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Properties

Label 16.16.1100984940...3125.2
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 1652141^{3}$
Root discriminant $48.99$
Ramified primes $5, 1652141$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1651

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -910, 22158, -136791, 130478, 127414, -158038, -41070, 63489, 6644, -12113, -642, 1183, 38, -56, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 56*x^14 + 38*x^13 + 1183*x^12 - 642*x^11 - 12113*x^10 + 6644*x^9 + 63489*x^8 - 41070*x^7 - 158038*x^6 + 127414*x^5 + 130478*x^4 - 136791*x^3 + 22158*x^2 - 910*x + 11)
 
gp: K = bnfinit(x^16 - x^15 - 56*x^14 + 38*x^13 + 1183*x^12 - 642*x^11 - 12113*x^10 + 6644*x^9 + 63489*x^8 - 41070*x^7 - 158038*x^6 + 127414*x^5 + 130478*x^4 - 136791*x^3 + 22158*x^2 - 910*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 56 x^{14} + 38 x^{13} + 1183 x^{12} - 642 x^{11} - 12113 x^{10} + 6644 x^{9} + 63489 x^{8} - 41070 x^{7} - 158038 x^{6} + 127414 x^{5} + 130478 x^{4} - 136791 x^{3} + 22158 x^{2} - 910 x + 11 \)

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:  \(1100984940802011528564453125=5^{12}\cdot 1652141^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.99$
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:  $5, 1652141$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{28725251203076996548079848901} a^{15} - \frac{10860403770264887233260768596}{28725251203076996548079848901} a^{14} + \frac{8767080888729078476194062565}{28725251203076996548079848901} a^{13} + \frac{4831080062073219830070374467}{28725251203076996548079848901} a^{12} - \frac{12882993928926964871337270044}{28725251203076996548079848901} a^{11} + \frac{768060993325714634920900317}{28725251203076996548079848901} a^{10} + \frac{10485696126305814555308320053}{28725251203076996548079848901} a^{9} + \frac{5569269899873433188757360871}{28725251203076996548079848901} a^{8} - \frac{13295214669143257016660436902}{28725251203076996548079848901} a^{7} - \frac{8650049447067213866771049210}{28725251203076996548079848901} a^{6} - \frac{5792216191719685479656105353}{28725251203076996548079848901} a^{5} + \frac{11722210577636017032115075708}{28725251203076996548079848901} a^{4} + \frac{3069967976702614861599074337}{28725251203076996548079848901} a^{3} - \frac{7593436955622110100637557516}{28725251203076996548079848901} a^{2} - \frac{5403393920307769368610986203}{28725251203076996548079848901} a + \frac{1481993396550097520048807265}{28725251203076996548079848901}$

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:  \( 227934621.417 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1651:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 44 conjugacy class representatives for t16n1651
Character table for t16n1651 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.1032588125.1

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 $16$ $16$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
1652141Data not computed