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Properties

Label 16.8.54742762016...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 5^{6}\cdot 13^{8}$
Root discriminant $26.37$
Ramified primes $2, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times C_4).C_2^4$ (as 16T205)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-647, -460, 2016, 1128, -2773, -1956, 1310, 1596, 115, -616, -296, 112, 103, -8, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 16*x^14 - 8*x^13 + 103*x^12 + 112*x^11 - 296*x^10 - 616*x^9 + 115*x^8 + 1596*x^7 + 1310*x^6 - 1956*x^5 - 2773*x^4 + 1128*x^3 + 2016*x^2 - 460*x - 647)
 
gp: K = bnfinit(x^16 - 16*x^14 - 8*x^13 + 103*x^12 + 112*x^11 - 296*x^10 - 616*x^9 + 115*x^8 + 1596*x^7 + 1310*x^6 - 1956*x^5 - 2773*x^4 + 1128*x^3 + 2016*x^2 - 460*x - 647, 1)
 

Normalized defining polynomial

\( x^{16} - 16 x^{14} - 8 x^{13} + 103 x^{12} + 112 x^{11} - 296 x^{10} - 616 x^{9} + 115 x^{8} + 1596 x^{7} + 1310 x^{6} - 1956 x^{5} - 2773 x^{4} + 1128 x^{3} + 2016 x^{2} - 460 x - 647 \)

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:  \(54742762016210944000000=2^{32}\cdot 5^{6}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.37$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{312481046936931931} a^{15} - \frac{115884053977144550}{312481046936931931} a^{14} - \frac{15441423622762452}{312481046936931931} a^{13} + \frac{79292455478923938}{312481046936931931} a^{12} - \frac{39777240938627727}{312481046936931931} a^{11} - \frac{232528879704795}{312481046936931931} a^{10} + \frac{80523056048123009}{312481046936931931} a^{9} + \frac{54260935612895929}{312481046936931931} a^{8} + \frac{29849928395218210}{312481046936931931} a^{7} - \frac{52009883566889827}{312481046936931931} a^{6} - \frac{127954160265848212}{312481046936931931} a^{5} + \frac{18126777770996520}{312481046936931931} a^{4} + \frac{903764049488876}{312481046936931931} a^{3} - \frac{5168309251825077}{312481046936931931} a^{2} - \frac{24386321255457656}{312481046936931931} a - \frac{125962170460445841}{312481046936931931}$

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:  $11$
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:  \( 318064.445657 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).C_2^4$ (as 16T205):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $(C_2\times C_4).C_2^4$
Character table for $(C_2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.2924646400.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$5$5.8.6.3$x^{8} + 25 x^{4} + 200$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
13Data not computed