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Properties

Label 16.2.88918500023...9987.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,41^{15}\cdot 83^{3}$
Root discriminant $74.44$
Ramified primes $41, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-251312, -735228, -755227, -211133, 146265, 77422, -38281, -35891, -9060, -1677, -94, 675, 297, -24, -29, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 29*x^14 - 24*x^13 + 297*x^12 + 675*x^11 - 94*x^10 - 1677*x^9 - 9060*x^8 - 35891*x^7 - 38281*x^6 + 77422*x^5 + 146265*x^4 - 211133*x^3 - 755227*x^2 - 735228*x - 251312)
 
gp: K = bnfinit(x^16 - x^15 - 29*x^14 - 24*x^13 + 297*x^12 + 675*x^11 - 94*x^10 - 1677*x^9 - 9060*x^8 - 35891*x^7 - 38281*x^6 + 77422*x^5 + 146265*x^4 - 211133*x^3 - 755227*x^2 - 735228*x - 251312, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 29 x^{14} - 24 x^{13} + 297 x^{12} + 675 x^{11} - 94 x^{10} - 1677 x^{9} - 9060 x^{8} - 35891 x^{7} - 38281 x^{6} + 77422 x^{5} + 146265 x^{4} - 211133 x^{3} - 755227 x^{2} - 735228 x - 251312 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-889185000234066487453260499987=-\,41^{15}\cdot 83^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.44$
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:  $41, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{88433227302426038467945148618372} a^{15} + \frac{2925811584408751919123014166889}{88433227302426038467945148618372} a^{14} + \frac{2119624278488186648317978378749}{88433227302426038467945148618372} a^{13} - \frac{9077597779144484406018236683445}{44216613651213019233972574309186} a^{12} + \frac{16550677438056059662789249559475}{88433227302426038467945148618372} a^{11} + \frac{13119248569850510759865911384469}{88433227302426038467945148618372} a^{10} - \frac{1333723584917718730025646086159}{44216613651213019233972574309186} a^{9} + \frac{11158456134038342069053139924783}{88433227302426038467945148618372} a^{8} + \frac{4680656836311658576230838060157}{44216613651213019233972574309186} a^{7} - \frac{10285624834741185767925248478475}{88433227302426038467945148618372} a^{6} + \frac{9306954347011230319078407774947}{88433227302426038467945148618372} a^{5} - \frac{5293370015504112391443923842771}{44216613651213019233972574309186} a^{4} - \frac{23985798776771783036462588541991}{88433227302426038467945148618372} a^{3} + \frac{8077033478071669512469028437913}{88433227302426038467945148618372} a^{2} + \frac{14026784989911420194005421374569}{88433227302426038467945148618372} a - \frac{92399036647442818531245559676}{22108306825606509616986287154593}$

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:  $8$
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:  \( 556518262.188 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.16164604732123.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
41Data not computed
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$