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Properties

Label 16.4.16853306931...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{16}\cdot 5^{8}\cdot 71^{2}\cdot 1901^{4}$
Root discriminant $50.31$
Ramified primes $2, 5, 71, 1901$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1884

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3613801, 0, 1041748, 0, -1126950, 0, -160043, 0, 37995, 0, 905, 0, -47, 0, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 21*x^14 - 47*x^12 + 905*x^10 + 37995*x^8 - 160043*x^6 - 1126950*x^4 + 1041748*x^2 + 3613801)
 
gp: K = bnfinit(x^16 + 21*x^14 - 47*x^12 + 905*x^10 + 37995*x^8 - 160043*x^6 - 1126950*x^4 + 1041748*x^2 + 3613801, 1)
 

Normalized defining polynomial

\( x^{16} + 21 x^{14} - 47 x^{12} + 905 x^{10} + 37995 x^{8} - 160043 x^{6} - 1126950 x^{4} + 1041748 x^{2} + 3613801 \)

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:  \(1685330693180842009600000000=2^{16}\cdot 5^{8}\cdot 71^{2}\cdot 1901^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.31$
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:  $2, 5, 71, 1901$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{3}{17} a^{10} + \frac{2}{17} a^{8} + \frac{5}{17} a^{6} - \frac{3}{17} a^{4} + \frac{3}{17} a^{2} + \frac{8}{17}$, $\frac{1}{17} a^{13} + \frac{3}{17} a^{11} + \frac{2}{17} a^{9} + \frac{5}{17} a^{7} - \frac{3}{17} a^{5} + \frac{3}{17} a^{3} + \frac{8}{17} a$, $\frac{1}{24751347648854828662527481} a^{14} - \frac{722912188898055624924179}{24751347648854828662527481} a^{12} - \frac{2345933590870277873407537}{24751347648854828662527481} a^{10} - \frac{2103354265156201874591149}{24751347648854828662527481} a^{8} + \frac{3020440270517812175710869}{24751347648854828662527481} a^{6} - \frac{6474838825838968423631652}{24751347648854828662527481} a^{4} - \frac{5881283503117410079879372}{24751347648854828662527481} a^{2} + \frac{6019550434182318654414}{13020172356051987723581}$, $\frac{1}{24751347648854828662527481} a^{15} - \frac{722912188898055624924179}{24751347648854828662527481} a^{13} - \frac{2345933590870277873407537}{24751347648854828662527481} a^{11} - \frac{2103354265156201874591149}{24751347648854828662527481} a^{9} + \frac{3020440270517812175710869}{24751347648854828662527481} a^{7} - \frac{6474838825838968423631652}{24751347648854828662527481} a^{5} - \frac{5881283503117410079879372}{24751347648854828662527481} a^{3} + \frac{6019550434182318654414}{13020172356051987723581} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 3638997.79141 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1884:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147456
The 136 conjugacy class representatives for t16n1884 are not computed
Character table for t16n1884 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.6.84356875.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.8.0.1$x^{8} - 7 x + 13$$1$$8$$0$$C_8$$[\ ]^{8}$
1901Data not computed