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Properties

Label 16.4.28643813255...2081.1
Degree $16$
Signature $[4, 6]$
Discriminant $11^{4}\cdot 89^{14}$
Root discriminant $92.48$
Ramified primes $11, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, 147840, 0, -1604663, 0, -101684, 0, 24772, 0, 4422, 0, 580, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 44*x^14 + 580*x^12 + 4422*x^10 + 24772*x^8 - 101684*x^6 - 1604663*x^4 + 147840*x^2 + 4096)
 
gp: K = bnfinit(x^16 + 44*x^14 + 580*x^12 + 4422*x^10 + 24772*x^8 - 101684*x^6 - 1604663*x^4 + 147840*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{16} + 44 x^{14} + 580 x^{12} + 4422 x^{10} + 24772 x^{8} - 101684 x^{6} - 1604663 x^{4} + 147840 x^{2} + 4096 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28643813255402702732772283462081=11^{4}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.48$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{5} - \frac{3}{16} a^{3} + \frac{1}{8} a^{2} + \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{3}{16} a^{6} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{5}{16} a^{2} + \frac{1}{8} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} + \frac{5}{32} a^{6} + \frac{5}{32} a^{5} + \frac{5}{32} a^{4} - \frac{11}{32} a^{3} + \frac{5}{32} a^{2} + \frac{1}{16} a - \frac{1}{2}$, $\frac{1}{13032149914814833664} a^{14} - \frac{4113331812106245}{3258037478703708416} a^{12} - \frac{128050311591693263}{3258037478703708416} a^{10} - \frac{310832377257056925}{6516074957407416832} a^{8} - \frac{513173117911182223}{3258037478703708416} a^{6} - \frac{491931643021839469}{3258037478703708416} a^{4} + \frac{5580725891160775753}{13032149914814833664} a^{2} - \frac{27114345200436613}{203627342418981776}$, $\frac{1}{208514398637037338624} a^{15} - \frac{1}{26064299829629667328} a^{14} - \frac{411368016650069797}{52128599659259334656} a^{13} + \frac{4113331812106245}{6516074957407416832} a^{12} + \frac{279204373246270289}{52128599659259334656} a^{11} + \frac{128050311591693263}{6516074957407416832} a^{10} + \frac{5390733210474432803}{104257199318518669312} a^{9} - \frac{1318186362094797283}{13032149914814833664} a^{8} + \frac{1523100306278635537}{52128599659259334656} a^{7} + \frac{513173117911182223}{6516074957407416832} a^{6} - \frac{12302317503322782477}{52128599659259334656} a^{5} + \frac{491931643021839469}{6516074957407416832} a^{4} - \frac{103563529645413456183}{208514398637037338624} a^{3} + \frac{10709461502357766327}{26064299829629667328} a^{2} + \frac{176512997218545163}{3258037478703708416} a - \frac{176512997218545163}{407254684837963552}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 16780906748.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 8.4.5351991522359009.1, 8.2.5466794200571.1, 8.6.486544683850819.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$