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Properties

Label 16.4.45453408428...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 37^{6}$
Root discriminant $30.10$
Ramified primes $2, 5, 11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1191

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -247, -505, 474, 55, 1798, 2859, 1126, 418, 142, 51, -11, -35, -17, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 17*x^13 - 35*x^12 - 11*x^11 + 51*x^10 + 142*x^9 + 418*x^8 + 1126*x^7 + 2859*x^6 + 1798*x^5 + 55*x^4 + 474*x^3 - 505*x^2 - 247*x + 361)
 
gp: K = bnfinit(x^16 - x^15 - 17*x^13 - 35*x^12 - 11*x^11 + 51*x^10 + 142*x^9 + 418*x^8 + 1126*x^7 + 2859*x^6 + 1798*x^5 + 55*x^4 + 474*x^3 - 505*x^2 - 247*x + 361, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 17 x^{13} - 35 x^{12} - 11 x^{11} + 51 x^{10} + 142 x^{9} + 418 x^{8} + 1126 x^{7} + 2859 x^{6} + 1798 x^{5} + 55 x^{4} + 474 x^{3} - 505 x^{2} - 247 x + 361 \)

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:  \(454534084285444900000000=2^{8}\cdot 5^{8}\cdot 11^{6}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.10$
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:  $2, 5, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{14} + \frac{3}{22} a^{13} + \frac{1}{11} a^{11} - \frac{5}{22} a^{10} + \frac{1}{22} a^{8} + \frac{3}{22} a^{7} - \frac{1}{2} a^{6} + \frac{1}{22} a^{5} + \frac{3}{22} a^{4} - \frac{3}{11} a^{3} - \frac{5}{22} a^{2} - \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{453178911703943894848100986} a^{15} - \frac{3561266332659570686528309}{226589455851971947424050493} a^{14} - \frac{756630927680068801391983}{11925760834314313022318447} a^{13} - \frac{55783551766732823672785453}{226589455851971947424050493} a^{12} - \frac{110687630176320836405988118}{226589455851971947424050493} a^{11} + \frac{183657019846686793743710413}{453178911703943894848100986} a^{10} - \frac{16560346639077040833765176}{226589455851971947424050493} a^{9} - \frac{107039471341706594099497029}{453178911703943894848100986} a^{8} - \frac{1445428463920998021271423}{23851521668628626044636894} a^{7} + \frac{101941552096727189040660769}{226589455851971947424050493} a^{6} - \frac{73360607393541854655884690}{226589455851971947424050493} a^{5} - \frac{183796959373947266320126673}{453178911703943894848100986} a^{4} + \frac{18698876839207069899187559}{41198082882176717713463726} a^{3} - \frac{46990089633308889312726600}{226589455851971947424050493} a^{2} - \frac{25050408805572203409772849}{453178911703943894848100986} a + \frac{10840378341210263294115675}{23851521668628626044636894}$

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:  $9$
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:  \( 457579.3002 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1191:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1191 are not computed
Character table for t16n1191 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.4.103530625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.6.2$x^{8} + 333 x^{4} + 34225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$