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Properties

Label 16.0.32674345198...5296.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 41^{7}$
Root discriminant $14.36$
Ramified primes $2, 41$
Class number $1$
Class group Trivial
Galois group $C_4.C_2^2\wr C_2$ (as 16T385)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -5, 14, -6, 0, 18, -78, 148, -154, 91, -14, -24, 20, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 20*x^13 - 24*x^12 - 14*x^11 + 91*x^10 - 154*x^9 + 148*x^8 - 78*x^7 + 18*x^6 - 6*x^4 + 14*x^3 - 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 20*x^13 - 24*x^12 - 14*x^11 + 91*x^10 - 154*x^9 + 148*x^8 - 78*x^7 + 18*x^6 - 6*x^4 + 14*x^3 - 5*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 20 x^{13} - 24 x^{12} - 14 x^{11} + 91 x^{10} - 154 x^{9} + 148 x^{8} - 78 x^{7} + 18 x^{6} - 6 x^{4} + 14 x^{3} - 5 x^{2} - 2 x + 1 \)

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:  \(3267434519824695296=2^{24}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.36$
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:  $2, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{415922} a^{15} - \frac{82177}{415922} a^{14} - \frac{14621}{415922} a^{13} + \frac{2647}{12233} a^{12} + \frac{131369}{415922} a^{11} + \frac{7921}{415922} a^{10} + \frac{4923}{207961} a^{9} + \frac{81047}{415922} a^{8} - \frac{43026}{207961} a^{7} + \frac{25139}{415922} a^{6} - \frac{60347}{207961} a^{5} + \frac{161399}{415922} a^{4} + \frac{82933}{207961} a^{3} - \frac{66635}{415922} a^{2} - \frac{89971}{415922} a + \frac{72706}{207961}$

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:  \( -\frac{620}{941} a^{15} + \frac{1413}{1882} a^{14} + \frac{3190}{941} a^{13} - \frac{20327}{1882} a^{12} + \frac{6062}{941} a^{11} + \frac{34935}{1882} a^{10} - \frac{48244}{941} a^{9} + \frac{55779}{941} a^{8} - \frac{28808}{941} a^{7} - \frac{11689}{941} a^{6} + \frac{22662}{941} a^{5} - \frac{13673}{941} a^{4} + \frac{8734}{941} a^{3} - \frac{8433}{941} a^{2} - \frac{2342}{941} a + \frac{4481}{1882} \) (order $4$)
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:  \( 894.032911839 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2\wr C_2$ (as 16T385):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_4.C_2^2\wr C_2$
Character table for $C_4.C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.656.1, 8.0.17643776.1

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

Sibling fields

Degree 32 siblings: data not computed
Arithmetically equvalently 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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.16.2$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 20$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
41Data not computed