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Properties

Label 16.4.59190897027...8125.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{4}\cdot 5^{15}\cdot 41^{6}\cdot 71^{2}$
Root discriminant $40.81$
Ramified primes $3, 5, 41, 71$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92111, -2048, -32500, 44260, -5735, 31406, -32268, 19795, -2745, -3595, 1787, -81, -235, 80, 5, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 5*x^14 + 80*x^13 - 235*x^12 - 81*x^11 + 1787*x^10 - 3595*x^9 - 2745*x^8 + 19795*x^7 - 32268*x^6 + 31406*x^5 - 5735*x^4 + 44260*x^3 - 32500*x^2 - 2048*x + 92111)
 
gp: K = bnfinit(x^16 - 7*x^15 + 5*x^14 + 80*x^13 - 235*x^12 - 81*x^11 + 1787*x^10 - 3595*x^9 - 2745*x^8 + 19795*x^7 - 32268*x^6 + 31406*x^5 - 5735*x^4 + 44260*x^3 - 32500*x^2 - 2048*x + 92111, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 5 x^{14} + 80 x^{13} - 235 x^{12} - 81 x^{11} + 1787 x^{10} - 3595 x^{9} - 2745 x^{8} + 19795 x^{7} - 32268 x^{6} + 31406 x^{5} - 5735 x^{4} + 44260 x^{3} - 32500 x^{2} - 2048 x + 92111 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(59190897027263214111328125=3^{4}\cdot 5^{15}\cdot 41^{6}\cdot 71^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.81$
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:  $3, 5, 41, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{30598145988818064668402957096344175161789} a^{15} - \frac{2418817817181841588382836856103177216116}{30598145988818064668402957096344175161789} a^{14} + \frac{669730443202075827959999682611161945785}{30598145988818064668402957096344175161789} a^{13} + \frac{14521829572286658756967298830867635150504}{30598145988818064668402957096344175161789} a^{12} - \frac{7079744717079760683117132072091164184607}{30598145988818064668402957096344175161789} a^{11} - \frac{10206763162158514221563009743979020908945}{30598145988818064668402957096344175161789} a^{10} - \frac{10870547197311761745736718003338970176574}{30598145988818064668402957096344175161789} a^{9} + \frac{458835701311630629906282310568531589471}{30598145988818064668402957096344175161789} a^{8} - \frac{2047730252987476777200323139287920179123}{30598145988818064668402957096344175161789} a^{7} + \frac{12558397178622163891877670889516055788924}{30598145988818064668402957096344175161789} a^{6} - \frac{316835724221649310421176554057682169962}{1055108482373036712703550244701523281441} a^{5} - \frac{576563479413213546224664417583680678278}{30598145988818064668402957096344175161789} a^{4} - \frac{275584030755342236279726721600642152551}{30598145988818064668402957096344175161789} a^{3} + \frac{17224318268512669791498927877709527282}{2781649635347096788036632463304015923799} a^{2} - \frac{8548560364171546757204448111433095723556}{30598145988818064668402957096344175161789} a + \frac{6118338956737713245312468158294932299895}{30598145988818064668402957096344175161789}$

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:  $9$
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:  \( 1874859.74762 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1113:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1113
Character table for t16n1113 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.4.131328125.1

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 $16$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ $16$ R $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71Data not computed