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Properties

Label 16.0.55346356857...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $83.45$
Ramified primes $5, 41, 661$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![148680785341, -108971734827, 53368743541, -14727902028, 4754727171, -853453216, 221564424, -33125809, 6994857, -605119, 9024, 19974, -5229, 472, -29, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 29*x^14 + 472*x^13 - 5229*x^12 + 19974*x^11 + 9024*x^10 - 605119*x^9 + 6994857*x^8 - 33125809*x^7 + 221564424*x^6 - 853453216*x^5 + 4754727171*x^4 - 14727902028*x^3 + 53368743541*x^2 - 108971734827*x + 148680785341)
 
gp: K = bnfinit(x^16 - 7*x^15 - 29*x^14 + 472*x^13 - 5229*x^12 + 19974*x^11 + 9024*x^10 - 605119*x^9 + 6994857*x^8 - 33125809*x^7 + 221564424*x^6 - 853453216*x^5 + 4754727171*x^4 - 14727902028*x^3 + 53368743541*x^2 - 108971734827*x + 148680785341, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 29 x^{14} + 472 x^{13} - 5229 x^{12} + 19974 x^{11} + 9024 x^{10} - 605119 x^{9} + 6994857 x^{8} - 33125809 x^{7} + 221564424 x^{6} - 853453216 x^{5} + 4754727171 x^{4} - 14727902028 x^{3} + 53368743541 x^{2} - 108971734827 x + 148680785341 \)

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:  \(5534635685714755139654541015625=5^{14}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.45$
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:  $5, 41, 661$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{2}{9} a^{12} - \frac{2}{9} a^{11} + \frac{1}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{15} - \frac{11363793710722596677869854867695366283548629914637997789612640900230070883460}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{14} + \frac{4011028697087775551514302012294429789882168447832464091089328410432779325720}{9750977699141157092075999477650162789771129501993813449727824457156637250429} a^{13} + \frac{158751161504190084277789774923700541761444001104237279166572401451951933069588}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{12} + \frac{2519274465376043947455520938153012832666840716137016706836123205034811372916}{133263361888262480258371992861218891460205436527248783812946934247807375755863} a^{11} + \frac{11464999222425269221794739253254265025350149004161397779938141991436361740701}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{10} + \frac{177899291953901596319034124479707852686697108367951380523828864060062873474411}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{9} - \frac{71467755293244179302544055265941538005042770068397014202247052590399090744710}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{8} - \frac{149953169059858834487745101628038662118389473979898784976859776618290531149786}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{7} + \frac{36793184559238643341402646504735269458224504841687922171103155353422169550208}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{6} - \frac{190691844958291299210820967073512799655859749645375068150859602804618626250649}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{5} - \frac{31686717655045838285572109513364121506264229530719410585428293573780827145086}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{4} - \frac{47603646844588118862171528893338394870321223463070936023945723728215995793942}{133263361888262480258371992861218891460205436527248783812946934247807375755863} a^{3} + \frac{66055339106114973448551618493412167768476539192490093659364387692019899777150}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a^{2} + \frac{82478524113340758515068744552924821870002579183121403928954240535278696600693}{399790085664787440775115978583656674380616309581746351438840802743422127267589} a - \frac{11390258500895080011531370085235514997480104314758813653315489447869247724392}{133263361888262480258371992861218891460205436527248783812946934247807375755863}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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{2250889781110013764222656141345672455277888944659175054578}{67397511668924398101388881121750429351823283817859238559597520971029} a^{15} - \frac{19954609222701397962982299485802837814167552981846443979739}{67397511668924398101388881121750429351823283817859238559597520971029} a^{14} - \frac{131013364934774015317273286271494884969089811225234397497396}{67397511668924398101388881121750429351823283817859238559597520971029} a^{13} + \frac{1228260538311153530924288969208014937511784104400738655427778}{67397511668924398101388881121750429351823283817859238559597520971029} a^{12} - \frac{10381896740908100067967098014272016038825342680061192908993731}{67397511668924398101388881121750429351823283817859238559597520971029} a^{11} + \frac{35286342163775927072506894789111773440095363795899019510553167}{67397511668924398101388881121750429351823283817859238559597520971029} a^{10} + \frac{68390757418527709958263342138879529315855937910688205849816688}{22465837222974799367129627040583476450607761272619746186532506990343} a^{9} - \frac{128490564095991945980124429569753105609227229294820019291893191}{7488612407658266455709875680194492150202587090873248728844168996781} a^{8} + \frac{16339881291443993093720327856616486079822847535705512765088126607}{67397511668924398101388881121750429351823283817859238559597520971029} a^{7} - \frac{14400959742211201502500402098551251609570366634062900014044241926}{22465837222974799367129627040583476450607761272619746186532506990343} a^{6} + \frac{435717313135563583509780429372651352108466534499023828494550321818}{67397511668924398101388881121750429351823283817859238559597520971029} a^{5} - \frac{1228606467519027133935041783800352498648384934373610580863213650046}{67397511668924398101388881121750429351823283817859238559597520971029} a^{4} + \frac{8318320392374347575023673985443335224951133576243634887325205484846}{67397511668924398101388881121750429351823283817859238559597520971029} a^{3} - \frac{16830130506611142065502168441030080947797069716839868606661855118237}{67397511668924398101388881121750429351823283817859238559597520971029} a^{2} + \frac{58244641403129353830256978534309957907663157735232791066314732261020}{67397511668924398101388881121750429351823283817859238559597520971029} a - \frac{62878997405927176742984019195408344090674700914530856593353977112590}{67397511668924398101388881121750429351823283817859238559597520971029} \) (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:  \( 292691113.931 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T852:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 41 conjugacy class representatives for t16n852
Character table for t16n852 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.26265625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.2$x^{4} - 1476$$4$$1$$3$$C_4$$[\ ]_{4}$
661Data not computed