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Properties

Label 16.8.25213549702...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 19\cdot 29^{4}\cdot 31^{5}$
Root discriminant $51.59$
Ramified primes $2, 5, 19, 29, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1719

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19069, 64102, 15662, -72424, -11254, -19442, 18416, 4542, -392, 2874, -1962, 254, -187, 74, -8, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 8*x^14 + 74*x^13 - 187*x^12 + 254*x^11 - 1962*x^10 + 2874*x^9 - 392*x^8 + 4542*x^7 + 18416*x^6 - 19442*x^5 - 11254*x^4 - 72424*x^3 + 15662*x^2 + 64102*x + 19069)
 
gp: K = bnfinit(x^16 - 4*x^15 - 8*x^14 + 74*x^13 - 187*x^12 + 254*x^11 - 1962*x^10 + 2874*x^9 - 392*x^8 + 4542*x^7 + 18416*x^6 - 19442*x^5 - 11254*x^4 - 72424*x^3 + 15662*x^2 + 64102*x + 19069, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 8 x^{14} + 74 x^{13} - 187 x^{12} + 254 x^{11} - 1962 x^{10} + 2874 x^{9} - 392 x^{8} + 4542 x^{7} + 18416 x^{6} - 19442 x^{5} - 11254 x^{4} - 72424 x^{3} + 15662 x^{2} + 64102 x + 19069 \)

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:  \(2521354970203350630400000000=2^{24}\cdot 5^{8}\cdot 19\cdot 29^{4}\cdot 31^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.59$
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:  $2, 5, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{95} a^{14} - \frac{1}{95} a^{13} - \frac{9}{95} a^{12} - \frac{31}{95} a^{11} - \frac{13}{95} a^{10} + \frac{1}{95} a^{9} - \frac{47}{95} a^{8} + \frac{18}{95} a^{7} + \frac{1}{19} a^{6} - \frac{1}{19} a^{5} + \frac{4}{95} a^{3} + \frac{5}{19} a^{2} - \frac{27}{95} a - \frac{5}{19}$, $\frac{1}{17092567262748331372367215438470126365} a^{15} + \frac{4762398823425843183660273570964492}{3418513452549666274473443087694025273} a^{14} - \frac{1174601710940118798526607311520183696}{17092567262748331372367215438470126365} a^{13} - \frac{973092711511744893171470551759060968}{17092567262748331372367215438470126365} a^{12} + \frac{193887667790099444152814777545755762}{743155098380362233581183279933483755} a^{11} + \frac{2812601310463596374466224177636055992}{17092567262748331372367215438470126365} a^{10} + \frac{5146180255212237858520876810156097082}{17092567262748331372367215438470126365} a^{9} - \frac{5663499830446327090781282251318644063}{17092567262748331372367215438470126365} a^{8} + \frac{1403969366572957381114780753743322155}{3418513452549666274473443087694025273} a^{7} - \frac{1110563417105011892412083599284262294}{3418513452549666274473443087694025273} a^{6} + \frac{5434867882846398955786041771689118979}{17092567262748331372367215438470126365} a^{5} - \frac{7525381868110886292896269807215867648}{17092567262748331372367215438470126365} a^{4} - \frac{663802417835770138802366452055501341}{3418513452549666274473443087694025273} a^{3} + \frac{5269498174560782922919021467047687431}{17092567262748331372367215438470126365} a^{2} + \frac{5466185950129519275272231770866123131}{17092567262748331372367215438470126365} a - \frac{4224417016787805844890741392275987367}{17092567262748331372367215438470126365}$

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:  \( 60534748.4109 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1719:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 152 conjugacy class representatives for t16n1719 are not computed
Character table for t16n1719 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.359600.2, 4.4.12400.1, 4.4.725.1, 8.8.129312160000.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.0.1$x^{4} - 2 x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$