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Properties

Label 16.10.6520302999...8687.1
Degree $16$
Signature $[10, 3]$
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![-2311, 17114, -30961, -14482, 63642, -39068, -12710, 24976, -13412, 3286, 1119, -1346, 284, 98, -35, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 35*x^14 + 98*x^13 + 284*x^12 - 1346*x^11 + 1119*x^10 + 3286*x^9 - 13412*x^8 + 24976*x^7 - 12710*x^6 - 39068*x^5 + 63642*x^4 - 14482*x^3 - 30961*x^2 + 17114*x - 2311)
 
gp: K = bnfinit(x^16 - 2*x^15 - 35*x^14 + 98*x^13 + 284*x^12 - 1346*x^11 + 1119*x^10 + 3286*x^9 - 13412*x^8 + 24976*x^7 - 12710*x^6 - 39068*x^5 + 63642*x^4 - 14482*x^3 - 30961*x^2 + 17114*x - 2311, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 35 x^{14} + 98 x^{13} + 284 x^{12} - 1346 x^{11} + 1119 x^{10} + 3286 x^{9} - 13412 x^{8} + 24976 x^{7} - 12710 x^{6} - 39068 x^{5} + 63642 x^{4} - 14482 x^{3} - 30961 x^{2} + 17114 x - 2311 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 3]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{8} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6}$, $\frac{1}{44838} a^{14} + \frac{847}{14946} a^{13} - \frac{91}{14946} a^{12} + \frac{935}{44838} a^{11} - \frac{281}{14946} a^{10} + \frac{766}{22419} a^{9} - \frac{5155}{22419} a^{8} - \frac{10745}{44838} a^{7} - \frac{82}{423} a^{6} - \frac{1559}{44838} a^{5} + \frac{3938}{22419} a^{4} - \frac{10213}{44838} a^{3} - \frac{3145}{14946} a^{2} - \frac{6713}{14946} a + \frac{6820}{22419}$, $\frac{1}{2188830969309219017378922} a^{15} + \frac{4987440526321850821}{1094415484654609508689461} a^{14} + \frac{1427443044856681535287}{81067813678119222865886} a^{13} - \frac{98463762910645018093603}{2188830969309219017378922} a^{12} + \frac{134285313835523532422779}{2188830969309219017378922} a^{11} - \frac{141175690434852065333299}{2188830969309219017378922} a^{10} - \frac{145453659387862608900679}{2188830969309219017378922} a^{9} + \frac{18289161569590806031157}{81067813678119222865886} a^{8} - \frac{184446538686322820573609}{2188830969309219017378922} a^{7} - \frac{702484597579417321409833}{2188830969309219017378922} a^{6} - \frac{88159825007667501805940}{364805161551536502896487} a^{5} - \frac{404352544376055316315249}{1094415484654609508689461} a^{4} - \frac{310507652217684727916579}{2188830969309219017378922} a^{3} - \frac{320289069727951015189859}{729610323103073005792974} a^{2} - \frac{69053870733229957580342}{1094415484654609508689461} a - \frac{426710562374337336433624}{1094415484654609508689461}$

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:  $12$
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:  \( 1257077823.15 \) (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