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Properties

Label 16.8.91603437103...1609.2
Degree $16$
Signature $[8, 4]$
Discriminant $13^{6}\cdot 17^{12}\cdot 2389^{4}$
Root discriminant $153.15$
Ramified primes $13, 17, 2389$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1392

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-181612, 199950, -230331, -1077988, 394497, 643684, -369711, -68697, 73976, -29093, 9904, 2505, -1011, 0, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 3*x^14 - 1011*x^12 + 2505*x^11 + 9904*x^10 - 29093*x^9 + 73976*x^8 - 68697*x^7 - 369711*x^6 + 643684*x^5 + 394497*x^4 - 1077988*x^3 - 230331*x^2 + 199950*x - 181612)
 
gp: K = bnfinit(x^16 - 2*x^15 - 3*x^14 - 1011*x^12 + 2505*x^11 + 9904*x^10 - 29093*x^9 + 73976*x^8 - 68697*x^7 - 369711*x^6 + 643684*x^5 + 394497*x^4 - 1077988*x^3 - 230331*x^2 + 199950*x - 181612, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 3 x^{14} - 1011 x^{12} + 2505 x^{11} + 9904 x^{10} - 29093 x^{9} + 73976 x^{8} - 68697 x^{7} - 369711 x^{6} + 643684 x^{5} + 394497 x^{4} - 1077988 x^{3} - 230331 x^{2} + 199950 x - 181612 \)

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:  \(91603437103816431399094670113021609=13^{6}\cdot 17^{12}\cdot 2389^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.15$
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:  $13, 17, 2389$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{26143840909934683479934611267311023660947397516} a^{15} - \frac{149339540416962479350941053064020662461722478}{6535960227483670869983652816827755915236849379} a^{14} - \frac{666308725561206893624752933985385601551427700}{6535960227483670869983652816827755915236849379} a^{13} + \frac{5682897983017180055594656169454559843726791471}{26143840909934683479934611267311023660947397516} a^{12} + \frac{1257020190181062148665812983921120160395361197}{6535960227483670869983652816827755915236849379} a^{11} + \frac{11130794201176885496440798649713064866257354637}{26143840909934683479934611267311023660947397516} a^{10} - \frac{3171527909948081089290074848110447296531497993}{6535960227483670869983652816827755915236849379} a^{9} - \frac{2908541207163621174628380629873768554790994043}{6535960227483670869983652816827755915236849379} a^{8} + \frac{7448585991948523957452945750769162801580733317}{26143840909934683479934611267311023660947397516} a^{7} + \frac{2248202399709544429526636447738756046148026027}{6535960227483670869983652816827755915236849379} a^{6} + \frac{3324371415874681387651778602692703784764768301}{26143840909934683479934611267311023660947397516} a^{5} - \frac{2281818562824656379421160305323044705484541996}{6535960227483670869983652816827755915236849379} a^{4} - \frac{5589834895064751310956194856210768475752374739}{13071920454967341739967305633655511830473698758} a^{3} - \frac{5743092470631384305241704570237795096217669647}{26143840909934683479934611267311023660947397516} a^{2} - \frac{2966949353812123980387450766142390753312931937}{6535960227483670869983652816827755915236849379} a - \frac{2855712332934738359184969541760465259044816923}{6535960227483670869983652816827755915236849379}$

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:  \( 1294348567980 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1392:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 80 conjugacy class representatives for t16n1392 are not computed
Character table for t16n1392 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.152583041.1, 4.4.8975473.1, 8.8.23281584400807681.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
2389Data not computed