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Properties

Label 16.0.16587211113...4784.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{6}$
Root discriminant $13.76$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois group $C_8:C_2^2$ (as 16T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, 34, 92, 198, 348, 502, 576, 518, 352, 166, 36, -18, -20, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{14} - 20 x^{13} - 18 x^{12} + 36 x^{11} + 166 x^{10} + 352 x^{9} + 518 x^{8} + 576 x^{7} + 502 x^{6} + 348 x^{5} + 198 x^{4} + 92 x^{3} + 34 x^{2} + 8 x + 1 \)

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:  \(1658721111359094784=2^{36}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.76$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{34} a^{14} - \frac{3}{34} a^{13} + \frac{3}{17} a^{12} + \frac{2}{17} a^{11} + \frac{5}{34} a^{10} - \frac{1}{34} a^{9} - \frac{3}{34} a^{8} - \frac{8}{17} a^{7} + \frac{13}{34} a^{6} + \frac{13}{34} a^{5} - \frac{4}{17} a^{4} - \frac{7}{17} a^{3} - \frac{5}{34} a^{2} - \frac{3}{34} a + \frac{11}{34}$, $\frac{1}{487934} a^{15} - \frac{736}{243967} a^{14} + \frac{57181}{487934} a^{13} - \frac{8094}{243967} a^{12} - \frac{25673}{243967} a^{11} + \frac{52035}{487934} a^{10} - \frac{23560}{243967} a^{9} - \frac{5871}{243967} a^{8} + \frac{192701}{487934} a^{7} - \frac{39768}{243967} a^{6} - \frac{127123}{487934} a^{5} - \frac{48871}{243967} a^{4} + \frac{111456}{243967} a^{3} - \frac{220373}{487934} a^{2} - \frac{14808}{243967} a - \frac{44842}{243967}$

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:  \( \frac{259881}{487934} a^{15} + \frac{38477}{487934} a^{14} - \frac{1611437}{487934} a^{13} - \frac{5339603}{487934} a^{12} - \frac{5269823}{487934} a^{11} + \frac{4651518}{243967} a^{10} + \frac{44445819}{487934} a^{9} + \frac{2831010}{14351} a^{8} + \frac{142986239}{487934} a^{7} + \frac{160078999}{487934} a^{6} + \frac{140535689}{487934} a^{5} + \frac{97065813}{487934} a^{4} + \frac{54431221}{487934} a^{3} + \frac{12339536}{243967} a^{2} + \frac{8200939}{487934} a + \frac{834596}{243967} \) (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:  \( 1382.13723796 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_8:C_2^2$
Character table for $C_8:C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), 4.0.272.1, 4.0.4352.2, \(\Q(\zeta_{8})\), 8.0.18939904.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$