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Properties

Label 16.6.23685588318...4528.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{32}\cdot 223^{5}$
Root discriminant $21.67$
Ramified primes $2, 223$
Class number $1$
Class group Trivial
Galois group 16T1759

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 34, 28, 36, 0, -50, 32, -135, 160, -58, -8, -2, 12, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^13 - 2*x^12 - 8*x^11 - 58*x^10 + 160*x^9 - 135*x^8 + 32*x^7 - 50*x^6 + 36*x^4 + 28*x^3 + 34*x^2 - 4*x - 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^13 - 2*x^12 - 8*x^11 - 58*x^10 + 160*x^9 - 135*x^8 + 32*x^7 - 50*x^6 + 36*x^4 + 28*x^3 + 34*x^2 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{13} - 2 x^{12} - 8 x^{11} - 58 x^{10} + 160 x^{9} - 135 x^{8} + 32 x^{7} - 50 x^{6} + 36 x^{4} + 28 x^{3} + 34 x^{2} - 4 x - 1 \)

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:  \(-2368558831812663574528=-\,2^{32}\cdot 223^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.67$
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:  $2, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{6} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{7} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{976817874968} a^{15} - \frac{11563623835}{244204468742} a^{14} + \frac{116419972757}{976817874968} a^{13} + \frac{112589295731}{976817874968} a^{12} + \frac{21210533730}{122102234371} a^{11} - \frac{14127075999}{75139836536} a^{10} - \frac{198542445679}{976817874968} a^{9} - \frac{65779216029}{488408937484} a^{8} - \frac{7959235974}{122102234371} a^{7} + \frac{22264084071}{976817874968} a^{6} - \frac{18003831685}{75139836536} a^{5} - \frac{7226929367}{37569918268} a^{4} + \frac{346613997077}{976817874968} a^{3} + \frac{455171163661}{976817874968} a^{2} - \frac{11724394187}{37569918268} a + \frac{267527899245}{976817874968}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 58915.7828067 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1759:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12288
The 93 conjugacy class representatives for t16n1759 are not computed
Character table for t16n1759 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14272.1, 8.8.3259039744.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
223Data not computed