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Properties

Label 16.12.1732855303...8849.1
Degree $16$
Signature $[12, 2]$
Discriminant $61^{12}\cdot 97^{14}$
Root discriminant $1195.15$
Ramified primes $61, 97$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2927618217243747, -1620760775169645, 203116946704020, 96649385827419, -29332809376400, -664207203366, 764166087629, -29475464342, -5522964746, 489716073, -21896116, -2827226, 382131, 7176, -1200, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 1200*x^14 + 7176*x^13 + 382131*x^12 - 2827226*x^11 - 21896116*x^10 + 489716073*x^9 - 5522964746*x^8 - 29475464342*x^7 + 764166087629*x^6 - 664207203366*x^5 - 29332809376400*x^4 + 96649385827419*x^3 + 203116946704020*x^2 - 1620760775169645*x + 2927618217243747)
 
gp: K = bnfinit(x^16 - 4*x^15 - 1200*x^14 + 7176*x^13 + 382131*x^12 - 2827226*x^11 - 21896116*x^10 + 489716073*x^9 - 5522964746*x^8 - 29475464342*x^7 + 764166087629*x^6 - 664207203366*x^5 - 29332809376400*x^4 + 96649385827419*x^3 + 203116946704020*x^2 - 1620760775169645*x + 2927618217243747, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 1200 x^{14} + 7176 x^{13} + 382131 x^{12} - 2827226 x^{11} - 21896116 x^{10} + 489716073 x^{9} - 5522964746 x^{8} - 29475464342 x^{7} + 764166087629 x^{6} - 664207203366 x^{5} - 29332809376400 x^{4} + 96649385827419 x^{3} + 203116946704020 x^{2} - 1620760775169645 x + 2927618217243747 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17328553034618181867273005889005225191371184588849=61^{12}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1195.15$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{12} + \frac{1}{9} a^{10} - \frac{7}{18} a^{9} + \frac{4}{9} a^{7} + \frac{7}{18} a^{6} - \frac{2}{9} a^{5} - \frac{2}{9} a^{4} - \frac{5}{18} a^{3} + \frac{2}{9} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{13} + \frac{1}{9} a^{11} - \frac{1}{18} a^{10} - \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{1}{18} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{5}{18} a^{4} - \frac{1}{9} a^{3} - \frac{1}{6} a$, $\frac{1}{112698} a^{14} - \frac{2597}{112698} a^{13} + \frac{2273}{112698} a^{12} + \frac{9901}{112698} a^{11} + \frac{2039}{112698} a^{10} + \frac{35573}{112698} a^{9} - \frac{8405}{37566} a^{8} - \frac{1805}{37566} a^{7} + \frac{31717}{112698} a^{6} + \frac{11335}{37566} a^{5} + \frac{15569}{112698} a^{4} + \frac{33319}{112698} a^{3} + \frac{3481}{37566} a^{2} + \frac{665}{4174} a - \frac{105}{4174}$, $\frac{1}{597789376114090996835891304267069761836063093205192746982516783478034870688268663309540403692621855138} a^{15} + \frac{178129132973222439917093600711332776476235285820519072204450127358592652541449718881693875468582}{298894688057045498417945652133534880918031546602596373491258391739017435344134331654770201846310927569} a^{14} - \frac{2314763836743976286282335495429255030170811584595859305148115965124616746829853359291072985383033753}{99631562685681832805981884044511626972677182200865457830419463913005811781378110551590067282103642523} a^{13} + \frac{1134271216186677377410408612078894875128166544996791145904323751066604061564262147627321317752448345}{199263125371363665611963768089023253945354364401730915660838927826011623562756221103180134564207285046} a^{12} + \frac{11717735163544494007289019736900212362021064364343706974929311926355114690424960210735980079444423105}{99631562685681832805981884044511626972677182200865457830419463913005811781378110551590067282103642523} a^{11} - \frac{37221579214919868129521343329241476431903915898231490694521382438727925500935630273951603076735980549}{298894688057045498417945652133534880918031546602596373491258391739017435344134331654770201846310927569} a^{10} - \frac{179900450686269476024179658152922244212730674060824495688396193587774077340417265487687373650565751403}{597789376114090996835891304267069761836063093205192746982516783478034870688268663309540403692621855138} a^{9} + \frac{819010788430764371819043931616606330375779580647183733584338908749619832323943081025060775457808865}{11070173631742425867331320449390180774741909133429495314491051545889534642375345616843340809122626947} a^{8} + \frac{850785075317804458192186195510346878334518634087845610081200769792517520653518190039613339629040685}{298894688057045498417945652133534880918031546602596373491258391739017435344134331654770201846310927569} a^{7} + \frac{57246752458640764958122750080612095447429072099737778128744097150563213155059055578077836491405573665}{597789376114090996835891304267069761836063093205192746982516783478034870688268663309540403692621855138} a^{6} + \frac{96308121534186418350349547907645059902668300512300282615782866098604045124041341841745512826275335326}{298894688057045498417945652133534880918031546602596373491258391739017435344134331654770201846310927569} a^{5} + \frac{27804392296348981571904107460298115701054533915322542131698330205179752738417662427559791350685502510}{99631562685681832805981884044511626972677182200865457830419463913005811781378110551590067282103642523} a^{4} - \frac{82087631741564575802244283197015566509672030354877145254110933824123930526628412700595795252991233735}{597789376114090996835891304267069761836063093205192746982516783478034870688268663309540403692621855138} a^{3} + \frac{20791407169203532397595422398026142067191494474984025920820941029945404591102794030460155649526762140}{99631562685681832805981884044511626972677182200865457830419463913005811781378110551590067282103642523} a^{2} - \frac{9324829065493705724354992035005333313173390753638340730147209738633274533580629512910701545085868798}{33210520895227277601993961348170542324225727400288485943473154637668603927126036850530022427367880841} a + \frac{4556975907850020879852498697291943186232662423066137312162662946582325080866438036778437812889511599}{11070173631742425867331320449390180774741909133429495314491051545889534642375345616843340809122626947}$

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

Class group and class number

$C_{4}$, 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:  $13$
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:  \( 3765433852790000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.1118720199956720578033.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
97Data not computed