↑

Properties

Label 16.6.65203029999...8687.2
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

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17251, 112076, -41221, -304112, -14955, 180456, -22060, -63378, 17884, 11534, -4655, -921, 572, 35, -33, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 33*x^14 + 35*x^13 + 572*x^12 - 921*x^11 - 4655*x^10 + 11534*x^9 + 17884*x^8 - 63378*x^7 - 22060*x^6 + 180456*x^5 - 14955*x^4 - 304112*x^3 - 41221*x^2 + 112076*x - 17251)
 
gp: K = bnfinit(x^16 - x^15 - 33*x^14 + 35*x^13 + 572*x^12 - 921*x^11 - 4655*x^10 + 11534*x^9 + 17884*x^8 - 63378*x^7 - 22060*x^6 + 180456*x^5 - 14955*x^4 - 304112*x^3 - 41221*x^2 + 112076*x - 17251, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 33 x^{14} + 35 x^{13} + 572 x^{12} - 921 x^{11} - 4655 x^{10} + 11534 x^{9} + 17884 x^{8} - 63378 x^{7} - 22060 x^{6} + 180456 x^{5} - 14955 x^{4} - 304112 x^{3} - 41221 x^{2} + 112076 x - 17251 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{578912673511139507121417525393908095333} a^{15} - \frac{203166008146083580100861242546491932546}{578912673511139507121417525393908095333} a^{14} - \frac{119054915208672583414079575449624311021}{578912673511139507121417525393908095333} a^{13} + \frac{157509770250997568586730568987805349226}{578912673511139507121417525393908095333} a^{12} + \frac{270113334977378633289405136549050626935}{578912673511139507121417525393908095333} a^{11} - \frac{9476596558374581510428572568252855578}{578912673511139507121417525393908095333} a^{10} - \frac{223342179524316871697007511230106704208}{578912673511139507121417525393908095333} a^{9} - \frac{104114826806414207256180991791704017919}{578912673511139507121417525393908095333} a^{8} - \frac{179143526217082778498658739275720174526}{578912673511139507121417525393908095333} a^{7} + \frac{73625432803778143672322098763484358212}{578912673511139507121417525393908095333} a^{6} + \frac{149173560029429716988824236699897343389}{578912673511139507121417525393908095333} a^{5} - \frac{8830752944350586338165743194937579459}{578912673511139507121417525393908095333} a^{4} - \frac{90644001502215645082485867777746945816}{578912673511139507121417525393908095333} a^{3} - \frac{200062513354836299070143843164676360715}{578912673511139507121417525393908095333} a^{2} - \frac{112540360045031460416031819890511682487}{578912673511139507121417525393908095333} a - \frac{5388759143337520028601712827138769130}{44531744116241500547801348107223699641}$

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