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Properties

Label 16.0.82944000000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{4}\cdot 5^{14}$
Root discriminant $15.22$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 46, 69, 6, -119, -182, -54, 98, 162, 32, -29, -58, 11, 4, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 + 4*x^13 + 11*x^12 - 58*x^11 - 29*x^10 + 32*x^9 + 162*x^8 + 98*x^7 - 54*x^6 - 182*x^5 - 119*x^4 + 6*x^3 + 69*x^2 + 46*x + 11)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 + 4*x^13 + 11*x^12 - 58*x^11 - 29*x^10 + 32*x^9 + 162*x^8 + 98*x^7 - 54*x^6 - 182*x^5 - 119*x^4 + 6*x^3 + 69*x^2 + 46*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} + 4 x^{13} + 11 x^{12} - 58 x^{11} - 29 x^{10} + 32 x^{9} + 162 x^{8} + 98 x^{7} - 54 x^{6} - 182 x^{5} - 119 x^{4} + 6 x^{3} + 69 x^{2} + 46 x + 11 \)

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:  \(8294400000000000000=2^{24}\cdot 3^{4}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.22$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{13} - \frac{1}{11} a^{12} + \frac{4}{11} a^{11} + \frac{3}{11} a^{10} + \frac{3}{11} a^{9} + \frac{4}{11} a^{8} + \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{3}{11} a^{5} + \frac{1}{11} a^{4} + \frac{3}{11} a^{3} + \frac{1}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{1136416721} a^{15} - \frac{43453963}{1136416721} a^{14} + \frac{67576697}{1136416721} a^{13} + \frac{197081836}{1136416721} a^{12} + \frac{56920958}{1136416721} a^{11} + \frac{298624048}{1136416721} a^{10} + \frac{422147240}{1136416721} a^{9} + \frac{134606039}{1136416721} a^{8} - \frac{24890381}{103310611} a^{7} - \frac{498068909}{1136416721} a^{6} - \frac{484745298}{1136416721} a^{5} + \frac{440540112}{1136416721} a^{4} - \frac{228024428}{1136416721} a^{3} - \frac{402205018}{1136416721} a^{2} - \frac{219916502}{1136416721} a - \frac{22447846}{103310611}$

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:  \( \frac{174906122}{1136416721} a^{15} - \frac{59303944}{103310611} a^{14} - \frac{762737724}{1136416721} a^{13} + \frac{4378426037}{1136416721} a^{12} + \frac{1940184860}{1136416721} a^{11} - \frac{3889865790}{1136416721} a^{10} - \frac{23345708982}{1136416721} a^{9} - \frac{4694820830}{1136416721} a^{8} + \frac{22501200574}{1136416721} a^{7} + \frac{59628961235}{1136416721} a^{6} + \frac{42129005522}{1136416721} a^{5} + \frac{4670922467}{1136416721} a^{4} - \frac{32263330196}{1136416721} a^{3} - \frac{35582847879}{1136416721} a^{2} - \frac{20546463368}{1136416721} a - \frac{468060820}{103310611} \) (order $10$)
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:  \( 3884.1943821 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-2}) \), 4.4.8000.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 8.0.64000000.1

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 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 R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed