↑

Properties

Label 16.8.83646323707...5776.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 41^{7}$
Root discriminant $20.31$
Ramified primes $2, 41$
Class number $1$
Class group Trivial
Galois group $C_4.D_4:C_4$ (as 16T289)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -28, -68, 112, 258, 44, -306, -296, 24, 144, 114, -16, -13, -20, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 20*x^13 - 13*x^12 - 16*x^11 + 114*x^10 + 144*x^9 + 24*x^8 - 296*x^7 - 306*x^6 + 44*x^5 + 258*x^4 + 112*x^3 - 68*x^2 - 28*x + 1)
 
gp: K = bnfinit(x^16 + 2*x^14 - 20*x^13 - 13*x^12 - 16*x^11 + 114*x^10 + 144*x^9 + 24*x^8 - 296*x^7 - 306*x^6 + 44*x^5 + 258*x^4 + 112*x^3 - 68*x^2 - 28*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 20 x^{13} - 13 x^{12} - 16 x^{11} + 114 x^{10} + 144 x^{9} + 24 x^{8} - 296 x^{7} - 306 x^{6} + 44 x^{5} + 258 x^{4} + 112 x^{3} - 68 x^{2} - 28 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(836463237075121995776=2^{32}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{37} a^{14} + \frac{18}{37} a^{13} - \frac{1}{37} a^{12} - \frac{4}{37} a^{11} - \frac{17}{37} a^{10} - \frac{13}{37} a^{9} - \frac{8}{37} a^{7} - \frac{9}{37} a^{6} + \frac{12}{37} a^{5} + \frac{4}{37} a^{4} + \frac{3}{37} a^{3} + \frac{3}{37} a^{2} - \frac{1}{37} a + \frac{6}{37}$, $\frac{1}{423252673751269} a^{15} - \frac{1316590974411}{423252673751269} a^{14} + \frac{198852422410937}{423252673751269} a^{13} + \frac{50959893837159}{423252673751269} a^{12} + \frac{65847099540938}{423252673751269} a^{11} + \frac{869101520258}{11439261452737} a^{10} - \frac{15346725069043}{423252673751269} a^{9} - \frac{154355256442429}{423252673751269} a^{8} + \frac{36250397742026}{423252673751269} a^{7} - \frac{49820336994420}{423252673751269} a^{6} - \frac{26861235226875}{423252673751269} a^{5} - \frac{184303603038672}{423252673751269} a^{4} + \frac{110039113586365}{423252673751269} a^{3} + \frac{72865548240742}{423252673751269} a^{2} - \frac{74728804273467}{423252673751269} a - \frac{125820432552387}{423252673751269}$

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:  $11$
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:  \( 25508.578859 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.282300416.1

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 $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$41$41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
41.8.7.8$x^{8} + 11477376$$8$$1$$7$$C_8$$[\ ]_{8}$