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Properties

Label 16.12.6767652375...1904.1
Degree $16$
Signature $[12, 2]$
Discriminant $2^{62}\cdot 1889^{5}\cdot 247007^{2}$
Root discriminant $731.82$
Ramified primes $2, 1889, 247007$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T1186

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6580128555447173793296, 0, -42597270982345604736, 0, -88192020563282736, 0, 1040831008579456, 0, -1266129331670, 0, -3409772064, 0, 7689860, 0, -4896, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4896*x^14 + 7689860*x^12 - 3409772064*x^10 - 1266129331670*x^8 + 1040831008579456*x^6 - 88192020563282736*x^4 - 42597270982345604736*x^2 + 6580128555447173793296)
 
gp: K = bnfinit(x^16 - 4896*x^14 + 7689860*x^12 - 3409772064*x^10 - 1266129331670*x^8 + 1040831008579456*x^6 - 88192020563282736*x^4 - 42597270982345604736*x^2 + 6580128555447173793296, 1)
 

Normalized defining polynomial

\( x^{16} - 4896 x^{14} + 7689860 x^{12} - 3409772064 x^{10} - 1266129331670 x^{8} + 1040831008579456 x^{6} - 88192020563282736 x^{4} - 42597270982345604736 x^{2} + 6580128555447173793296 \)

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:  \(6767652375800250507037432977960083710388731904=2^{62}\cdot 1889^{5}\cdot 247007^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $731.82$
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:  $2, 1889, 247007$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{7556} a^{12} - \frac{559}{3778} a^{10} + \frac{815}{3778} a^{8} + \frac{347}{1889} a^{6} - \frac{483}{3778} a^{4} + \frac{378}{1889} a^{2}$, $\frac{1}{7556} a^{13} + \frac{771}{7556} a^{11} + \frac{815}{3778} a^{9} + \frac{347}{1889} a^{7} - \frac{483}{3778} a^{5} - \frac{1133}{3778} a^{3}$, $\frac{1}{1752824110048538223526679021636870587301906842757798574078896601998984} a^{14} - \frac{10332021034322558601513322469643447485825115327174946745955235980}{219103013756067277940834877704608823412738355344724821759862075249873} a^{12} - \frac{19614501967464717770711531401065596777553239939394193803344894477228}{219103013756067277940834877704608823412738355344724821759862075249873} a^{10} + \frac{26949501129012459358078670352393022069329310305958711339375032753927}{219103013756067277940834877704608823412738355344724821759862075249873} a^{8} - \frac{158309104086738647000132990134624719998057042779886111307578714477571}{876412055024269111763339510818435293650953421378899287039448300999492} a^{6} + \frac{2398819211596539282724755708485501588894295359885017948750177537416}{219103013756067277940834877704608823412738355344724821759862075249873} a^{4} + \frac{48399784621783814894815480533493708938930622726550331378709101994}{115988890289077436707694482638755332669527980595407528724119679857} a^{2} - \frac{684310526905177650482520496621900736910270045802028058}{248585146153395752429171469394331836821422321276285874959}$, $\frac{1}{1752824110048538223526679021636870587301906842757798574078896601998984} a^{15} - \frac{10332021034322558601513322469643447485825115327174946745955235980}{219103013756067277940834877704608823412738355344724821759862075249873} a^{13} - \frac{19614501967464717770711531401065596777553239939394193803344894477228}{219103013756067277940834877704608823412738355344724821759862075249873} a^{11} + \frac{26949501129012459358078670352393022069329310305958711339375032753927}{219103013756067277940834877704608823412738355344724821759862075249873} a^{9} - \frac{158309104086738647000132990134624719998057042779886111307578714477571}{876412055024269111763339510818435293650953421378899287039448300999492} a^{7} + \frac{2398819211596539282724755708485501588894295359885017948750177537416}{219103013756067277940834877704608823412738355344724821759862075249873} a^{5} + \frac{48399784621783814894815480533493708938930622726550331378709101994}{115988890289077436707694482638755332669527980595407528724119679857} a^{3} - \frac{684310526905177650482520496621900736910270045802028058}{248585146153395752429171469394331836821422321276285874959} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 70498197375200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1186:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 46 conjugacy class representatives for t16n1186
Character table for t16n1186 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.31.36$x^{8} + 12 x^{4} + 2$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
2.8.31.36$x^{8} + 12 x^{4} + 2$$8$$1$$31$$(C_8:C_2):C_2$$[2, 3, 7/2, 4, 5]$
1889Data not computed
247007Data not computed