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Properties

Label 16.4.166581322179183497.1
Degree $16$
Signature $[4, 6]$
Discriminant $1609\cdot 15809\cdot 31393\cdot 208609$
Root discriminant $11.92$
Ramified primes $1609, 15809, 31393, 208609$
Class number $1$
Class group Trivial
Galois group $S_{16}$ (as 16T1954)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 3, -14, 21, -16, 3, 10, -19, 26, -28, 20, -6, -5, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 7*x^14 - 5*x^13 - 6*x^12 + 20*x^11 - 28*x^10 + 26*x^9 - 19*x^8 + 10*x^7 + 3*x^6 - 16*x^5 + 21*x^4 - 14*x^3 + 3*x^2 + x - 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 7*x^14 - 5*x^13 - 6*x^12 + 20*x^11 - 28*x^10 + 26*x^9 - 19*x^8 + 10*x^7 + 3*x^6 - 16*x^5 + 21*x^4 - 14*x^3 + 3*x^2 + x - 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 7 x^{14} - 5 x^{13} - 6 x^{12} + 20 x^{11} - 28 x^{10} + 26 x^{9} - 19 x^{8} + 10 x^{7} + 3 x^{6} - 16 x^{5} + 21 x^{4} - 14 x^{3} + 3 x^{2} + x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(166581322179183497=1609\cdot 15809\cdot 31393\cdot 208609\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.92$
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:  $1609, 15809, 31393, 208609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{22481} a^{15} - \frac{3219}{22481} a^{14} + \frac{7832}{22481} a^{13} - \frac{1165}{22481} a^{12} - \frac{8858}{22481} a^{11} - \frac{4937}{22481} a^{10} + \frac{841}{22481} a^{9} - \frac{6069}{22481} a^{8} - \frac{1692}{22481} a^{7} - \frac{612}{22481} a^{6} - \frac{10745}{22481} a^{5} - \frac{8138}{22481} a^{4} - \frac{4193}{22481} a^{3} - \frac{8119}{22481} a^{2} + \frac{2147}{22481} a - \frac{937}{22481}$

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:  $9$
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:  \( 131.22830925 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_{16}$ (as 16T1954):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20922789888000
The 231 conjugacy class representatives for $S_{16}$ are not computed
Character table for $S_{16}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ $16$ $16$ $16$ ${\href{/LocalNumberField/11.11.0.1}{11} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ $16$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $16$ $16$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
1609Data not computed
15809Data not computed
31393Data not computed
208609Data not computed