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Properties

Label 16.4.51371070174...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 5^{8}\cdot 283^{6}$
Root discriminant $26.27$
Ramified primes $2, 5, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1665

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![271, 539, -3700, 6490, -8004, 7067, -4897, 3053, -1684, 1043, -518, 219, -54, -9, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 - 9*x^13 - 54*x^12 + 219*x^11 - 518*x^10 + 1043*x^9 - 1684*x^8 + 3053*x^7 - 4897*x^6 + 7067*x^5 - 8004*x^4 + 6490*x^3 - 3700*x^2 + 539*x + 271)
 
gp: K = bnfinit(x^16 - 2*x^15 + 9*x^14 - 9*x^13 - 54*x^12 + 219*x^11 - 518*x^10 + 1043*x^9 - 1684*x^8 + 3053*x^7 - 4897*x^6 + 7067*x^5 - 8004*x^4 + 6490*x^3 - 3700*x^2 + 539*x + 271, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 9 x^{14} - 9 x^{13} - 54 x^{12} + 219 x^{11} - 518 x^{10} + 1043 x^{9} - 1684 x^{8} + 3053 x^{7} - 4897 x^{6} + 7067 x^{5} - 8004 x^{4} + 6490 x^{3} - 3700 x^{2} + 539 x + 271 \)

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:  \(51371070174496900000000=2^{8}\cdot 5^{8}\cdot 283^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.27$
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:  $2, 5, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{605781479585755164208292744} a^{15} - \frac{75244169392663537098186077}{605781479585755164208292744} a^{14} + \frac{1473893928861455261383052}{75722684948219395526036593} a^{13} + \frac{108863183044151275720298415}{605781479585755164208292744} a^{12} + \frac{172865766288781493214794869}{605781479585755164208292744} a^{11} - \frac{47658598833870985940862993}{151445369896438791052073186} a^{10} - \frac{29075254602276100721288357}{302890739792877582104146372} a^{9} + \frac{43668362667171285160842129}{605781479585755164208292744} a^{8} + \frac{253181805308537985432070161}{605781479585755164208292744} a^{7} - \frac{20272195283791662304446295}{302890739792877582104146372} a^{6} + \frac{242225523202135645567189625}{605781479585755164208292744} a^{5} - \frac{1847061999482320031924209}{75722684948219395526036593} a^{4} - \frac{26288268035241327958763945}{151445369896438791052073186} a^{3} - \frac{57412324477427672250438373}{302890739792877582104146372} a^{2} - \frac{11150862691674064611378771}{302890739792877582104146372} a + \frac{21680386768232350508047765}{605781479585755164208292744}$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 133705.188345 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1665:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 78 conjugacy class representatives for t16n1665 are not computed
Character table for t16n1665 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.283.1, 8.4.50055625.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
5Data not computed
283Data not computed