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Properties

Label 16.6.61695001296...0000.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,2^{24}\cdot 5^{8}\cdot 11^{3}\cdot 29^{4}$
Root discriminant $23.01$
Ramified primes $2, 5, 11, 29$
Class number $1$
Class group Trivial
Galois group 16T1567

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, -32, 104, 80, -431, 422, 106, -516, 356, -22, -102, 36, 22, -10, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 10*x^13 + 22*x^12 + 36*x^11 - 102*x^10 - 22*x^9 + 356*x^8 - 516*x^7 + 106*x^6 + 422*x^5 - 431*x^4 + 80*x^3 + 104*x^2 - 32*x - 11)
 
gp: K = bnfinit(x^16 - 2*x^15 - 10*x^13 + 22*x^12 + 36*x^11 - 102*x^10 - 22*x^9 + 356*x^8 - 516*x^7 + 106*x^6 + 422*x^5 - 431*x^4 + 80*x^3 + 104*x^2 - 32*x - 11, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 10 x^{13} + 22 x^{12} + 36 x^{11} - 102 x^{10} - 22 x^{9} + 356 x^{8} - 516 x^{7} + 106 x^{6} + 422 x^{5} - 431 x^{4} + 80 x^{3} + 104 x^{2} - 32 x - 11 \)

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:  \(-6169500129689600000000=-\,2^{24}\cdot 5^{8}\cdot 11^{3}\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.01$
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:  $2, 5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29} a^{14} + \frac{6}{29} a^{13} + \frac{3}{29} a^{12} + \frac{5}{29} a^{11} + \frac{14}{29} a^{10} + \frac{10}{29} a^{9} - \frac{14}{29} a^{8} - \frac{4}{29} a^{7} - \frac{3}{29} a^{6} - \frac{12}{29} a^{5} + \frac{5}{29} a^{3} - \frac{14}{29} a^{2} + \frac{4}{29} a + \frac{12}{29}$, $\frac{1}{7671285411131021} a^{15} - \frac{76913773118478}{7671285411131021} a^{14} - \frac{2580304555125422}{7671285411131021} a^{13} - \frac{768827740639595}{7671285411131021} a^{12} - \frac{955588285635349}{7671285411131021} a^{11} + \frac{1735131119331319}{7671285411131021} a^{10} + \frac{1564027693047087}{7671285411131021} a^{9} + \frac{2041555176571320}{7671285411131021} a^{8} - \frac{262590144400904}{7671285411131021} a^{7} + \frac{223638362281784}{7671285411131021} a^{6} - \frac{2262555291093458}{7671285411131021} a^{5} + \frac{1669811635074241}{7671285411131021} a^{4} - \frac{2046510313804643}{7671285411131021} a^{3} + \frac{164576962871280}{7671285411131021} a^{2} + \frac{251930639999503}{7671285411131021} a + \frac{1711320596543755}{7671285411131021}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 65861.7086074 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1567:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 106 conjugacy class representatives for t16n1567 are not computed
Character table for t16n1567 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.6.1480160000.3

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$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}$
11.4.3.2$x^{4} - 11$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$