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Properties

Label 16.0.22138542742...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{6}\cdot 661^{4}$
Root discriminant $68.25$
Ramified primes $5, 41, 661$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T852

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5151426151, -981739260, -598932388, 213506931, -194441392, 6108554, 18725575, -6434275, 4146511, -526129, 244525, -20077, 8025, -455, 138, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 138*x^14 - 455*x^13 + 8025*x^12 - 20077*x^11 + 244525*x^10 - 526129*x^9 + 4146511*x^8 - 6434275*x^7 + 18725575*x^6 + 6108554*x^5 - 194441392*x^4 + 213506931*x^3 - 598932388*x^2 - 981739260*x + 5151426151)
 
gp: K = bnfinit(x^16 - 4*x^15 + 138*x^14 - 455*x^13 + 8025*x^12 - 20077*x^11 + 244525*x^10 - 526129*x^9 + 4146511*x^8 - 6434275*x^7 + 18725575*x^6 + 6108554*x^5 - 194441392*x^4 + 213506931*x^3 - 598932388*x^2 - 981739260*x + 5151426151, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 138 x^{14} - 455 x^{13} + 8025 x^{12} - 20077 x^{11} + 244525 x^{10} - 526129 x^{9} + 4146511 x^{8} - 6434275 x^{7} + 18725575 x^{6} + 6108554 x^{5} - 194441392 x^{4} + 213506931 x^{3} - 598932388 x^{2} - 981739260 x + 5151426151 \)

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:  \(221385427428590205586181640625=5^{12}\cdot 41^{6}\cdot 661^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.25$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{15} - \frac{8537175899331484733650182272077971605790720914232535174759488975579540248}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{14} - \frac{8478626562391760793078561641359476344751505246287302918640372063879409867}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{13} - \frac{5603222065847530202919978364290001580454511247384048180917927232772902119}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{12} + \frac{5083563934037732268604336992983119818654647160274822739428963454739105723}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{11} - \frac{3429270786024086061318650461974447999750910270855724503617462065742679723}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{10} - \frac{8673314672444624113616501766707699941766263253678833580176044395669219399}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{9} - \frac{5277383801489277881765556491019029122829368975611639785679173213737256093}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{8} + \frac{123468951289981363831231362605201978236853344277396207414443928788200522}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{7} - \frac{8690638739719326399738512795735064392739137841215569931348379078728926670}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{6} + \frac{892839951773170801113220581374206624709423108715509927089766474263698248}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{5} - \frac{3081914938676616321554464865638236967911745638524932625680357610290703669}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{4} - \frac{7777590330694532129712016118310405458348238460587590871653394002390273647}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{3} + \frac{7493175601345778612704505546229577555386454597179551838100826932709456308}{18991848912087251965578406907999675888646741473396615628985950834544349159} a^{2} - \frac{6236851601041792857846768387776868352574712520461738668812196032276477808}{18991848912087251965578406907999675888646741473396615628985950834544349159} a + \frac{6871503572563978828539484216563191575617265735187380032607143612644666715}{18991848912087251965578406907999675888646741473396615628985950834544349159}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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{7323312871579862676935145998896313904027880210501287098}{27265849509921315684839317953762048777983186397651760606368763109} a^{15} + \frac{9413350884512015538006618639208157198895677516717038107}{27265849509921315684839317953762048777983186397651760606368763109} a^{14} + \frac{979448895588500962120667920083723492582099159530624596803}{27265849509921315684839317953762048777983186397651760606368763109} a^{13} + \frac{1471275141926567473117628784801453910816735969165148170399}{27265849509921315684839317953762048777983186397651760606368763109} a^{12} + \frac{58107956052518980721784766416602597399617684346859899112208}{27265849509921315684839317953762048777983186397651760606368763109} a^{11} + \frac{103847929710261619322170368813163121862874582640959223067933}{27265849509921315684839317953762048777983186397651760606368763109} a^{10} + \frac{1987674001269368120407327430670866942461200788380006188062487}{27265849509921315684839317953762048777983186397651760606368763109} a^{9} + \frac{3023047894372248797614730439156621886138770623144770641832017}{27265849509921315684839317953762048777983186397651760606368763109} a^{8} + \frac{39034855873240209279395117393770825707474735567613651025664331}{27265849509921315684839317953762048777983186397651760606368763109} a^{7} + \frac{47070860554042187420594164376683635194127192508633136730286715}{27265849509921315684839317953762048777983186397651760606368763109} a^{6} + \frac{371077374412285243485294373589181877764615297965357149168675778}{27265849509921315684839317953762048777983186397651760606368763109} a^{5} + \frac{12821604412906339288910613143049329493770094512357509054649603}{27265849509921315684839317953762048777983186397651760606368763109} a^{4} + \frac{779644403653929655632465452464342039738277820395392349537590782}{27265849509921315684839317953762048777983186397651760606368763109} a^{3} - \frac{4241169336494087876824081629517940500353618539079470809896357435}{27265849509921315684839317953762048777983186397651760606368763109} a^{2} - \frac{9776012755824782479569710944689773207031142436455767001629740562}{27265849509921315684839317953762048777983186397651760606368763109} a - \frac{828338776000796845672841946410000217406442650062762205886977543}{27265849509921315684839317953762048777983186397651760606368763109} \) (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:  \( 98292125.5155 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
41Data not computed
661Data not computed