↑

Properties

Label 16.0.34846083869...5488.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{79}\cdot 7^{8}$
Root discriminant $81.07$
Ramified primes $2, 7$
Class number $54788$ (GRH)
Class group $[2, 27394]$ (GRH)
Galois group $C_{16}$ (as 16T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11529602, 0, 52706752, 0, 39530064, 0, 11294304, 0, 1584660, 0, 120736, 0, 5096, 0, 112, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 112*x^14 + 5096*x^12 + 120736*x^10 + 1584660*x^8 + 11294304*x^6 + 39530064*x^4 + 52706752*x^2 + 11529602)
 
gp: K = bnfinit(x^16 + 112*x^14 + 5096*x^12 + 120736*x^10 + 1584660*x^8 + 11294304*x^6 + 39530064*x^4 + 52706752*x^2 + 11529602, 1)
 

Normalized defining polynomial

\( x^{16} + 112 x^{14} + 5096 x^{12} + 120736 x^{10} + 1584660 x^{8} + 11294304 x^{6} + 39530064 x^{4} + 52706752 x^{2} + 11529602 \)

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:  \(3484608386920116940487669055488=2^{79}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.07$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(448=2^{6}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{448}(1,·)$, $\chi_{448}(69,·)$, $\chi_{448}(393,·)$, $\chi_{448}(13,·)$, $\chi_{448}(337,·)$, $\chi_{448}(405,·)$, $\chi_{448}(281,·)$, $\chi_{448}(349,·)$, $\chi_{448}(225,·)$, $\chi_{448}(293,·)$, $\chi_{448}(169,·)$, $\chi_{448}(237,·)$, $\chi_{448}(113,·)$, $\chi_{448}(181,·)$, $\chi_{448}(57,·)$, $\chi_{448}(125,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{7} a^{2}$, $\frac{1}{7} a^{3}$, $\frac{1}{49} a^{4}$, $\frac{1}{49} a^{5}$, $\frac{1}{343} a^{6}$, $\frac{1}{343} a^{7}$, $\frac{1}{2401} a^{8}$, $\frac{1}{2401} a^{9}$, $\frac{1}{16807} a^{10}$, $\frac{1}{16807} a^{11}$, $\frac{1}{117649} a^{12}$, $\frac{1}{117649} a^{13}$, $\frac{1}{823543} a^{14}$, $\frac{1}{823543} a^{15}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{27394}$, which has order $54788$ (assuming GRH)

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:  \( -1 \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 15753.94986242651 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16$ $16$ R $16$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ $16$ $16$

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
$7$7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$