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Properties

Label 16.16.2154295895...1664.2
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![694209877033441561, 0, -56980396020618580, 0, 1596828320830072, 0, -21949434606900, 0, 167072156785, 0, -729133076, 0, 1778562, 0, -2184, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2184*x^14 + 1778562*x^12 - 729133076*x^10 + 167072156785*x^8 - 21949434606900*x^6 + 1596828320830072*x^4 - 56980396020618580*x^2 + 694209877033441561)
 
gp: K = bnfinit(x^16 - 2184*x^14 + 1778562*x^12 - 729133076*x^10 + 167072156785*x^8 - 21949434606900*x^6 + 1596828320830072*x^4 - 56980396020618580*x^2 + 694209877033441561, 1)
 

Normalized defining polynomial

\( x^{16} - 2184 x^{14} + 1778562 x^{12} - 729133076 x^{10} + 167072156785 x^{8} - 21949434606900 x^{6} + 1596828320830072 x^{4} - 56980396020618580 x^{2} + 694209877033441561 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{937} a^{10} - \frac{310}{937} a^{8} + \frac{136}{937} a^{6} + \frac{33}{937} a^{4} - \frac{204}{937} a^{2}$, $\frac{1}{937} a^{11} - \frac{310}{937} a^{9} + \frac{136}{937} a^{7} + \frac{33}{937} a^{5} - \frac{204}{937} a^{3}$, $\frac{1}{877969} a^{12} - \frac{310}{877969} a^{10} + \frac{319653}{877969} a^{8} - \frac{163942}{877969} a^{6} - \frac{14259}{877969} a^{4} - \frac{165}{937} a^{2}$, $\frac{1}{877969} a^{13} - \frac{310}{877969} a^{11} + \frac{319653}{877969} a^{9} - \frac{163942}{877969} a^{7} - \frac{14259}{877969} a^{5} - \frac{165}{937} a^{3}$, $\frac{1}{3997317059415230028305919560250760717642666692827} a^{14} + \frac{1400562678636832369388123724650020311124433}{3997317059415230028305919560250760717642666692827} a^{12} - \frac{1490911097313984954750823062341806538466712032}{3997317059415230028305919560250760717642666692827} a^{10} + \frac{69056445634866505621088683252119050200295167283}{173796393887618696882866067836989596419246377949} a^{8} + \frac{1938641133523241141991628654559061354227678375339}{3997317059415230028305919560250760717642666692827} a^{6} + \frac{1445882780330231549329378084070797675862015195}{4266080106099498429355303692903693401966559971} a^{4} + \frac{1339244389896116971707121050562160451496777}{4552913667128600244776204581540761368160683} a^{2} + \frac{8842352765755684922519473700133941503}{66562092178895048972620350309801923483}$, $\frac{1}{51965121772397990367976954283259889329354667006751} a^{15} + \frac{1158407975386356392593595189944023416585114}{3997317059415230028305919560250760717642666692827} a^{13} + \frac{11339199616654410535028521448440058997010092662}{51965121772397990367976954283259889329354667006751} a^{11} + \frac{550276853634792044518807225463486411736793694389}{2259353120539043059477258881880864753450202913337} a^{9} + \frac{14012098572237867428491370949730086217168491000886}{51965121772397990367976954283259889329354667006751} a^{7} + \frac{27435493177937226031548829253316563615957827934}{55459041379293479581618948007748014225565279623} a^{5} + \frac{17733620817742325323761458572566993809206643}{59187877672671803182090659560029897786088879} a^{3} - \frac{124281831592034413022721226919469905463}{865307198325635636644064554027425005279} a$

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:  \( 1703568361330000 \) (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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
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}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
937Data not computed