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Properties

Label 16.0.58013247259...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{8}\cdot 41^{6}$
Root discriminant $72.48$
Ramified primes $5, 29, 41$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T516)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4086896, -6447144, 4355720, -473844, 1941235, 366285, 181877, 216989, 11192, 7257, 3193, 23, 248, 15, 29, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 29*x^14 + 15*x^13 + 248*x^12 + 23*x^11 + 3193*x^10 + 7257*x^9 + 11192*x^8 + 216989*x^7 + 181877*x^6 + 366285*x^5 + 1941235*x^4 - 473844*x^3 + 4355720*x^2 - 6447144*x + 4086896)
 
gp: K = bnfinit(x^16 - x^15 + 29*x^14 + 15*x^13 + 248*x^12 + 23*x^11 + 3193*x^10 + 7257*x^9 + 11192*x^8 + 216989*x^7 + 181877*x^6 + 366285*x^5 + 1941235*x^4 - 473844*x^3 + 4355720*x^2 - 6447144*x + 4086896, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 29 x^{14} + 15 x^{13} + 248 x^{12} + 23 x^{11} + 3193 x^{10} + 7257 x^{9} + 11192 x^{8} + 216989 x^{7} + 181877 x^{6} + 366285 x^{5} + 1941235 x^{4} - 473844 x^{3} + 4355720 x^{2} - 6447144 x + 4086896 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(580132472595479362207275390625=5^{12}\cdot 29^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.48$
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:  $5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{60} a^{12} + \frac{7}{60} a^{11} + \frac{1}{10} a^{10} + \frac{1}{5} a^{9} - \frac{7}{60} a^{8} + \frac{7}{30} a^{7} - \frac{2}{5} a^{6} - \frac{9}{20} a^{5} + \frac{17}{60} a^{4} + \frac{4}{15} a^{3} - \frac{7}{15} a^{2} + \frac{2}{15} a - \frac{1}{15}$, $\frac{1}{9480} a^{13} - \frac{7}{2370} a^{12} + \frac{107}{1185} a^{11} + \frac{379}{3160} a^{10} + \frac{473}{9480} a^{9} - \frac{41}{9480} a^{8} - \frac{2119}{9480} a^{7} + \frac{1351}{3160} a^{6} + \frac{1231}{4740} a^{5} + \frac{163}{790} a^{4} - \frac{1301}{3160} a^{3} + \frac{1319}{4740} a^{2} + \frac{1159}{2370} a - \frac{109}{237}$, $\frac{1}{104280} a^{14} + \frac{1}{52140} a^{13} + \frac{83}{26070} a^{12} + \frac{7699}{104280} a^{11} + \frac{10409}{104280} a^{10} + \frac{22681}{104280} a^{9} + \frac{1649}{9480} a^{8} + \frac{13637}{104280} a^{7} + \frac{7787}{26070} a^{6} - \frac{713}{8690} a^{5} + \frac{48299}{104280} a^{4} - \frac{20333}{52140} a^{3} - \frac{471}{8690} a^{2} + \frac{11797}{26070} a - \frac{208}{1185}$, $\frac{1}{998380320672753046525392677529396480605400} a^{15} + \frac{1383170043740083228405223568341478361}{998380320672753046525392677529396480605400} a^{14} - \frac{50780627966548073631870414317071654079}{998380320672753046525392677529396480605400} a^{13} + \frac{912935649210030645813733682617773511787}{998380320672753046525392677529396480605400} a^{12} + \frac{423991361575944662968306553986137331013}{3781743638911943358050729839126501820475} a^{11} - \frac{519442005910310207757400194388591838713}{998380320672753046525392677529396480605400} a^{10} - \frac{162675411757948902363424118983596013571353}{998380320672753046525392677529396480605400} a^{9} + \frac{161488953337283925994749713966723642786371}{998380320672753046525392677529396480605400} a^{8} - \frac{17817338789552485147416784762234016522789}{249595080168188261631348169382349120151350} a^{7} - \frac{111004635042013876341101614858378780035983}{998380320672753046525392677529396480605400} a^{6} - \frac{182381142499905367900933026095282650066619}{998380320672753046525392677529396480605400} a^{5} + \frac{319506588559642170018402584638240512797957}{998380320672753046525392677529396480605400} a^{4} + \frac{66664139485639217535725783138457676947149}{998380320672753046525392677529396480605400} a^{3} - \frac{15509877558109782564419144262251419446562}{124797540084094130815674084691174560075675} a^{2} + \frac{3032156888596994151210443124206250159869}{6568291583373375306088109720588134740825} a - \frac{4383517356103284214227258409672820251021}{11345230916735830074152189517379505461425}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  $7$
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:  \( 33646217.6607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T516):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.29725.2, 4.0.3625.1, 4.4.5125.1, 8.0.22089390625.5

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$