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Properties

Label 16.6.65203029999...8687.3
Degree $16$
Signature $[6, 5]$
Discriminant $-\,17^{12}\cdot 47^{9}$
Root discriminant $73.01$
Ramified primes $17, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4759, -4173, -5137, 28413, -42541, 39909, -12193, -1733, 11469, -8756, 3516, -930, -211, 181, -21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 21*x^14 + 181*x^13 - 211*x^12 - 930*x^11 + 3516*x^10 - 8756*x^9 + 11469*x^8 - 1733*x^7 - 12193*x^6 + 39909*x^5 - 42541*x^4 + 28413*x^3 - 5137*x^2 - 4173*x + 4759)
 
gp: K = bnfinit(x^16 - 6*x^15 - 21*x^14 + 181*x^13 - 211*x^12 - 930*x^11 + 3516*x^10 - 8756*x^9 + 11469*x^8 - 1733*x^7 - 12193*x^6 + 39909*x^5 - 42541*x^4 + 28413*x^3 - 5137*x^2 - 4173*x + 4759, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 21 x^{14} + 181 x^{13} - 211 x^{12} - 930 x^{11} + 3516 x^{10} - 8756 x^{9} + 11469 x^{8} - 1733 x^{7} - 12193 x^{6} + 39909 x^{5} - 42541 x^{4} + 28413 x^{3} - 5137 x^{2} - 4173 x + 4759 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-652030299991134977060343848687=-\,17^{12}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.01$
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:  $17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{26} a^{14} - \frac{1}{13} a^{13} + \frac{3}{26} a^{12} - \frac{1}{26} a^{11} - \frac{1}{13} a^{10} + \frac{5}{26} a^{9} - \frac{6}{13} a^{8} + \frac{1}{26} a^{7} - \frac{11}{26} a^{5} + \frac{9}{26} a^{4} + \frac{4}{13} a^{3} - \frac{5}{13} a^{2} + \frac{3}{26} a + \frac{1}{13}$, $\frac{1}{12043762984133866990026247723959578} a^{15} - \frac{72446080804774595413173870440209}{6021881492066933495013123861979789} a^{14} - \frac{305167193408721912080304211901641}{12043762984133866990026247723959578} a^{13} + \frac{1228381230585972415441782944247933}{12043762984133866990026247723959578} a^{12} - \frac{2608753652298037362843539445533854}{6021881492066933495013123861979789} a^{11} + \frac{636957941106676422314572155270083}{12043762984133866990026247723959578} a^{10} + \frac{1014892491983899584341620580664292}{6021881492066933495013123861979789} a^{9} - \frac{4542470077999428784422157720740219}{12043762984133866990026247723959578} a^{8} - \frac{1639955437604689948150788101422257}{6021881492066933495013123861979789} a^{7} - \frac{1880819375811229997245911056989869}{12043762984133866990026247723959578} a^{6} + \frac{3696971609715192564551880078795831}{12043762984133866990026247723959578} a^{5} - \frac{70328383730208710272250359152986}{6021881492066933495013123861979789} a^{4} + \frac{861636168345365672312858765983085}{6021881492066933495013123861979789} a^{3} - \frac{4204626748170732646027052450399683}{12043762984133866990026247723959578} a^{2} - \frac{1360353434843633312671503189871992}{6021881492066933495013123861979789} a + \frac{1732012280573546547413483879782995}{6021881492066933495013123861979789}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $10$
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:  \( 468564403.701 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.6.2506034826287.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
47Data not computed