↑

Properties

Label 16.0.16244794399...856.17
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $18.33$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $C_8:C_2^2$ (as 16T35)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144, 0, -192, 0, 208, 0, -160, 0, 92, 0, -16, 0, 20, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 20*x^12 - 16*x^10 + 92*x^8 - 160*x^6 + 208*x^4 - 192*x^2 + 144)
 
gp: K = bnfinit(x^16 - 4*x^14 + 20*x^12 - 16*x^10 + 92*x^8 - 160*x^6 + 208*x^4 - 192*x^2 + 144, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} + 20 x^{12} - 16 x^{10} + 92 x^{8} - 160 x^{6} + 208 x^{4} - 192 x^{2} + 144 \)

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:  \(162447943996702457856=2^{32}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.33$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{24} a^{10} - \frac{1}{12} a^{6} + \frac{1}{6} a^{2}$, $\frac{1}{48} a^{11} - \frac{1}{24} a^{7} - \frac{1}{4} a^{6} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{12} - \frac{1}{24} a^{8} - \frac{1}{4} a^{7} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{144} a^{13} + \frac{1}{144} a^{11} + \frac{1}{36} a^{9} - \frac{13}{72} a^{7} - \frac{1}{4} a^{6} - \frac{2}{9} a^{5} - \frac{11}{36} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{53712} a^{14} - \frac{157}{26856} a^{12} - \frac{139}{6714} a^{10} - \frac{1093}{26856} a^{8} - \frac{1}{4} a^{7} - \frac{722}{3357} a^{6} + \frac{1159}{13428} a^{4} - \frac{193}{2238} a^{2} - \frac{77}{746}$, $\frac{1}{107424} a^{15} - \frac{157}{53712} a^{13} - \frac{139}{13428} a^{11} + \frac{283}{6714} a^{9} - \frac{361}{3357} a^{7} - \frac{1}{4} a^{6} + \frac{1129}{6714} a^{5} - \frac{193}{4476} a^{3} - \frac{225}{746} a$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{83}{13428} a^{14} + \frac{325}{13428} a^{12} - \frac{425}{3357} a^{10} + \frac{1279}{13428} a^{8} - \frac{2000}{3357} a^{6} + \frac{2275}{3357} a^{4} - \frac{1532}{1119} a^{2} + \frac{473}{373} \) (order $6$)
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:  \( 20952.2918331 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8:C_2^2$ (as 16T35):

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

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-21}) \), 4.2.9408.2 x2, 4.0.1008.1 x2, \(\Q(\sqrt{-3}, \sqrt{7})\), 8.2.4248502272.1 x2, 8.2.4248502272.2 x2, 8.0.796594176.14

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$