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Properties

Label 16.4.51371070174...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 5^{8}\cdot 283^{6}$
Root discriminant $26.27$
Ramified primes $2, 5, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1665

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, -312, 1435, -1529, -1410, 3377, -3904, 1055, -516, -169, 77, -23, -14, 4, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 + 4*x^13 - 14*x^12 - 23*x^11 + 77*x^10 - 169*x^9 - 516*x^8 + 1055*x^7 - 3904*x^6 + 3377*x^5 - 1410*x^4 - 1529*x^3 + 1435*x^2 - 312*x - 11)
 
gp: K = bnfinit(x^16 - 3*x^15 + 6*x^14 + 4*x^13 - 14*x^12 - 23*x^11 + 77*x^10 - 169*x^9 - 516*x^8 + 1055*x^7 - 3904*x^6 + 3377*x^5 - 1410*x^4 - 1529*x^3 + 1435*x^2 - 312*x - 11, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 6 x^{14} + 4 x^{13} - 14 x^{12} - 23 x^{11} + 77 x^{10} - 169 x^{9} - 516 x^{8} + 1055 x^{7} - 3904 x^{6} + 3377 x^{5} - 1410 x^{4} - 1529 x^{3} + 1435 x^{2} - 312 x - 11 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(51371070174496900000000=2^{8}\cdot 5^{8}\cdot 283^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.27$
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:  $2, 5, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{212} a^{14} + \frac{19}{212} a^{13} - \frac{45}{212} a^{12} + \frac{67}{212} a^{11} + \frac{93}{212} a^{10} + \frac{17}{53} a^{9} - \frac{17}{53} a^{8} - \frac{61}{212} a^{7} - \frac{35}{106} a^{6} - \frac{18}{53} a^{5} - \frac{3}{106} a^{4} - \frac{87}{212} a^{3} - \frac{43}{106} a^{2} + \frac{35}{106} a + \frac{61}{212}$, $\frac{1}{57119473443810053653667528} a^{15} - \frac{14450954290154633133504}{7139934180476256706708441} a^{14} - \frac{311280894240196111260661}{28559736721905026826833764} a^{13} + \frac{9884224399100669621466359}{28559736721905026826833764} a^{12} + \frac{2068896151348446433010839}{14279868360952513413416882} a^{11} - \frac{18741419754575902516502851}{57119473443810053653667528} a^{10} - \frac{3047851917910999857578751}{14279868360952513413416882} a^{9} - \frac{2857381486300314224748165}{57119473443810053653667528} a^{8} + \frac{1916389695442769594025843}{8159924777687150521952504} a^{7} - \frac{102013337600228200416177}{1679984513053236872166692} a^{6} + \frac{10488237640372559285545345}{28559736721905026826833764} a^{5} - \frac{2606624229252066093558231}{8159924777687150521952504} a^{4} - \frac{11189216913694411331719269}{57119473443810053653667528} a^{3} - \frac{2833804789917236458332402}{7139934180476256706708441} a^{2} - \frac{12630280871844233344800197}{57119473443810053653667528} a + \frac{2511928467058009245823809}{57119473443810053653667528}$

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:  \( 131295.094957 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1665:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 78 conjugacy class representatives for t16n1665 are not computed
Character table for t16n1665 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.283.1, 8.4.50055625.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
5Data not computed
283Data not computed