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Properties

Label 16.12.1218218632...8656.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{50}\cdot 7^{8}\cdot 137^{2}$
Root discriminant $42.69$
Ramified primes $2, 7, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1439

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 0, -756, 0, 2944, 0, -908, 0, -638, 0, 236, 0, 16, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49)
 
gp: K = bnfinit(x^16 - 12*x^14 + 16*x^12 + 236*x^10 - 638*x^8 - 908*x^6 + 2944*x^4 - 756*x^2 + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 12 x^{14} + 16 x^{12} + 236 x^{10} - 638 x^{8} - 908 x^{6} + 2944 x^{4} - 756 x^{2} + 49 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(121821863230510940507078656=2^{50}\cdot 7^{8}\cdot 137^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.69$
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:  $2, 7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} + \frac{1}{4}$, $\frac{1}{4} a^{9} + \frac{1}{4} a$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{3} a^{6} - \frac{1}{4} a^{4} - \frac{1}{12} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{7}{24} a + \frac{1}{24}$, $\frac{1}{591864} a^{14} - \frac{12059}{591864} a^{12} + \frac{21499}{591864} a^{10} + \frac{40927}{591864} a^{8} - \frac{25495}{591864} a^{6} + \frac{157229}{591864} a^{4} - \frac{7351}{197288} a^{2} - \frac{38927}{84552}$, $\frac{1}{4143048} a^{15} + \frac{6301}{2071524} a^{13} - \frac{1}{24} a^{12} - \frac{124359}{1381016} a^{11} - \frac{1}{12} a^{10} - \frac{90511}{2071524} a^{9} - \frac{1}{8} a^{8} - \frac{370193}{1381016} a^{7} - \frac{1}{6} a^{6} - \frac{254309}{2071524} a^{5} + \frac{1}{8} a^{4} - \frac{712561}{4143048} a^{3} + \frac{5}{12} a^{2} - \frac{14121}{98644} a + \frac{1}{24}$

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:  $13$
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:  \( 61882651.1808 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1439:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1439 are not computed
Character table for t16n1439 is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.86228860928.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$