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Properties

Label 16.6.47285046214...0000.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{16}\cdot 5^{10}\cdot 71\cdot 101^{4}$
Root discriminant $22.63$
Ramified primes $2, 5, 71, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1776

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![109, -444, 640, -538, 429, -124, -204, 398, -334, 150, -46, -32, 35, -30, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 30*x^13 + 35*x^12 - 32*x^11 - 46*x^10 + 150*x^9 - 334*x^8 + 398*x^7 - 204*x^6 - 124*x^5 + 429*x^4 - 538*x^3 + 640*x^2 - 444*x + 109)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 30*x^13 + 35*x^12 - 32*x^11 - 46*x^10 + 150*x^9 - 334*x^8 + 398*x^7 - 204*x^6 - 124*x^5 + 429*x^4 - 538*x^3 + 640*x^2 - 444*x + 109, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 30 x^{13} + 35 x^{12} - 32 x^{11} - 46 x^{10} + 150 x^{9} - 334 x^{8} + 398 x^{7} - 204 x^{6} - 124 x^{5} + 429 x^{4} - 538 x^{3} + 640 x^{2} - 444 x + 109 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-4728504621440000000000=-\,2^{16}\cdot 5^{10}\cdot 71\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.63$
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:  $2, 5, 71, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{704617930059704} a^{15} + \frac{8522219493193}{704617930059704} a^{14} + \frac{41806163903711}{704617930059704} a^{13} - \frac{46302926736027}{704617930059704} a^{12} - \frac{10479227923660}{88077241257463} a^{11} - \frac{14046662878459}{88077241257463} a^{10} - \frac{67160676816679}{352308965029852} a^{9} + \frac{23674833338087}{176154482514926} a^{8} + \frac{28170849828007}{352308965029852} a^{7} + \frac{494312007859}{88077241257463} a^{6} - \frac{50387629978587}{176154482514926} a^{5} - \frac{37536595292979}{88077241257463} a^{4} - \frac{247857655437403}{704617930059704} a^{3} - \frac{22351042687269}{704617930059704} a^{2} - \frac{138460930237849}{704617930059704} a + \frac{324035718658579}{704617930059704}$

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:  $10$
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:  \( 48151.6090261 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1776:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 136 conjugacy class representatives for t16n1776 are not computed
Character table for t16n1776 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.8160800000.1

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 $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
71.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$