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Properties

Label 16.16.4292702462...3872.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{53}\cdot 383^{2}\cdot 1217^{6}$
Root discriminant $299.95$
Ramified primes $2, 383, 1217$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1113

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![85165710013832, 0, -18901405129856, 0, 1672904683652, 0, -75799317308, 0, 1884364081, 0, -25837780, 0, 189762, 0, -696, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 696*x^14 + 189762*x^12 - 25837780*x^10 + 1884364081*x^8 - 75799317308*x^6 + 1672904683652*x^4 - 18901405129856*x^2 + 85165710013832)
 
gp: K = bnfinit(x^16 - 696*x^14 + 189762*x^12 - 25837780*x^10 + 1884364081*x^8 - 75799317308*x^6 + 1672904683652*x^4 - 18901405129856*x^2 + 85165710013832, 1)
 

Normalized defining polynomial

\( x^{16} - 696 x^{14} + 189762 x^{12} - 25837780 x^{10} + 1884364081 x^{8} - 75799317308 x^{6} + 1672904683652 x^{4} - 18901405129856 x^{2} + 85165710013832 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4292702462281032366933505959381142863872=2^{53}\cdot 383^{2}\cdot 1217^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $299.95$
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:  $2, 383, 1217$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{200291197790670001415950231525845850669484} a^{14} + \frac{15870472753339462367744367819374646334487}{200291197790670001415950231525845850669484} a^{12} - \frac{8334627208632269214916240943620075050669}{200291197790670001415950231525845850669484} a^{10} + \frac{44314020709025025586120295860187699898381}{200291197790670001415950231525845850669484} a^{8} - \frac{4004319673555564199740383795544853334386}{50072799447667500353987557881461462667371} a^{6} + \frac{114379198508351836045632524361365746125}{259444556723665804942940714411717423147} a^{4} - \frac{18875798898994884142879410395865624390051}{50072799447667500353987557881461462667371} a^{2} - \frac{3799084636319494356633800567298568}{107426770549649118673422334768888661}$, $\frac{1}{1402038384534690009911651620680920954686388} a^{15} - \frac{84275126141995538340230747943548279000255}{1402038384534690009911651620680920954686388} a^{13} - \frac{8334627208632269214916240943620075050669}{1402038384534690009911651620680920954686388} a^{11} - \frac{155977177081644975829829935665658150771103}{1402038384534690009911651620680920954686388} a^{9} - \frac{4004319673555564199740383795544853334386}{350509596133672502477912905170230238671597} a^{7} - \frac{1587353500048956962509319952159290469779}{3632223794131321269201170001764043924058} a^{5} - \frac{18875798898994884142879410395865624390051}{350509596133672502477912905170230238671597} a^{3} + \frac{103627685913329624316788534201590093}{751987393847543830713956343382220627} a$

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:  $15$
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:  \( 2420220989490000 \) (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

\(\Q(\sqrt{2}) \), 4.4.77888.1, 8.8.776517189632.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ $16$

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$2.8.28.14$x^{8} + 8 x^{5} + 14$$8$$1$$28$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
2.8.25.2$x^{8} + 10 x^{4} + 20 x^{2} + 2$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
383Data not computed
1217Data not computed