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Properties

Label 16.0.24471271802...0000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{14}\cdot 9929^{3}$
Root discriminant $38.62$
Ramified primes $2, 5, 9929$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1872

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26951, -22942, 94071, -33138, 101581, -36406, 57849, -22064, 20667, -6854, 4189, -1036, 481, -78, 31, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 31*x^14 - 78*x^13 + 481*x^12 - 1036*x^11 + 4189*x^10 - 6854*x^9 + 20667*x^8 - 22064*x^7 + 57849*x^6 - 36406*x^5 + 101581*x^4 - 33138*x^3 + 94071*x^2 - 22942*x + 26951)
 
gp: K = bnfinit(x^16 - 2*x^15 + 31*x^14 - 78*x^13 + 481*x^12 - 1036*x^11 + 4189*x^10 - 6854*x^9 + 20667*x^8 - 22064*x^7 + 57849*x^6 - 36406*x^5 + 101581*x^4 - 33138*x^3 + 94071*x^2 - 22942*x + 26951, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 31 x^{14} - 78 x^{13} + 481 x^{12} - 1036 x^{11} + 4189 x^{10} - 6854 x^{9} + 20667 x^{8} - 22064 x^{7} + 57849 x^{6} - 36406 x^{5} + 101581 x^{4} - 33138 x^{3} + 94071 x^{2} - 22942 x + 26951 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24471271802225000000000000=2^{12}\cdot 5^{14}\cdot 9929^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.62$
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:  $2, 5, 9929$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{87744001385981763256547640328638038} a^{15} - \frac{9002931712368674620225092387910150}{43872000692990881628273820164319019} a^{14} + \frac{16654768950110325767782953094082385}{43872000692990881628273820164319019} a^{13} + \frac{32188460487190470971443783457647251}{87744001385981763256547640328638038} a^{12} - \frac{10142101304701862585765158726037883}{43872000692990881628273820164319019} a^{11} - \frac{20851340166061154834531177871490321}{43872000692990881628273820164319019} a^{10} - \frac{14319378506994831074531351040869245}{87744001385981763256547640328638038} a^{9} - \frac{103928722808470105808336175720569}{242386744160170616730794586543199} a^{8} - \frac{21795513811649477715850645218263900}{43872000692990881628273820164319019} a^{7} + \frac{35366599926900925344338757922679123}{87744001385981763256547640328638038} a^{6} + \frac{8261261627321285620005889609879425}{43872000692990881628273820164319019} a^{5} + \frac{18805191985869230887239039976140461}{43872000692990881628273820164319019} a^{4} - \frac{27716584618850427845919757613099041}{87744001385981763256547640328638038} a^{3} + \frac{5473720489493002202460263512837528}{43872000692990881628273820164319019} a^{2} - \frac{7559670122656118292169356701565730}{43872000692990881628273820164319019} a - \frac{27447297159045816463870001640062341}{87744001385981763256547640328638038}$

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:  $7$
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:  \( 901816.955065 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1872:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1872 are not computed
Character table for t16n1872 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.155140625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16$ R $16$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.10$x^{12} - 6 x^{10} + 23 x^{8} - 28 x^{6} - 9 x^{4} - 30 x^{2} - 15$$2$$6$$12$12T58$[2, 2, 2, 2]^{6}$
5Data not computed
9929Data not computed