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Properties

Label 19.19.1584354688...3641.1
Degree $19$
Signature $[19, 0]$
Discriminant $419^{18}$
Root discriminant $304.93$
Ramified prime $419$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{19}$ (as 19T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41768519, -92941225, 129492371, 325674937, -65770266, -328284116, -3418584, 153105664, 7325624, -37345852, -1201976, 4834266, 70692, -335304, -1727, 12055, 37, -198, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 198*x^17 + 37*x^16 + 12055*x^15 - 1727*x^14 - 335304*x^13 + 70692*x^12 + 4834266*x^11 - 1201976*x^10 - 37345852*x^9 + 7325624*x^8 + 153105664*x^7 - 3418584*x^6 - 328284116*x^5 - 65770266*x^4 + 325674937*x^3 + 129492371*x^2 - 92941225*x - 41768519)
 
gp: K = bnfinit(x^19 - x^18 - 198*x^17 + 37*x^16 + 12055*x^15 - 1727*x^14 - 335304*x^13 + 70692*x^12 + 4834266*x^11 - 1201976*x^10 - 37345852*x^9 + 7325624*x^8 + 153105664*x^7 - 3418584*x^6 - 328284116*x^5 - 65770266*x^4 + 325674937*x^3 + 129492371*x^2 - 92941225*x - 41768519, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 198 x^{17} + 37 x^{16} + 12055 x^{15} - 1727 x^{14} - 335304 x^{13} + 70692 x^{12} + 4834266 x^{11} - 1201976 x^{10} - 37345852 x^{9} + 7325624 x^{8} + 153105664 x^{7} - 3418584 x^{6} - 328284116 x^{5} - 65770266 x^{4} + 325674937 x^{3} + 129492371 x^{2} - 92941225 x - 41768519 \)

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

Invariants

Degree:  $19$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[19, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(158435468857090504879482314342339950574787173641=419^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $304.93$
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:  $419$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(419\)
Dirichlet character group:    $\lbrace$$\chi_{419}(1,·)$, $\chi_{419}(343,·)$, $\chi_{419}(7,·)$, $\chi_{419}(136,·)$, $\chi_{419}(329,·)$, $\chi_{419}(330,·)$, $\chi_{419}(139,·)$, $\chi_{419}(208,·)$, $\chi_{419}(215,·)$, $\chi_{419}(135,·)$, $\chi_{419}(199,·)$, $\chi_{419}(107,·)$, $\chi_{419}(306,·)$, $\chi_{419}(47,·)$, $\chi_{419}(49,·)$, $\chi_{419}(114,·)$, $\chi_{419}(248,·)$, $\chi_{419}(379,·)$, $\chi_{419}(60,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{2}{13} a^{9} + \frac{3}{13} a^{6} - \frac{6}{13} a^{5} - \frac{4}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{13} a^{11} - \frac{4}{13} a^{9} + \frac{3}{13} a^{7} + \frac{1}{13} a^{5} - \frac{4}{13} a^{3} + \frac{3}{13} a$, $\frac{1}{13} a^{12} + \frac{5}{13} a^{9} + \frac{3}{13} a^{8} + \frac{2}{13} a^{5} - \frac{4}{13} a^{4} + \frac{6}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{169} a^{15} + \frac{4}{169} a^{14} - \frac{5}{169} a^{13} + \frac{5}{169} a^{12} - \frac{3}{169} a^{11} - \frac{80}{169} a^{9} + \frac{28}{169} a^{8} - \frac{35}{169} a^{7} - \frac{6}{13} a^{6} - \frac{19}{169} a^{5} - \frac{20}{169} a^{4} + \frac{24}{169} a^{3} - \frac{82}{169} a^{2} - \frac{6}{13} a$, $\frac{1}{11999} a^{16} + \frac{5}{11999} a^{15} + \frac{298}{11999} a^{14} - \frac{1}{923} a^{13} + \frac{288}{11999} a^{12} - \frac{367}{11999} a^{11} + \frac{128}{11999} a^{10} + \frac{420}{923} a^{9} + \frac{513}{11999} a^{8} - \frac{1205}{11999} a^{7} - \frac{2684}{11999} a^{6} + \frac{21}{923} a^{5} + \frac{3930}{11999} a^{4} - \frac{1137}{11999} a^{3} + \frac{2258}{11999} a^{2} + \frac{327}{923} a$, $\frac{1}{224417297} a^{17} - \frac{1779}{224417297} a^{16} + \frac{611350}{224417297} a^{15} + \frac{206542}{224417297} a^{14} + \frac{3145563}{224417297} a^{13} + \frac{450802}{224417297} a^{12} - \frac{7192561}{224417297} a^{11} - \frac{869915}{224417297} a^{10} + \frac{1523810}{224417297} a^{9} + \frac{515548}{3160807} a^{8} + \frac{59005469}{224417297} a^{7} + \frac{94517974}{224417297} a^{6} + \frac{30066210}{224417297} a^{5} - \frac{45897182}{224417297} a^{4} + \frac{20677680}{224417297} a^{3} - \frac{13335032}{224417297} a^{2} - \frac{6185623}{17262869} a - \frac{97}{317}$, $\frac{1}{1042834163522617761547794080884145199946303936259041} a^{18} + \frac{1900600924393924061442167165129885490020786}{1042834163522617761547794080884145199946303936259041} a^{17} + \frac{255625872441637489603664284456736166549130931}{14687805120036869880954846209635847886567661074071} a^{16} - \frac{2815499194635496076799823106420017205405060187373}{1042834163522617761547794080884145199946303936259041} a^{15} + \frac{36719646538407826183041760452195321370787473335730}{1042834163522617761547794080884145199946303936259041} a^{14} + \frac{27560233653924477161702260452184612686419923236545}{1042834163522617761547794080884145199946303936259041} a^{13} - \frac{20772073013087303763363386354774336802539010685049}{1042834163522617761547794080884145199946303936259041} a^{12} + \frac{35792355767131171182619213946124709172697392558625}{1042834163522617761547794080884145199946303936259041} a^{11} - \frac{15535502398592942626711578349519450526724901708270}{1042834163522617761547794080884145199946303936259041} a^{10} - \frac{309810982968240948803594848572405343222185193622010}{1042834163522617761547794080884145199946303936259041} a^{9} + \frac{38870511466818832626736915388436547852333041711419}{1042834163522617761547794080884145199946303936259041} a^{8} - \frac{385459208844720805062275446345449050191692390463149}{1042834163522617761547794080884145199946303936259041} a^{7} - \frac{178376834822160388438593689865108119331203700001275}{1042834163522617761547794080884145199946303936259041} a^{6} + \frac{497468026768791970790540614663346480821241981612413}{1042834163522617761547794080884145199946303936259041} a^{5} - \frac{67434928010540201373164059169947974010075016323131}{1042834163522617761547794080884145199946303936259041} a^{4} + \frac{10733897598078383776047826698348401170944759267713}{80218012578662904734445698529549630765100302789157} a^{3} + \frac{15920237766862484284822197939898284442353482697941}{1042834163522617761547794080884145199946303936259041} a^{2} - \frac{22023973750975333291921423201198984603511901118614}{80218012578662904734445698529549630765100302789157} a + \frac{72576132929794252898904974389918736234227055}{1473052363858877733522700452275182818831377101}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $18$
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:  \( 1240119351367981000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{19}$ (as 19T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 19
The 19 conjugacy class representatives for $C_{19}$
Character table for $C_{19}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $19$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{19}$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{19}$

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
419Data not computed