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Properties

Label 16.0.21592487076...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 5^{4}\cdot 17^{12}\cdot 47^{2}$
Root discriminant $68.14$
Ramified primes $2, 5, 17, 47$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1380625, 0, -1903500, 0, 776600, 0, -107160, 0, 12301, 0, 3648, 0, 496, 0, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 + 496*x^12 + 3648*x^10 + 12301*x^8 - 107160*x^6 + 776600*x^4 - 1903500*x^2 + 1380625)
 
gp: K = bnfinit(x^16 + 32*x^14 + 496*x^12 + 3648*x^10 + 12301*x^8 - 107160*x^6 + 776600*x^4 - 1903500*x^2 + 1380625, 1)
 

Normalized defining polynomial

\( x^{16} + 32 x^{14} + 496 x^{12} + 3648 x^{10} + 12301 x^{8} - 107160 x^{6} + 776600 x^{4} - 1903500 x^{2} + 1380625 \)

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:  \(215924870769789347131555840000=2^{28}\cdot 5^{4}\cdot 17^{12}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.14$
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:  $2, 5, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{8} + \frac{1}{10} a^{6} - \frac{1}{5} a^{4} + \frac{1}{10} a^{2}$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{9} + \frac{1}{10} a^{7} - \frac{1}{5} a^{5} + \frac{1}{10} a^{3}$, $\frac{1}{600} a^{12} - \frac{1}{200} a^{10} - \frac{31}{150} a^{8} - \frac{37}{600} a^{6} + \frac{37}{75} a^{4} - \frac{19}{120} a^{2} + \frac{5}{24}$, $\frac{1}{600} a^{13} - \frac{1}{200} a^{11} - \frac{31}{150} a^{9} - \frac{37}{600} a^{7} + \frac{37}{75} a^{5} - \frac{19}{120} a^{3} + \frac{5}{24} a$, $\frac{1}{54189810259942767000} a^{14} + \frac{2320448428726103}{13547452564985691750} a^{12} + \frac{837129642046013531}{54189810259942767000} a^{10} - \frac{4127262614656474999}{18063270086647589000} a^{8} + \frac{4748361402589547347}{18063270086647589000} a^{6} - \frac{72424095577681873}{230594937276352200} a^{4} + \frac{50984263921706954}{270949051299713835} a^{2} + \frac{1031953533674869}{9223797491054088}$, $\frac{1}{54189810259942767000} a^{15} + \frac{2320448428726103}{13547452564985691750} a^{13} + \frac{837129642046013531}{54189810259942767000} a^{11} - \frac{4127262614656474999}{18063270086647589000} a^{9} + \frac{4748361402589547347}{18063270086647589000} a^{7} - \frac{72424095577681873}{230594937276352200} a^{5} + \frac{50984263921706954}{270949051299713835} a^{3} + \frac{1031953533674869}{9223797491054088} a$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $7$
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:  \( 60187606.0742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n781 are not computed
Character table for t16n781 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.2312.1, 4.0.39304.1, 8.0.1544804416.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.8.20.4$x^{8} + 72 x^{4} + 656$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
$5$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17Data not computed
47Data not computed