↑

Properties

Label 16.8.52646317342...8937.1
Degree $16$
Signature $[8, 4]$
Discriminant $73^{15}\cdot 79^{12}$
Root discriminant $1479.40$
Ramified primes $73, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20138270326434543616, 1432339139767432576, 266577755600244992, -386139934252072, -2108604337447076, -45621986975278, -12918699536515, -7995133208, 61421290143, -41499142, 204791405, 2780, -850971, 5886, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 5886*x^13 - 850971*x^12 + 2780*x^11 + 204791405*x^10 - 41499142*x^9 + 61421290143*x^8 - 7995133208*x^7 - 12918699536515*x^6 - 45621986975278*x^5 - 2108604337447076*x^4 - 386139934252072*x^3 + 266577755600244992*x^2 + 1432339139767432576*x + 20138270326434543616)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 5886*x^13 - 850971*x^12 + 2780*x^11 + 204791405*x^10 - 41499142*x^9 + 61421290143*x^8 - 7995133208*x^7 - 12918699536515*x^6 - 45621986975278*x^5 - 2108604337447076*x^4 - 386139934252072*x^3 + 266577755600244992*x^2 + 1432339139767432576*x + 20138270326434543616, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 5886 x^{13} - 850971 x^{12} + 2780 x^{11} + 204791405 x^{10} - 41499142 x^{9} + 61421290143 x^{8} - 7995133208 x^{7} - 12918699536515 x^{6} - 45621986975278 x^{5} - 2108604337447076 x^{4} - 386139934252072 x^{3} + 266577755600244992 x^{2} + 1432339139767432576 x + 20138270326434543616 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(526463173424807410745021702403427128295303807818937=73^{15}\cdot 79^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1479.40$
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:  $73, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{632} a^{8} - \frac{1}{316} a^{7} + \frac{51}{316} a^{6} - \frac{23}{158} a^{5} - \frac{31}{158} a^{4} + \frac{49}{316} a^{3} + \frac{51}{632} a^{2} - \frac{57}{316} a - \frac{37}{79}$, $\frac{1}{632} a^{9} + \frac{19}{632} a^{7} - \frac{125}{632} a^{6} - \frac{71}{632} a^{5} + \frac{87}{632} a^{4} + \frac{21}{79} a^{3} + \frac{67}{632} a^{2} + \frac{133}{316} a + \frac{5}{79}$, $\frac{1}{632} a^{10} - \frac{1}{79} a^{7} + \frac{31}{158} a^{6} + \frac{9}{316} a^{5} + \frac{75}{632} a^{4} + \frac{45}{158} a^{3} + \frac{83}{316} a^{2} - \frac{41}{158} a - \frac{8}{79}$, $\frac{1}{1264} a^{11} - \frac{1}{1264} a^{10} - \frac{1}{1264} a^{9} + \frac{9}{632} a^{7} + \frac{141}{632} a^{6} - \frac{55}{1264} a^{5} - \frac{105}{1264} a^{4} - \frac{425}{1264} a^{3} + \frac{203}{632} a^{2} + \frac{43}{158} a - \frac{28}{79}$, $\frac{1}{5056} a^{12} + \frac{1}{5056} a^{11} - \frac{1}{5056} a^{10} + \frac{1}{2528} a^{9} + \frac{1}{2528} a^{8} - \frac{111}{2528} a^{7} - \frac{111}{5056} a^{6} + \frac{377}{5056} a^{5} - \frac{49}{5056} a^{4} + \frac{71}{1264} a^{3} + \frac{77}{1264} a^{2} + \frac{24}{79} a + \frac{21}{79}$, $\frac{1}{5056} a^{13} - \frac{1}{2528} a^{11} + \frac{3}{5056} a^{10} + \frac{295}{5056} a^{7} - \frac{3}{316} a^{6} - \frac{89}{2528} a^{5} + \frac{997}{5056} a^{4} + \frac{149}{316} a^{3} - \frac{155}{1264} a^{2} - \frac{107}{316} a - \frac{30}{79}$, $\frac{1}{38842892065792} a^{14} + \frac{1414933175}{19421446032896} a^{13} + \frac{3316927015}{38842892065792} a^{12} + \frac{963642599}{9710723016448} a^{11} - \frac{11264540619}{38842892065792} a^{10} + \frac{10418151787}{19421446032896} a^{9} + \frac{28316981873}{38842892065792} a^{8} + \frac{344310760959}{9710723016448} a^{7} - \frac{2548185026465}{38842892065792} a^{6} - \frac{2167215058749}{19421446032896} a^{5} - \frac{6407212799423}{38842892065792} a^{4} + \frac{4811883570685}{9710723016448} a^{3} + \frac{3912433880815}{9710723016448} a^{2} - \frac{1014704695401}{2427680754112} a - \frac{13377491697}{151730047132}$, $\frac{1}{207765513549033874231353056091164786374183475594368035209462774648418854703220814717049947834368} a^{15} + \frac{14162107335138616741832158872348201624333354054468242825303017917522969663952431}{1314971604740720722983247190450410040342933389837772374743435282584929460146967181753480682496} a^{14} - \frac{18284813785685141955522369549875849842725418350837384315430615374660667002785734572919208593}{207765513549033874231353056091164786374183475594368035209462774648418854703220814717049947834368} a^{13} + \frac{2500053525430610848870011868840719180345252580556851712645270642098436181798791395607546647}{25970689193629234278919132011395598296772934449296004401182846831052356837902601839631243479296} a^{12} + \frac{76622368791278742575578759761434043465025245265488504393065287065593824993219000946013466133}{207765513549033874231353056091164786374183475594368035209462774648418854703220814717049947834368} a^{11} + \frac{55392631881984870041246893686951928755232330087807772855147064396633863745085709483093814205}{103882756774516937115676528045582393187091737797184017604731387324209427351610407358524973917184} a^{10} + \frac{43272786958666932648429048736144270318122051325465049182294248042293789509440644633769183945}{207765513549033874231353056091164786374183475594368035209462774648418854703220814717049947834368} a^{9} + \frac{9079479832538739981272928890998412054010742114801449379296255430130230631332675643769867319}{12985344596814617139459566005697799148386467224648002200591423415526178418951300919815621739648} a^{8} + \frac{7174921014988989792267825044096091460615460127153219993127330145187870150481080458010432530047}{207765513549033874231353056091164786374183475594368035209462774648418854703220814717049947834368} a^{7} - \frac{9220463792743420389673191540956076383716377420677830957601198328558989793389946473551636194111}{103882756774516937115676528045582393187091737797184017604731387324209427351610407358524973917184} a^{6} - \frac{28332032254483224743177615707302684990167586400689393798323152420704341451154281744673090702231}{207765513549033874231353056091164786374183475594368035209462774648418854703220814717049947834368} a^{5} - \frac{2641268074527908926262928209073885251964390150978301550036915667868244314523921282611727494515}{25970689193629234278919132011395598296772934449296004401182846831052356837902601839631243479296} a^{4} + \frac{20726683041093718176407995998887521328531729006202774954386101241180507514858778162314699120355}{51941378387258468557838264022791196593545868898592008802365693662104713675805203679262486958592} a^{3} - \frac{401306290429338582529777089334255880152367251385580879339212356086509680355943316775931018307}{6492672298407308569729783002848899574193233612324001100295711707763089209475650459907810869824} a^{2} - \frac{1551403386768532466449919481255965730201789975233456723126996080711631559811335787782053769609}{3246336149203654284864891501424449787096616806162000550147855853881544604737825229953905434912} a - \frac{65181928874363078512174708207420801282895663414962210184167820002789563778934793254740819093}{202896009325228392804055718839028111693538550385125034384240990867596537796114076872119089682}$

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:  $11$
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:  \( 15933210727600000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.8.430297067158108196857.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ $16$ $16$ $16$ $16$

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
73Data not computed
$79$79.8.6.3$x^{8} - 79 x^{4} + 18723$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
79.8.6.3$x^{8} - 79 x^{4} + 18723$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$