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Properties

Label 16.10.6520302999...8687.2
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![-103, 1321, 3002, -17369, -11930, 19365, 15508, -2043, -2077, 1092, -267, -586, 96, 89, -20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 20*x^14 + 89*x^13 + 96*x^12 - 586*x^11 - 267*x^10 + 1092*x^9 - 2077*x^8 - 2043*x^7 + 15508*x^6 + 19365*x^5 - 11930*x^4 - 17369*x^3 + 3002*x^2 + 1321*x - 103)
 
gp: K = bnfinit(x^16 - 6*x^15 - 20*x^14 + 89*x^13 + 96*x^12 - 586*x^11 - 267*x^10 + 1092*x^9 - 2077*x^8 - 2043*x^7 + 15508*x^6 + 19365*x^5 - 11930*x^4 - 17369*x^3 + 3002*x^2 + 1321*x - 103, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 20 x^{14} + 89 x^{13} + 96 x^{12} - 586 x^{11} - 267 x^{10} + 1092 x^{9} - 2077 x^{8} - 2043 x^{7} + 15508 x^{6} + 19365 x^{5} - 11930 x^{4} - 17369 x^{3} + 3002 x^{2} + 1321 x - 103 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{14464743189577666428664639396021} a^{15} + \frac{3247670139614147621975334660372}{14464743189577666428664639396021} a^{14} + \frac{385910939712985483327369665889}{14464743189577666428664639396021} a^{13} + \frac{6107619985616075404845602876562}{14464743189577666428664639396021} a^{12} + \frac{4903535805928011746764469846895}{14464743189577666428664639396021} a^{11} - \frac{600793152224221252579155240951}{14464743189577666428664639396021} a^{10} - \frac{6117225640600806904470578346336}{14464743189577666428664639396021} a^{9} + \frac{3848752957687235464453253989300}{14464743189577666428664639396021} a^{8} - \frac{1054449771342799535834254619113}{14464743189577666428664639396021} a^{7} - \frac{7138130466904328754659826354187}{14464743189577666428664639396021} a^{6} - \frac{1983317673812992775098671866278}{14464743189577666428664639396021} a^{5} - \frac{1459966965923714155904341310090}{14464743189577666428664639396021} a^{4} + \frac{2739995266837555274032690478027}{14464743189577666428664639396021} a^{3} + \frac{2054561716032122248245847405192}{14464743189577666428664639396021} a^{2} - \frac{5611486262668458134666391823478}{14464743189577666428664639396021} a + \frac{6989744258861858290786012755986}{14464743189577666428664639396021}$

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