↑

Properties

Label 16.8.43843732772...6048.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 3^{8}\cdot 13^{9}\cdot 1327^{8}$
Root discriminant $534.10$
Ramified primes $2, 3, 13, 1327$
Class number $96$ (GRH)
Class group $[96]$ (GRH)
Galois group 16T1675

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3379364677, 0, 3934885019, 0, 63138950, 0, -86396905, 0, -2409767, 0, 225613, 0, 2120, 0, -137, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677)
 
gp: K = bnfinit(x^16 - 137*x^14 + 2120*x^12 + 225613*x^10 - 2409767*x^8 - 86396905*x^6 + 63138950*x^4 + 3934885019*x^2 + 3379364677, 1)
 

Normalized defining polynomial

\( x^{16} - 137 x^{14} + 2120 x^{12} + 225613 x^{10} - 2409767 x^{8} - 86396905 x^{6} + 63138950 x^{4} + 3934885019 x^{2} + 3379364677 \)

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:  \(43843732772164778616036288947079464176386048=2^{16}\cdot 3^{8}\cdot 13^{9}\cdot 1327^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $534.10$
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:  $2, 3, 13, 1327$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{6} + \frac{1}{3} a^{4} + \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{2}{9} a$, $\frac{1}{184316523282801120990185138694447} a^{14} - \frac{6873320299425966982217583744305}{184316523282801120990185138694447} a^{12} + \frac{9389248301212232144247553919158}{61438841094267040330061712898149} a^{10} - \frac{8447518103641496112411099089231}{184316523282801120990185138694447} a^{8} - \frac{30640914383209651399619165287457}{184316523282801120990185138694447} a^{6} - \frac{1566491778088710592398334335180}{20479613698089013443353904299383} a^{4} + \frac{20560979394942300253478108317124}{184316523282801120990185138694447} a^{2} + \frac{50413866555028625338282300816322}{184316523282801120990185138694447}$, $\frac{1}{2971735304888602473724754991170568981} a^{15} + \frac{9759658089322621085671288521457795}{990578434962867491241584997056856327} a^{13} - \frac{60384991547363288412306416224961309}{990578434962867491241584997056856327} a^{11} - \frac{79448869052990924642882205876395888}{2971735304888602473724754991170568981} a^{9} + \frac{476994122079862201567651481251366556}{990578434962867491241584997056856327} a^{7} - \frac{49237690805540254449599980938722272}{990578434962867491241584997056856327} a^{5} - \frac{607671016284000353604386924167274635}{2971735304888602473724754991170568981} a^{3} - \frac{37290224997833809947530268231770591}{990578434962867491241584997056856327} a$

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

Class group and class number

$C_{96}$, which has order $96$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 67240256894900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1675:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 54 conjugacy class representatives for t16n1675 are not computed
Character table for t16n1675 is not computed

Intermediate fields

\(\Q(\sqrt{51753}) \), 4.4.3981.1, 8.8.7173681975339714081.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1327Data not computed