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Properties

Label 16.6.19653336357...0000.2
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{8}\cdot 5^{11}\cdot 11^{6}\cdot 31^{6}$
Root discriminant $38.09$
Ramified primes $2, 5, 11, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1774

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -13, -16, 281, 52, 417, 738, 61, -529, -361, -2, 133, 72, -6, -16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 - 6*x^13 + 72*x^12 + 133*x^11 - 2*x^10 - 361*x^9 - 529*x^8 + 61*x^7 + 738*x^6 + 417*x^5 + 52*x^4 + 281*x^3 - 16*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 16*x^14 - 6*x^13 + 72*x^12 + 133*x^11 - 2*x^10 - 361*x^9 - 529*x^8 + 61*x^7 + 738*x^6 + 417*x^5 + 52*x^4 + 281*x^3 - 16*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 16 x^{14} - 6 x^{13} + 72 x^{12} + 133 x^{11} - 2 x^{10} - 361 x^{9} - 529 x^{8} + 61 x^{7} + 738 x^{6} + 417 x^{5} + 52 x^{4} + 281 x^{3} - 16 x^{2} - 13 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-19653336357700512500000000=-\,2^{8}\cdot 5^{11}\cdot 11^{6}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.09$
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:  $2, 5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{2} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{3}{10} a^{7} + \frac{3}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} + \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{2}{5} a^{10} + \frac{3}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a - \frac{1}{5}$, $\frac{1}{10} a^{14} + \frac{2}{5} a^{11} - \frac{1}{2} a^{10} - \frac{3}{10} a^{9} - \frac{3}{10} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{10} a^{3} - \frac{1}{2} a - \frac{1}{10}$, $\frac{1}{760136176475530} a^{15} + \frac{1824269129315}{152027235295106} a^{14} + \frac{3385003999983}{152027235295106} a^{13} + \frac{10481872275708}{380068088237765} a^{12} + \frac{35142176432937}{380068088237765} a^{11} + \frac{35776728692701}{380068088237765} a^{10} - \frac{44623455967518}{380068088237765} a^{9} - \frac{366510315443991}{760136176475530} a^{8} - \frac{64125799609943}{760136176475530} a^{7} + \frac{281365394008753}{760136176475530} a^{6} - \frac{114441369945829}{760136176475530} a^{5} + \frac{80420668734}{380068088237765} a^{4} + \frac{169026303870894}{380068088237765} a^{3} + \frac{26236185657658}{76013617647553} a^{2} - \frac{32146446350307}{152027235295106} a - \frac{142059637184383}{760136176475530}$

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:  $10$
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:  \( 4432203.05035 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1774:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1774 are not computed
Character table for t16n1774 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.8525.1, 8.8.123911940625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.8.8$x^{8} + 4 x^{5} + 8 x^{2} + 48$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$