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Properties

Label 16.2.65203029999...8687.6
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{12}\cdot 47^{9}$
Root discriminant $73.01$
Ramified primes $17, 47$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3727, -79, 3178, -32841, 63944, -81156, 88676, -81596, 57332, -27976, 8450, -1356, 162, -85, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 85*x^13 + 162*x^12 - 1356*x^11 + 8450*x^10 - 27976*x^9 + 57332*x^8 - 81596*x^7 + 88676*x^6 - 81156*x^5 + 63944*x^4 - 32841*x^3 + 3178*x^2 - 79*x + 3727)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 85*x^13 + 162*x^12 - 1356*x^11 + 8450*x^10 - 27976*x^9 + 57332*x^8 - 81596*x^7 + 88676*x^6 - 81156*x^5 + 63944*x^4 - 32841*x^3 + 3178*x^2 - 79*x + 3727, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 85 x^{13} + 162 x^{12} - 1356 x^{11} + 8450 x^{10} - 27976 x^{9} + 57332 x^{8} - 81596 x^{7} + 88676 x^{6} - 81156 x^{5} + 63944 x^{4} - 32841 x^{3} + 3178 x^{2} - 79 x + 3727 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{4}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9}$, $\frac{1}{193725804670157322688452854724567} a^{15} - \frac{938885340780609897467077625737}{21525089407795258076494761636063} a^{14} + \frac{2984975919011360577649853437240}{21525089407795258076494761636063} a^{13} - \frac{26800414350476894693716813084462}{193725804670157322688452854724567} a^{12} - \frac{38543908931562222029930841005612}{193725804670157322688452854724567} a^{11} - \frac{39990504917908018172143536514966}{193725804670157322688452854724567} a^{10} + \frac{3216026718675419678800411419757}{21525089407795258076494761636063} a^{9} + \frac{36148671931552291641884265251510}{193725804670157322688452854724567} a^{8} - \frac{11857313598089893780996850493121}{64575268223385774229484284908189} a^{7} + \frac{80139485675959402647375380314252}{193725804670157322688452854724567} a^{6} - \frac{95886175039577876345312830530026}{193725804670157322688452854724567} a^{5} + \frac{48259897921952493621494321304908}{193725804670157322688452854724567} a^{4} - \frac{3305226708784967292820650178031}{21525089407795258076494761636063} a^{3} + \frac{7075896990693754638063518681080}{21525089407795258076494761636063} a^{2} - \frac{59672176097504966015617927622762}{193725804670157322688452854724567} a - \frac{12809126688141590225754616359521}{193725804670157322688452854724567}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $8$
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:  \( 94680095.5605 \) (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