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Properties

Label 16.8.27375119589...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 13^{4}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}$
Root discriminant $21.87$
Ramified primes $5, 13, 19, 29, 31$
Class number $1$
Class group Trivial
Galois group 16T1116

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 59, 255, 86, -701, -483, 637, 444, -403, -15, 161, -114, 32, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 6*x^14 + 32*x^13 - 114*x^12 + 161*x^11 - 15*x^10 - 403*x^9 + 444*x^8 + 637*x^7 - 483*x^6 - 701*x^5 + 86*x^4 + 255*x^3 + 59*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 6*x^14 + 32*x^13 - 114*x^12 + 161*x^11 - 15*x^10 - 403*x^9 + 444*x^8 + 637*x^7 - 483*x^6 - 701*x^5 + 86*x^4 + 255*x^3 + 59*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 6 x^{14} + 32 x^{13} - 114 x^{12} + 161 x^{11} - 15 x^{10} - 403 x^{9} + 444 x^{8} + 637 x^{7} - 483 x^{6} - 701 x^{5} + 86 x^{4} + 255 x^{3} + 59 x^{2} + 9 x + 1 \)

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:  \(2737511958932953515625=5^{8}\cdot 13^{4}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.87$
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:  $5, 13, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{5}{13} a^{10} + \frac{2}{13} a^{9} - \frac{3}{13} a^{8} + \frac{1}{13} a^{7} + \frac{6}{13} a^{6} + \frac{3}{13} a^{5} - \frac{5}{13} a^{4} - \frac{1}{13} a^{3} - \frac{4}{13} a^{2} - \frac{6}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{12} + \frac{3}{13} a^{10} + \frac{3}{13} a^{8} + \frac{1}{13} a^{7} - \frac{1}{13} a^{6} + \frac{6}{13} a^{5} - \frac{2}{13} a^{4} + \frac{1}{13} a^{3} + \frac{1}{13} a^{2} - \frac{4}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{13} - \frac{2}{13} a^{10} - \frac{3}{13} a^{9} - \frac{3}{13} a^{8} - \frac{4}{13} a^{7} + \frac{1}{13} a^{6} + \frac{2}{13} a^{5} + \frac{3}{13} a^{4} + \frac{4}{13} a^{3} - \frac{5}{13} a^{2} + \frac{6}{13} a - \frac{2}{13}$, $\frac{1}{1131} a^{14} - \frac{20}{1131} a^{13} + \frac{5}{1131} a^{12} + \frac{14}{1131} a^{11} - \frac{323}{1131} a^{10} + \frac{401}{1131} a^{9} + \frac{103}{377} a^{8} + \frac{388}{1131} a^{7} + \frac{229}{1131} a^{6} - \frac{21}{377} a^{5} + \frac{517}{1131} a^{4} + \frac{281}{1131} a^{3} + \frac{515}{1131} a^{2} - \frac{134}{1131} a - \frac{187}{1131}$, $\frac{1}{39151921190811} a^{15} + \frac{502416731}{1186421854267} a^{14} + \frac{673450913944}{39151921190811} a^{13} + \frac{339442410524}{13050640396937} a^{12} - \frac{31123648372}{39151921190811} a^{11} + \frac{576707720138}{3559265562801} a^{10} + \frac{1017486079606}{39151921190811} a^{9} - \frac{126966355786}{1350066247959} a^{8} - \frac{3813790103961}{13050640396937} a^{7} + \frac{170495495969}{39151921190811} a^{6} + \frac{12495121861672}{39151921190811} a^{5} + \frac{321115860305}{1350066247959} a^{4} - \frac{1786316454794}{13050640396937} a^{3} - \frac{2320165202374}{39151921190811} a^{2} - \frac{1062068594491}{3559265562801} a + \frac{12589972067308}{39151921190811}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44270.6822206 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1116:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1116 are not computed
Character table for t16n1116 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.309593125.1, 8.4.52321238125.1, 8.4.88830625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$