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Properties

Label 16.16.2154295895...1664.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{48}\cdot 3^{8}\cdot 73^{6}\cdot 937^{4}$
Root discriminant $383.12$
Ramified primes $2, 3, 73, 937$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1189

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4107750751677169, 0, -666358713185624, 0, 35801259737778, 0, -897484048920, 0, 12182297082, 0, -95045720, 0, 425702, 0, -1016, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 1016*x^14 + 425702*x^12 - 95045720*x^10 + 12182297082*x^8 - 897484048920*x^6 + 35801259737778*x^4 - 666358713185624*x^2 + 4107750751677169)
 
gp: K = bnfinit(x^16 - 1016*x^14 + 425702*x^12 - 95045720*x^10 + 12182297082*x^8 - 897484048920*x^6 + 35801259737778*x^4 - 666358713185624*x^2 + 4107750751677169, 1)
 

Normalized defining polynomial

\( x^{16} - 1016 x^{14} + 425702 x^{12} - 95045720 x^{10} + 12182297082 x^{8} - 897484048920 x^{6} + 35801259737778 x^{4} - 666358713185624 x^{2} + 4107750751677169 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(215429589500286246610837301474983006961664=2^{48}\cdot 3^{8}\cdot 73^{6}\cdot 937^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $383.12$
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:  $2, 3, 73, 937$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{1874} a^{10} - \frac{79}{1874} a^{8} + \frac{152}{937} a^{6} - \frac{94}{937} a^{4} - \frac{663}{1874} a^{2} - \frac{1}{2}$, $\frac{1}{1874} a^{11} - \frac{79}{1874} a^{9} + \frac{152}{937} a^{7} - \frac{94}{937} a^{5} - \frac{663}{1874} a^{3} - \frac{1}{2} a$, $\frac{1}{128183474} a^{12} - \frac{508}{64091737} a^{10} + \frac{212851}{64091737} a^{8} + \frac{16568877}{64091737} a^{6} - \frac{59224685}{128183474} a^{4} + \frac{414}{937} a^{2}$, $\frac{1}{128183474} a^{13} - \frac{508}{64091737} a^{11} + \frac{212851}{64091737} a^{9} + \frac{16568877}{64091737} a^{7} - \frac{59224685}{128183474} a^{5} + \frac{414}{937} a^{3}$, $\frac{1}{2237178377048783455672401631111948702} a^{14} + \frac{713176043393317798156050340}{1118589188524391727836200815555974351} a^{12} + \frac{281624811321928061621695536823621}{1118589188524391727836200815555974351} a^{10} - \frac{93636034687509945947925902501477754}{1118589188524391727836200815555974351} a^{8} - \frac{262921193721844031188482828872341349}{2237178377048783455672401631111948702} a^{6} - \frac{457283023699816525239317313317263}{1193798493622616571863608127594423} a^{4} + \frac{136537155067750604931627020}{17452939191278147568947941223} a^{2} - \frac{3297896462021213710965621}{18626402552057788227265679}$, $\frac{1}{2237178377048783455672401631111948702} a^{15} + \frac{713176043393317798156050340}{1118589188524391727836200815555974351} a^{13} + \frac{281624811321928061621695536823621}{1118589188524391727836200815555974351} a^{11} - \frac{93636034687509945947925902501477754}{1118589188524391727836200815555974351} a^{9} - \frac{262921193721844031188482828872341349}{2237178377048783455672401631111948702} a^{7} - \frac{457283023699816525239317313317263}{1193798493622616571863608127594423} a^{5} + \frac{136537155067750604931627020}{17452939191278147568947941223} a^{3} - \frac{3297896462021213710965621}{18626402552057788227265679} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 2488658750900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1189:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1189 are not computed
Character table for t16n1189 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.10512.1, 8.8.28288548864.4

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.8.6.1$x^{8} - 14527 x^{4} + 78021889$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
937Data not computed