16 - 4*x^15 - 30*x^14 + 360*x^13 - 1751*x^12 + 146*x^11 - 12508*x^10 + 124450*x^9 - 1113909*x^8 + 3591426*x^7 - 3705988*x^6 - 7160878*x^5 + 126060809*x^4 - 403444758*x^3 + 338879286*x^2 + 76456848*x - 129643679, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 30 x^{14} + 360 x^{13} - 1751 x^{12} + 146 x^{11} - 12508 x^{10} + 124450 x^{9} - 1113909 x^{8} + 3591426 x^{7} - 3705988 x^{6} - 7160878 x^{5} + 126060809 x^{4} - 403444758 x^{3} + 338879286 x^{2} + 76456848 x - 129643679 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{13} - \frac{1}{25} a^{12} + \frac{2}{25} a^{10} + \frac{2}{25} a^{9} + \frac{2}{25} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{11}{25} a^{5} - \frac{1}{25} a^{3} - \frac{4}{25} a^{2} - \frac{7}{25} a + \frac{2}{25}$, $\frac{1}{721221056582734884959680916913267398789863211020124600725} a^{15} + \frac{1012883091283932730866709808849641050979957839190711176}{721221056582734884959680916913267398789863211020124600725} a^{14} + \frac{11769881570826592814811882288354920289533000907251867622}{721221056582734884959680916913267398789863211020124600725} a^{13} - \frac{3604669200238538669747664138280545922889801555164352632}{65565550598430444087243719719387945344533019183647690975} a^{12} - \frac{4266185858251224778462603343677400052335875134657529813}{721221056582734884959680916913267398789863211020124600725} a^{11} + \frac{32015672329665135088921103368795240474258916766367337001}{721221056582734884959680916913267398789863211020124600725} a^{10} + \frac{46227931225892049834791707360321595754782467660332009566}{721221056582734884959680916913267398789863211020124600725} a^{9} + \frac{9540052684037985009331207021676792337747472169266022514}{721221056582734884959680916913267398789863211020124600725} a^{8} + \frac{7393888480556325853787851395895670599987787698629802146}{28848842263309395398387236676530695951594528440804984029} a^{7} - \frac{213084201140876221170985124682040570704986599361227088219}{721221056582734884959680916913267398789863211020124600725} a^{6} + \frac{32209056605565729263503093149958099464172773965344147797}{721221056582734884959680916913267398789863211020124600725} a^{5} - \frac{288489965555538863288249089584830287484547744316045519186}{721221056582734884959680916913267398789863211020124600725} a^{4} - \frac{170955638665860305562187578071143868361864706828450821521}{721221056582734884959680916913267398789863211020124600725} a^{3} + \frac{63270316482568809996054520580150132124905442422879290606}{144244211316546976991936183382653479757972642204024920145} a^{2} - \frac{120044328440233489709399222688166197811081352325644572802}{721221056582734884959680916913267398789863211020124600725} a - \frac{12536890072837731265969708612881183378494794531581311306}{65565550598430444087243719719387945344533019183647690975}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 108229332651000 \) (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

\(\Q(\sqrt{5}) \), 4.4.1358692400.1, 4.4.426725.1, 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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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}$