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Properties

Label 16.0.24018728224...5473.2
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 97^{15}$
Root discriminant $1878.38$
Ramified primes $41, 97$
Class number $13002309512$ (GRH)
Class group $[13002309512]$ (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13133394052911819, -9598577056412940, 4651932548849904, -1527640572701475, 347355235431558, -53334252425244, 5130957769597, -278639592799, 13045857327, -994326206, 43785164, 2534658, -102364, -5528, 246, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 246*x^14 - 5528*x^13 - 102364*x^12 + 2534658*x^11 + 43785164*x^10 - 994326206*x^9 + 13045857327*x^8 - 278639592799*x^7 + 5130957769597*x^6 - 53334252425244*x^5 + 347355235431558*x^4 - 1527640572701475*x^3 + 4651932548849904*x^2 - 9598577056412940*x + 13133394052911819)
 
gp: K = bnfinit(x^16 - x^15 + 246*x^14 - 5528*x^13 - 102364*x^12 + 2534658*x^11 + 43785164*x^10 - 994326206*x^9 + 13045857327*x^8 - 278639592799*x^7 + 5130957769597*x^6 - 53334252425244*x^5 + 347355235431558*x^4 - 1527640572701475*x^3 + 4651932548849904*x^2 - 9598577056412940*x + 13133394052911819, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 246 x^{14} - 5528 x^{13} - 102364 x^{12} + 2534658 x^{11} + 43785164 x^{10} - 994326206 x^{9} + 13045857327 x^{8} - 278639592799 x^{7} + 5130957769597 x^{6} - 53334252425244 x^{5} + 347355235431558 x^{4} - 1527640572701475 x^{3} + 4651932548849904 x^{2} - 9598577056412940 x + 13133394052911819 \)

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:  \(24018728224219728067156294143658245146337540906275473=41^{14}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1878.38$
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:  $41, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3977=41\cdot 97\)
Dirichlet character group:    $\lbrace$$\chi_{3977}(1,·)$, $\chi_{3977}(2627,·)$, $\chi_{3977}(2500,·)$, $\chi_{3977}(1473,·)$, $\chi_{3977}(1034,·)$, $\chi_{3977}(79,·)$, $\chi_{3977}(2133,·)$, $\chi_{3977}(3927,·)$, $\chi_{3977}(2264,·)$, $\chi_{3977}(729,·)$, $\chi_{3977}(27,·)$, $\chi_{3977}(3868,·)$, $\chi_{3977}(2146,·)$, $\chi_{3977}(3320,·)$, $\chi_{3977}(1913,·)$, $\chi_{3977}(3775,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} - \frac{1}{27} a^{5} - \frac{2}{27} a^{4} + \frac{2}{27} a^{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{7} - \frac{1}{27} a^{6} - \frac{1}{27} a^{5} + \frac{2}{81} a^{4} + \frac{2}{27} a^{3}$, $\frac{1}{243} a^{11} - \frac{1}{243} a^{10} - \frac{4}{81} a^{7} - \frac{2}{81} a^{6} - \frac{34}{243} a^{5} + \frac{16}{243} a^{4} - \frac{4}{27} a^{3} - \frac{1}{27} a^{2} + \frac{1}{3} a$, $\frac{1}{729} a^{12} - \frac{4}{729} a^{10} + \frac{1}{243} a^{9} - \frac{4}{243} a^{8} - \frac{2}{81} a^{7} - \frac{31}{729} a^{6} + \frac{1}{9} a^{5} - \frac{107}{729} a^{4} + \frac{23}{243} a^{3} - \frac{10}{81} a^{2} + \frac{13}{27} a$, $\frac{1}{6561} a^{13} - \frac{1}{6561} a^{11} + \frac{1}{243} a^{10} + \frac{8}{2187} a^{9} + \frac{13}{729} a^{8} - \frac{67}{6561} a^{7} + \frac{22}{729} a^{6} + \frac{736}{6561} a^{5} - \frac{110}{729} a^{4} - \frac{86}{729} a^{3} - \frac{40}{81} a^{2} + \frac{7}{81} a - \frac{1}{3}$, $\frac{1}{5701509} a^{14} + \frac{1}{24057} a^{13} + \frac{2303}{5701509} a^{12} - \frac{601}{1900503} a^{11} - \frac{4585}{1900503} a^{10} - \frac{83}{23463} a^{9} + \frac{60035}{5701509} a^{8} - \frac{30985}{1900503} a^{7} + \frac{27935}{518319} a^{6} - \frac{45146}{1900503} a^{5} - \frac{71941}{633501} a^{4} - \frac{8252}{70389} a^{3} + \frac{6770}{23463} a^{2} + \frac{359}{23463} a - \frac{72}{869}$, $\frac{1}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{15} + \frac{878266025133807292451770380389680959689707825973556841257398372238570566650444354423648}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{14} + \frac{107768592089077586033229462582535033409107766895188206006500785039847318580935482143825355}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{13} - \frac{6125005266638537931464653607160904729081580523207210632499505665255410540333185575506929342}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{12} + \frac{7689261401417087251930424133769744924801958131262812780313748032124472508541810559659050}{36163754052547970918078666005084067260971543102853273979821539309095346511636934803415197761} a^{11} - \frac{3149288419367971808119393542206200637361096094863137259855566042815886090721561008160175174}{4665124272778688248432147914655844676665329060268072343396978570873299700001164589640560511169} a^{10} - \frac{65873057592927794773627533115816506742467200561702284656125645329018363697645903704419584959}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{9} - \frac{12968633977261677102847826727944238839865077891814767998754471166184426691981768669889521089}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{8} - \frac{40362360577065168823974694304186109301863801605942625414451550786162586113891133271200820903}{1272306619848733158663313067633412184545089743709474275471903246601809009091226706265607412137} a^{7} - \frac{666718572679412228258309975735090046782331559663233272225614675734168067422503020938061629254}{13995372818336064745296443743967534029995987180804217030190935712619899100003493768921681533507} a^{6} + \frac{482734194574618823561328945290769963186365057833779175562757680493400286833349073666647351825}{4665124272778688248432147914655844676665329060268072343396978570873299700001164589640560511169} a^{5} - \frac{198068080949238067586104741315723215432618881719433107810597262788458808773556376200258561046}{1555041424259562749477382638218614892221776353422690781132326190291099900000388196546853503723} a^{4} + \frac{70680410772675447386803080417920591558336675608593030631652076849105494972221390075003508225}{518347141419854249825794212739538297407258784474230260377442063430366633333462732182284501241} a^{3} - \frac{36533207317290024610557883602321402743720273299676094856875093844689494532203835965345325779}{172782380473284749941931404246512765802419594824743420125814021143455544444487577394094833747} a^{2} + \frac{14836944940625056876306019512665974993841613945855946654423754279065757282590078757950256374}{57594126824428249980643801415504255267473198274914473375271340381151848148162525798031611249} a + \frac{17783052043817990345439269819430570184254559564223589479786153093416621411811574189809390}{193919618937468855153682833048835876321458580050217082071620674684012956727819952181924617}$

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

Class group and class number

$C_{13002309512}$, which has order $13002309512$ (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:  \( -1 \) (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:  \( 1824756124090 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{16}$ (as 16T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.1534203313.1, 8.8.383800273765009032977233.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ $16$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ $16$

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
41Data not computed
97Data not computed