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Properties

Label 16.0.13348273655...4656.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 47^{4}\cdot 113^{4}$
Root discriminant $24.15$
Ramified primes $2, 47, 113$
Class number $1$
Class group Trivial
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288, -1248, 2776, -4112, 4632, -4360, 3680, -2744, 1759, -990, 539, -286, 146, -62, 23, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 23*x^14 - 62*x^13 + 146*x^12 - 286*x^11 + 539*x^10 - 990*x^9 + 1759*x^8 - 2744*x^7 + 3680*x^6 - 4360*x^5 + 4632*x^4 - 4112*x^3 + 2776*x^2 - 1248*x + 288)
 
gp: K = bnfinit(x^16 - 6*x^15 + 23*x^14 - 62*x^13 + 146*x^12 - 286*x^11 + 539*x^10 - 990*x^9 + 1759*x^8 - 2744*x^7 + 3680*x^6 - 4360*x^5 + 4632*x^4 - 4112*x^3 + 2776*x^2 - 1248*x + 288, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 23 x^{14} - 62 x^{13} + 146 x^{12} - 286 x^{11} + 539 x^{10} - 990 x^{9} + 1759 x^{8} - 2744 x^{7} + 3680 x^{6} - 4360 x^{5} + 4632 x^{4} - 4112 x^{3} + 2776 x^{2} - 1248 x + 288 \)

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:  \(13348273655014443974656=2^{24}\cdot 47^{4}\cdot 113^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.15$
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:  $2, 47, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{7}{16} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{48} a^{14} + \frac{1}{48} a^{13} - \frac{1}{24} a^{11} - \frac{5}{24} a^{9} + \frac{1}{48} a^{8} + \frac{1}{48} a^{7} - \frac{1}{3} a^{6} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{39069201842976} a^{15} - \frac{349156371811}{39069201842976} a^{14} + \frac{256627464605}{19534600921488} a^{13} + \frac{908208969731}{19534600921488} a^{12} - \frac{201557644415}{2441825115186} a^{11} - \frac{2384334270275}{19534600921488} a^{10} + \frac{559673549271}{13023067280992} a^{9} + \frac{2286490388765}{39069201842976} a^{8} - \frac{3096357595385}{6511533640496} a^{7} - \frac{2943175382173}{9767300460744} a^{6} - \frac{334080347023}{2441825115186} a^{5} - \frac{6650019241}{1627883410124} a^{4} + \frac{127767205975}{4883650230372} a^{3} + \frac{209970452311}{1627883410124} a^{2} - \frac{680518385773}{2441825115186} a - \frac{52733045366}{406970852531}$

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:  $7$
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:  \( 506496.645927 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.3008.1, 8.0.1022431232.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.2.2$x^{4} - 113 x^{2} + 127690$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$