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Properties

Label 16.0.19073412482...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 1652141^{4}$
Root discriminant $160.33$
Ramified primes $2, 5, 1652141$
Class number $14856864$ (GRH)
Class group $[2, 2, 3714216]$ (GRH)
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2729569883881, 0, 1786640146669, 0, 319403408552, 0, 20131740705, 0, 620292337, 0, 10503607, 0, 99711, 0, 496, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 496*x^14 + 99711*x^12 + 10503607*x^10 + 620292337*x^8 + 20131740705*x^6 + 319403408552*x^4 + 1786640146669*x^2 + 2729569883881)
 
gp: K = bnfinit(x^16 + 496*x^14 + 99711*x^12 + 10503607*x^10 + 620292337*x^8 + 20131740705*x^6 + 319403408552*x^4 + 1786640146669*x^2 + 2729569883881, 1)
 

Normalized defining polynomial

\( x^{16} + 496 x^{14} + 99711 x^{12} + 10503607 x^{10} + 620292337 x^{8} + 20131740705 x^{6} + 319403408552 x^{4} + 1786640146669 x^{2} + 2729569883881 \)

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:  \(190734124825347477084727321600000000=2^{16}\cdot 5^{8}\cdot 1652141^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.33$
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:  $2, 5, 1652141$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{894494758980242452225920866845342105477} a^{14} + \frac{273739346114743680906443852098262956545}{894494758980242452225920866845342105477} a^{12} - \frac{140157344627460228321693467695511939094}{894494758980242452225920866845342105477} a^{10} - \frac{31228943362972188613751322355379353455}{894494758980242452225920866845342105477} a^{8} - \frac{2779315177054272812935917286451646981}{894494758980242452225920866845342105477} a^{6} + \frac{430124488925707782085908222183959438257}{894494758980242452225920866845342105477} a^{4} + \frac{386214472629686630889398836180671264552}{894494758980242452225920866845342105477} a^{2} + \frac{198127431994878693951979510001861}{541415508107505625867235827235897}$, $\frac{1}{894494758980242452225920866845342105477} a^{15} + \frac{273739346114743680906443852098262956545}{894494758980242452225920866845342105477} a^{13} - \frac{140157344627460228321693467695511939094}{894494758980242452225920866845342105477} a^{11} - \frac{31228943362972188613751322355379353455}{894494758980242452225920866845342105477} a^{9} - \frac{2779315177054272812935917286451646981}{894494758980242452225920866845342105477} a^{7} + \frac{430124488925707782085908222183959438257}{894494758980242452225920866845342105477} a^{5} + \frac{386214472629686630889398836180671264552}{894494758980242452225920866845342105477} a^{3} + \frac{198127431994878693951979510001861}{541415508107505625867235827235897} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{3714216}$, which has order $14856864$ (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:  \( 6960.86418224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.1032588125.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1652141Data not computed