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Properties

Label 16.0.171163247133589504.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 113^{2}\cdot 3121$
Root discriminant $11.94$
Ramified primes $2, 113, 3121$
Class number $1$
Class group Trivial
Galois group 16T1769

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 28, -52, 51, -12, -34, 32, 44, -152, 224, -220, 158, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 224*x^10 - 152*x^9 + 44*x^8 + 32*x^7 - 34*x^6 - 12*x^5 + 51*x^4 - 52*x^3 + 28*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 158*x^12 - 220*x^11 + 224*x^10 - 152*x^9 + 44*x^8 + 32*x^7 - 34*x^6 - 12*x^5 + 51*x^4 - 52*x^3 + 28*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 158 x^{12} - 220 x^{11} + 224 x^{10} - 152 x^{9} + 44 x^{8} + 32 x^{7} - 34 x^{6} - 12 x^{5} + 51 x^{4} - 52 x^{3} + 28 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(171163247133589504=2^{32}\cdot 113^{2}\cdot 3121\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.94$
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:  $2, 113, 3121$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{7}{19} a^{13} - \frac{5}{19} a^{12} + \frac{7}{19} a^{11} - \frac{8}{19} a^{10} - \frac{1}{19} a^{9} + \frac{7}{19} a^{8} - \frac{1}{19} a^{7} + \frac{4}{19} a^{6} + \frac{9}{19} a^{5} + \frac{7}{19} a^{4} - \frac{9}{19} a^{3} + \frac{3}{19} a^{2} - \frac{7}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{15} + \frac{3}{19} a^{13} - \frac{9}{19} a^{12} + \frac{3}{19} a^{11} - \frac{9}{19} a^{8} - \frac{3}{19} a^{7} - \frac{1}{19} a^{6} - \frac{6}{19} a^{5} + \frac{2}{19} a^{4} - \frac{3}{19} a^{3} - \frac{5}{19} a^{2} - \frac{4}{19} a - \frac{8}{19}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{21}{19} a^{14} - \frac{147}{19} a^{13} + \frac{522}{19} a^{12} - \frac{1221}{19} a^{11} + \frac{2017}{19} a^{10} - \frac{2396}{19} a^{9} + \frac{1933}{19} a^{8} - \frac{781}{19} a^{7} - \frac{258}{19} a^{6} + \frac{569}{19} a^{5} - \frac{62}{19} a^{4} - \frac{474}{19} a^{3} + \frac{633}{19} a^{2} - \frac{356}{19} a + \frac{90}{19} \) (order $8$)
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:  \( 270.623636126 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1769:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 220 conjugacy class representatives for t16n1769 are not computed
Character table for t16n1769 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 8.0.7405568.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
3121Data not computed