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Properties

Label 16.0.13279393802...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{11}$
Root discriminant $76.33$
Ramified primes $5, 61$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group 16T1192

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![96701741, -11548887, 200316520, 121080785, 17144000, -13876421, -2801818, 1491835, 577600, -55530, -45528, -289, 2105, 120, -60, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 60*x^14 + 120*x^13 + 2105*x^12 - 289*x^11 - 45528*x^10 - 55530*x^9 + 577600*x^8 + 1491835*x^7 - 2801818*x^6 - 13876421*x^5 + 17144000*x^4 + 121080785*x^3 + 200316520*x^2 - 11548887*x + 96701741)
 
gp: K = bnfinit(x^16 - 3*x^15 - 60*x^14 + 120*x^13 + 2105*x^12 - 289*x^11 - 45528*x^10 - 55530*x^9 + 577600*x^8 + 1491835*x^7 - 2801818*x^6 - 13876421*x^5 + 17144000*x^4 + 121080785*x^3 + 200316520*x^2 - 11548887*x + 96701741, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 60 x^{14} + 120 x^{13} + 2105 x^{12} - 289 x^{11} - 45528 x^{10} - 55530 x^{9} + 577600 x^{8} + 1491835 x^{7} - 2801818 x^{6} - 13876421 x^{5} + 17144000 x^{4} + 121080785 x^{3} + 200316520 x^{2} - 11548887 x + 96701741 \)

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:  \(1327939380231806599761962890625=5^{15}\cdot 61^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.33$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{20801} a^{14} - \frac{9837}{20801} a^{13} + \frac{4330}{20801} a^{12} - \frac{5779}{20801} a^{11} + \frac{308}{1891} a^{10} - \frac{3285}{20801} a^{9} - \frac{2067}{20801} a^{8} - \frac{4563}{20801} a^{7} - \frac{5234}{20801} a^{6} + \frac{820}{20801} a^{5} + \frac{3019}{20801} a^{4} - \frac{9617}{20801} a^{3} + \frac{1429}{20801} a^{2} + \frac{6653}{20801} a + \frac{2}{671}$, $\frac{1}{5017344836747785536139484395096124303049623894696062787671} a^{15} - \frac{28889931165125891643044610109228677912305383231128}{5017344836747785536139484395096124303049623894696062787671} a^{14} - \frac{2369121075574518624418082179593410122388610750085172308596}{5017344836747785536139484395096124303049623894696062787671} a^{13} - \frac{1350650906240675214085038505523674056488125699386446126295}{5017344836747785536139484395096124303049623894696062787671} a^{12} + \frac{20023539146401188220636240976593867247611078809070113203}{82251554700783369444909580247477447590977440896656767011} a^{11} - \frac{2259552368308153272708626136168858410668191363074619193123}{5017344836747785536139484395096124303049623894696062787671} a^{10} + \frac{211959216150335577651079965763917576945696406517239198497}{456122257886162321467225854099647663913602172245096617061} a^{9} + \frac{965702129856545104599129259698354905024678313879237936690}{5017344836747785536139484395096124303049623894696062787671} a^{8} + \frac{2463687999497353067338201433467861874142708850779479623632}{5017344836747785536139484395096124303049623894696062787671} a^{7} + \frac{170723316833675338209534966815463980745403264081345876036}{456122257886162321467225854099647663913602172245096617061} a^{6} + \frac{454960114846402017251209058545499170176573650455007285710}{5017344836747785536139484395096124303049623894696062787671} a^{5} + \frac{1200503774683072118528684862017262787199211809533460797458}{5017344836747785536139484395096124303049623894696062787671} a^{4} + \frac{1142575022663404570709110082406218453338592046571268979499}{5017344836747785536139484395096124303049623894696062787671} a^{3} - \frac{64951546640183980373140464448164251023827900719313459679}{5017344836747785536139484395096124303049623894696062787671} a^{2} - \frac{2341160911276182706899075642939416822153500180395807301666}{5017344836747785536139484395096124303049623894696062787671} a + \frac{8867712194820268324092873071152015565183072700284200127}{161849833443476952778693045003100783969342706280518154441}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  \( -\frac{39595600715157461658635101582938539372503877}{82943658341700179136391932603133099189129356345506981} a^{15} + \frac{3942681868918381073932270223583448670981565029}{82943658341700179136391932603133099189129356345506981} a^{14} - \frac{21252104423240868163413726927395955424118081753}{82943658341700179136391932603133099189129356345506981} a^{13} - \frac{180264075288188327463776942858257507848435472437}{82943658341700179136391932603133099189129356345506981} a^{12} + \frac{1079867616725932396208176622419554575515746592997}{82943658341700179136391932603133099189129356345506981} a^{11} + \frac{5181767405189901586843946040198339034607925954034}{82943658341700179136391932603133099189129356345506981} a^{10} - \frac{22730387505571925801665026965568252329967355404980}{82943658341700179136391932603133099189129356345506981} a^{9} - \frac{115070742981787390520681601214004056849940984057772}{82943658341700179136391932603133099189129356345506981} a^{8} + \frac{287202253179028179612296435355164784646292184058305}{82943658341700179136391932603133099189129356345506981} a^{7} + \frac{1628481722150606980923655459188147622985017688206611}{82943658341700179136391932603133099189129356345506981} a^{6} - \frac{151666316787313905274130786613027241362723688644429}{7540332576518198103308357509375736289920850576864271} a^{5} - \frac{11031612430144019996354686758170116445717958509892392}{82943658341700179136391932603133099189129356345506981} a^{4} + \frac{553442173635154704719676896343131651111397707731060}{82943658341700179136391932603133099189129356345506981} a^{3} + \frac{94543239978853622341137730266572268821350799820010744}{82943658341700179136391932603133099189129356345506981} a^{2} + \frac{48200223293458177016672095493442949545225807178976532}{82943658341700179136391932603133099189129356345506981} a + \frac{3137225269701823679582597988953334324889948578032988}{2675601881990328359238449438810745135133205043403451} \) (order $10$)
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:  \( 74025590.4422 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
61Data not computed