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Properties

Label 16.8.23551829579...0144.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{40}\cdot 41^{6}\cdot 1249^{5}\cdot 620622294839^{4}$
Root discriminant $187{,}608.08$
Ramified primes $2, 41, 1249, 620622294839$
Class number Not computed
Class group Not computed
Galois group 16T1262

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![485919715103617715396797698138138113212445995123402968178129, 0, -462734523383447162851140406549194225919814502900297596, 0, 12233549536667813425509652749327206454640589402, 0, 3320365750251247350251659225492284179292, 0, -25860784484023364640720717188425, 0, -1564271640991422128285156, 0, -1080028460485598, 0, 188252080, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 188252080*x^14 - 1080028460485598*x^12 - 1564271640991422128285156*x^10 - 25860784484023364640720717188425*x^8 + 3320365750251247350251659225492284179292*x^6 + 12233549536667813425509652749327206454640589402*x^4 - 462734523383447162851140406549194225919814502900297596*x^2 + 485919715103617715396797698138138113212445995123402968178129)
 
gp: K = bnfinit(x^16 + 188252080*x^14 - 1080028460485598*x^12 - 1564271640991422128285156*x^10 - 25860784484023364640720717188425*x^8 + 3320365750251247350251659225492284179292*x^6 + 12233549536667813425509652749327206454640589402*x^4 - 462734523383447162851140406549194225919814502900297596*x^2 + 485919715103617715396797698138138113212445995123402968178129, 1)
 

Normalized defining polynomial

\( x^{16} + 188252080 x^{14} - 1080028460485598 x^{12} - 1564271640991422128285156 x^{10} - 25860784484023364640720717188425 x^{8} + 3320365750251247350251659225492284179292 x^{6} + 12233549536667813425509652749327206454640589402 x^{4} - 462734523383447162851140406549194225919814502900297596 x^{2} + 485919715103617715396797698138138113212445995123402968178129 \)

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:  \(2355182957996385665344213110300455032987816125842857635730792102196757438299209990144=2^{40}\cdot 41^{6}\cdot 1249^{5}\cdot 620622294839^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $187{,}608.08$
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:  $2, 41, 1249, 620622294839$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{2482489179356} a^{10} + \frac{47063020}{620622294839} a^{8} + \frac{474954829101}{2482489179356} a^{6} + \frac{144699564667}{620622294839} a^{4} + \frac{220010586025}{2482489179356} a^{2} - \frac{1}{2}$, $\frac{1}{2482489179356} a^{11} + \frac{47063020}{620622294839} a^{9} + \frac{474954829101}{2482489179356} a^{7} + \frac{144699564667}{620622294839} a^{5} + \frac{220010586025}{2482489179356} a^{3} - \frac{1}{2} a$, $\frac{1}{78897098521113861267413913956} a^{12} + \frac{47063020}{19724274630278465316853478489} a^{10} - \frac{540014230242799}{39448549260556930633706956978} a^{8} + \frac{19723492494457969605789335911}{39448549260556930633706956978} a^{6} - \frac{1130221864148305162903444673}{39448549260556930633706956978} a^{4} + \frac{75043506273947}{775157246253911} a^{2} + \frac{1}{4}$, $\frac{1}{78897098521113861267413913956} a^{13} + \frac{47063020}{19724274630278465316853478489} a^{11} - \frac{540014230242799}{39448549260556930633706956978} a^{9} + \frac{19723492494457969605789335911}{39448549260556930633706956978} a^{7} - \frac{1130221864148305162903444673}{39448549260556930633706956978} a^{5} + \frac{75043506273947}{775157246253911} a^{3} + \frac{1}{4} a$, $\frac{1}{131977244195710941458671687175931600874090956610204547949124508352644732493043550309651063776628292930453176080565585252805950779890873639270273118909510080419592980694884} a^{14} - \frac{177887053068678599168557008321994238236167340089402029375543581354709723537939597638123908135462404618283908852868691864666694705908730966}{38499779520335747216648683540236756381006696794108677931483228807655989642078048515067404835655861414951334912650404099418305361695120664897979322902424177485295501953} a^{12} - \frac{907103380690253065873684604998787506046004379747553102609010866437875958288058483326438104391225014633795047548193123478004943339278788799354751406594064082}{4713473006989676480666845970568985745503248450364448141040161012594454731894412511058966563451010461801899145734485187600212527853245487116795468532482502872128320739103} a^{10} - \frac{1267637896394591465808450311866207575800952644017250603947414812744481723890525039159961450886566821093290691953404118823278130396611116933058501671265917685454393671119}{18853892027958705922667383882275942982012993801457792564160644050377818927577650044235866253804041847207596582937940750400850111412981948467181874129930011488513282956412} a^{8} - \frac{6828712409478107100392008040505861881299228629373075482223507319675934148785700792880550223150992808804974251398501653658116852803139510863747241314142859803135545575531}{65988622097855470729335843587965800437045478305102273974562254176322366246521775154825531888314146465226588040282792626402975389945436819635136559454755040209796490347442} a^{6} + \frac{57928352791175021810954173948719743997134351218868772205666391326434823346600686899620155638994549707758994181559788101722702069295308361755626781056212382227}{212653082709424880997691346258451620114133456079885164902085929577957825629024107546642269903690292635953906868515196120430861020357385426425604276050109300156} a^{4} - \frac{2308710963363751207528755124453774514799558573954820272548530758727620580039065803526400415618042858824013651033688230832626347706933130900419}{6691107615846844455355475478357747508279598321538779671785281560952024421789591231681041178232029290458373513740342117576369465013230391895556} a^{2} + \frac{2279025166053824899808750829067343708751301888570135300877778021954849806652330328286422627825744537789118013704443108668533087}{8631935840350902162883397156914225959219570133525371701952355750458404178853470161877552410778261731278784045952935311507639196}$, $\frac{1}{131977244195710941458671687175931600874090956610204547949124508352644732493043550309651063776628292930453176080565585252805950779890873639270273118909510080419592980694884} a^{15} - \frac{177887053068678599168557008321994238236167340089402029375543581354709723537939597638123908135462404618283908852868691864666694705908730966}{38499779520335747216648683540236756381006696794108677931483228807655989642078048515067404835655861414951334912650404099418305361695120664897979322902424177485295501953} a^{13} - \frac{907103380690253065873684604998787506046004379747553102609010866437875958288058483326438104391225014633795047548193123478004943339278788799354751406594064082}{4713473006989676480666845970568985745503248450364448141040161012594454731894412511058966563451010461801899145734485187600212527853245487116795468532482502872128320739103} a^{11} - \frac{1267637896394591465808450311866207575800952644017250603947414812744481723890525039159961450886566821093290691953404118823278130396611116933058501671265917685454393671119}{18853892027958705922667383882275942982012993801457792564160644050377818927577650044235866253804041847207596582937940750400850111412981948467181874129930011488513282956412} a^{9} - \frac{6828712409478107100392008040505861881299228629373075482223507319675934148785700792880550223150992808804974251398501653658116852803139510863747241314142859803135545575531}{65988622097855470729335843587965800437045478305102273974562254176322366246521775154825531888314146465226588040282792626402975389945436819635136559454755040209796490347442} a^{7} + \frac{57928352791175021810954173948719743997134351218868772205666391326434823346600686899620155638994549707758994181559788101722702069295308361755626781056212382227}{212653082709424880997691346258451620114133456079885164902085929577957825629024107546642269903690292635953906868515196120430861020357385426425604276050109300156} a^{5} - \frac{2308710963363751207528755124453774514799558573954820272548530758727620580039065803526400415618042858824013651033688230832626347706933130900419}{6691107615846844455355475478357747508279598321538779671785281560952024421789591231681041178232029290458373513740342117576369465013230391895556} a^{3} + \frac{2279025166053824899808750829067343708751301888570135300877778021954849806652330328286422627825744537789118013704443108668533087}{8631935840350902162883397156914225959219570133525371701952355750458404178853470161877552410778261731278784045952935311507639196} a$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1262:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1262
Character table for t16n1262 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.8.8599834624.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.20.7$x^{8} + 72 x^{4} + 144$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.20.7$x^{8} + 72 x^{4} + 144$$4$$2$$20$$Q_8:C_2$$[2, 3, 7/2]^{2}$
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.8.0.1$x^{8} - x + 12$$1$$8$$0$$C_8$$[\ ]^{8}$
1249Data not computed
620622294839Data not computed