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Properties

Label 16.0.40804000000...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 101^{2}$
Root discriminant $14.56$
Ramified primes $2, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -54, 39, 166, -214, -212, 346, 168, -288, -68, 156, -8, -49, 14, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 9*x^14 + 14*x^13 - 49*x^12 - 8*x^11 + 156*x^10 - 68*x^9 - 288*x^8 + 168*x^7 + 346*x^6 - 212*x^5 - 214*x^4 + 166*x^3 + 39*x^2 - 54*x + 11)
 
gp: K = bnfinit(x^16 - 6*x^15 + 9*x^14 + 14*x^13 - 49*x^12 - 8*x^11 + 156*x^10 - 68*x^9 - 288*x^8 + 168*x^7 + 346*x^6 - 212*x^5 - 214*x^4 + 166*x^3 + 39*x^2 - 54*x + 11, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 9 x^{14} + 14 x^{13} - 49 x^{12} - 8 x^{11} + 156 x^{10} - 68 x^{9} - 288 x^{8} + 168 x^{7} + 346 x^{6} - 212 x^{5} - 214 x^{4} + 166 x^{3} + 39 x^{2} - 54 x + 11 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4080400000000000000=2^{16}\cdot 5^{14}\cdot 101^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.56$
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:  $2, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{13} + \frac{1}{11} a^{11} - \frac{4}{11} a^{10} - \frac{5}{11} a^{9} + \frac{2}{11} a^{8} - \frac{2}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} - \frac{3}{11} a^{4} - \frac{2}{11} a^{3} + \frac{5}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{25342229} a^{15} - \frac{18846}{25342229} a^{14} - \frac{2036396}{25342229} a^{13} + \frac{9085143}{25342229} a^{12} - \frac{2679503}{25342229} a^{11} - \frac{2187495}{25342229} a^{10} - \frac{10185271}{25342229} a^{9} + \frac{10666779}{25342229} a^{8} - \frac{2848356}{25342229} a^{7} - \frac{7206136}{25342229} a^{6} + \frac{7585672}{25342229} a^{5} - \frac{6927522}{25342229} a^{4} + \frac{8946433}{25342229} a^{3} + \frac{4975203}{25342229} a^{2} - \frac{3438605}{25342229} a + \frac{151755}{2303839}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{8935810}{25342229} a^{15} + \frac{51240335}{25342229} a^{14} - \frac{66069842}{25342229} a^{13} - \frac{142705116}{25342229} a^{12} + \frac{389873384}{25342229} a^{11} + \frac{17402118}{2303839} a^{10} - \frac{120368904}{2303839} a^{9} + \frac{187803271}{25342229} a^{8} + \frac{2595823840}{25342229} a^{7} - \frac{619355254}{25342229} a^{6} - \frac{3235894204}{25342229} a^{5} + \frac{787834546}{25342229} a^{4} + \frac{2094411102}{25342229} a^{3} - \frac{759823425}{25342229} a^{2} - \frac{593487110}{25342229} a + \frac{24954113}{2303839} \) (order $10$)
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:  \( 2357.22868989 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.2000.1, \(\Q(\zeta_{5})\), 4.2.400.1, 8.0.4000000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
5Data not computed
101Data not computed