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Properties

Label 16.8.14619841491...8016.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{32}\cdot 17^{8}\cdot 47^{4}$
Root discriminant $43.18$
Ramified primes $2, 17, 47$
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, -518, -65, 6650, -16775, 17940, -8510, 716, 316, 440, -194, -72, -7, 50, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 50*x^13 - 7*x^12 - 72*x^11 - 194*x^10 + 440*x^9 + 316*x^8 + 716*x^7 - 8510*x^6 + 17940*x^5 - 16775*x^4 + 6650*x^3 - 65*x^2 - 518*x + 49)
 
gp: K = bnfinit(x^16 - 2*x^15 - 15*x^14 + 50*x^13 - 7*x^12 - 72*x^11 - 194*x^10 + 440*x^9 + 316*x^8 + 716*x^7 - 8510*x^6 + 17940*x^5 - 16775*x^4 + 6650*x^3 - 65*x^2 - 518*x + 49, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 15 x^{14} + 50 x^{13} - 7 x^{12} - 72 x^{11} - 194 x^{10} + 440 x^{9} + 316 x^{8} + 716 x^{7} - 8510 x^{6} + 17940 x^{5} - 16775 x^{4} + 6650 x^{3} - 65 x^{2} - 518 x + 49 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(146198414913373828091478016=2^{32}\cdot 17^{8}\cdot 47^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.18$
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:  $2, 17, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{14} a^{13} + \frac{1}{7} a^{12} - \frac{1}{14} a^{11} + \frac{1}{7} a^{10} - \frac{5}{14} a^{9} + \frac{2}{7} a^{8} - \frac{5}{14} a^{7} - \frac{2}{7} a^{6} - \frac{3}{14} a^{5} - \frac{3}{7} a^{4} - \frac{5}{14} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{14} a^{14} + \frac{1}{7} a^{12} - \frac{3}{14} a^{11} - \frac{1}{7} a^{10} - \frac{1}{2} a^{9} - \frac{3}{7} a^{8} - \frac{1}{14} a^{7} - \frac{1}{7} a^{6} - \frac{1}{2} a^{5} - \frac{5}{14} a^{3} - \frac{1}{2} a^{2} - \frac{3}{14} a$, $\frac{1}{422611587983506054} a^{15} - \frac{6680028349439439}{422611587983506054} a^{14} - \frac{490859937183409}{211305793991753027} a^{13} + \frac{9257308591592457}{60373083997643722} a^{12} - \frac{18883712881639213}{211305793991753027} a^{11} - \frac{12535682794765261}{60373083997643722} a^{10} - \frac{56906162020207399}{211305793991753027} a^{9} + \frac{119346968115225873}{422611587983506054} a^{8} - \frac{57685011506904087}{211305793991753027} a^{7} + \frac{64316456979366731}{422611587983506054} a^{6} - \frac{476174451024871}{1180479296043313} a^{5} + \frac{47371720990806143}{422611587983506054} a^{4} - \frac{17030143111200867}{422611587983506054} a^{3} + \frac{45144472287727756}{211305793991753027} a^{2} - \frac{66290093471640053}{211305793991753027} a - \frac{10071846493432044}{30186541998821861}$

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:  $11$
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:  \( 32938422.328 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.257260716032.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.6.1$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}$
2.8.20.1$x^{8} + 4 x^{7} + 14 x^{4} + 4$$4$$2$$20$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
47Data not computed