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Properties

Label 16.2.13482376439...0944.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{32}\cdot 3^{15}\cdot 131\cdot 167$
Root discriminant $20.92$
Ramified primes $2, 3, 131, 167$
Class number $1$
Class group Trivial
Galois group 16T1643

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -32, 96, -208, 380, -600, 844, -1040, 1113, -1016, 784, -504, 266, -112, 36, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 266*x^12 - 504*x^11 + 784*x^10 - 1016*x^9 + 1113*x^8 - 1040*x^7 + 844*x^6 - 600*x^5 + 380*x^4 - 208*x^3 + 96*x^2 - 32*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 36*x^14 - 112*x^13 + 266*x^12 - 504*x^11 + 784*x^10 - 1016*x^9 + 1113*x^8 - 1040*x^7 + 844*x^6 - 600*x^5 + 380*x^4 - 208*x^3 + 96*x^2 - 32*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 36 x^{14} - 112 x^{13} + 266 x^{12} - 504 x^{11} + 784 x^{10} - 1016 x^{9} + 1113 x^{8} - 1040 x^{7} + 844 x^{6} - 600 x^{5} + 380 x^{4} - 208 x^{3} + 96 x^{2} - 32 x + 4 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1348237643948903890944=-\,2^{32}\cdot 3^{15}\cdot 131\cdot 167\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.92$
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:  $2, 3, 131, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$

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:  $8$
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:  \( 47776.04785 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.2.1728.1, 8.2.143327232.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.8.16.66$x^{8} + 8 x^{4} + 336$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
2.8.16.66$x^{8} + 8 x^{4} + 336$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
3Data not computed
$131$$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 3$$1$$1$$0$Trivial$[\ ]$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.2.0.1$x^{2} - x + 14$$1$$2$$0$$C_2$$[\ ]^{2}$
131.2.1.2$x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
131.4.0.1$x^{4} - x + 6$$1$$4$$0$$C_4$$[\ ]^{4}$
$167$167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$