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Properties

Label 19.19.1144459979...0721.1
Degree $19$
Signature $[19, 0]$
Discriminant $191^{18}$
Root discriminant $144.87$
Ramified prime $191$
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![117649, 1553447, 5125792, 2600136, -8950277, -8711114, 5023125, 6854098, -1153239, -2465084, 90110, 474003, 4822, -51184, -1124, 3044, 57, -90, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 90*x^17 + 57*x^16 + 3044*x^15 - 1124*x^14 - 51184*x^13 + 4822*x^12 + 474003*x^11 + 90110*x^10 - 2465084*x^9 - 1153239*x^8 + 6854098*x^7 + 5023125*x^6 - 8711114*x^5 - 8950277*x^4 + 2600136*x^3 + 5125792*x^2 + 1553447*x + 117649)
 
gp: K = bnfinit(x^19 - x^18 - 90*x^17 + 57*x^16 + 3044*x^15 - 1124*x^14 - 51184*x^13 + 4822*x^12 + 474003*x^11 + 90110*x^10 - 2465084*x^9 - 1153239*x^8 + 6854098*x^7 + 5023125*x^6 - 8711114*x^5 - 8950277*x^4 + 2600136*x^3 + 5125792*x^2 + 1553447*x + 117649, 1)
 

Normalized defining polynomial

\( x^{19} - x^{18} - 90 x^{17} + 57 x^{16} + 3044 x^{15} - 1124 x^{14} - 51184 x^{13} + 4822 x^{12} + 474003 x^{11} + 90110 x^{10} - 2465084 x^{9} - 1153239 x^{8} + 6854098 x^{7} + 5023125 x^{6} - 8711114 x^{5} - 8950277 x^{4} + 2600136 x^{3} + 5125792 x^{2} + 1553447 x + 117649 \)

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:  \(114445997944945591651333831028437092270721=191^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.87$
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:  $191$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(191\)
Dirichlet character group:    $\lbrace$$\chi_{191}(1,·)$, $\chi_{191}(69,·)$, $\chi_{191}(5,·)$, $\chi_{191}(6,·)$, $\chi_{191}(32,·)$, $\chi_{191}(136,·)$, $\chi_{191}(150,·)$, $\chi_{191}(153,·)$, $\chi_{191}(25,·)$, $\chi_{191}(154,·)$, $\chi_{191}(30,·)$, $\chi_{191}(52,·)$, $\chi_{191}(160,·)$, $\chi_{191}(36,·)$, $\chi_{191}(107,·)$, $\chi_{191}(177,·)$, $\chi_{191}(180,·)$, $\chi_{191}(121,·)$, $\chi_{191}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{49} a^{12} - \frac{1}{49} a^{11} + \frac{3}{49} a^{10} + \frac{2}{49} a^{9} - \frac{3}{49} a^{8} + \frac{20}{49} a^{6} + \frac{15}{49} a^{5} - \frac{24}{49} a^{4} - \frac{2}{49} a^{3} + \frac{10}{49} a^{2} - \frac{1}{7} a$, $\frac{1}{49} a^{13} + \frac{2}{49} a^{11} - \frac{2}{49} a^{10} - \frac{1}{49} a^{9} - \frac{3}{49} a^{8} - \frac{1}{49} a^{7} - \frac{2}{7} a^{6} - \frac{9}{49} a^{5} - \frac{19}{49} a^{4} + \frac{8}{49} a^{3} + \frac{3}{49} a^{2} + \frac{2}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{343} a^{15} + \frac{3}{343} a^{13} - \frac{3}{343} a^{12} + \frac{23}{343} a^{11} - \frac{8}{343} a^{10} + \frac{17}{343} a^{9} + \frac{1}{49} a^{8} + \frac{11}{343} a^{7} + \frac{143}{343} a^{6} + \frac{61}{343} a^{5} + \frac{106}{343} a^{4} - \frac{46}{343} a^{3} + \frac{3}{49} a^{2}$, $\frac{1}{343} a^{16} + \frac{3}{343} a^{14} - \frac{3}{343} a^{13} + \frac{2}{343} a^{12} + \frac{13}{343} a^{11} + \frac{3}{343} a^{10} + \frac{2}{49} a^{9} - \frac{24}{343} a^{8} - \frac{4}{343} a^{7} - \frac{16}{343} a^{6} + \frac{134}{343} a^{5} + \frac{66}{343} a^{4} + \frac{2}{49} a^{3} - \frac{16}{49} a^{2} - \frac{1}{7} a$, $\frac{1}{2401} a^{17} - \frac{2}{2401} a^{16} + \frac{3}{2401} a^{15} - \frac{16}{2401} a^{14} + \frac{1}{2401} a^{13} + \frac{9}{2401} a^{12} - \frac{135}{2401} a^{11} - \frac{27}{2401} a^{10} + \frac{53}{2401} a^{9} - \frac{166}{2401} a^{8} - \frac{1}{2401} a^{7} + \frac{264}{2401} a^{6} + \frac{988}{2401} a^{5} - \frac{965}{2401} a^{4} + \frac{8}{49} a^{3} + \frac{15}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{4327223165521294948263316675859741} a^{18} - \frac{491732326964188351515199255643}{4327223165521294948263316675859741} a^{17} - \frac{3331576872142052539766510866815}{4327223165521294948263316675859741} a^{16} + \frac{692768881823507288348466975674}{4327223165521294948263316675859741} a^{15} + \frac{25167318703770153041158185493852}{4327223165521294948263316675859741} a^{14} - \frac{29180078317179517535127387611114}{4327223165521294948263316675859741} a^{13} + \frac{4805903830560264512313024655985}{618174737931613564037616667979963} a^{12} + \frac{15970275892829005628127945848674}{4327223165521294948263316675859741} a^{11} - \frac{65729688008939871389812327444101}{4327223165521294948263316675859741} a^{10} - \frac{12447685091752672460672247731297}{4327223165521294948263316675859741} a^{9} - \frac{30765499074682577069307074186981}{4327223165521294948263316675859741} a^{8} - \frac{104169132650738606151760368489540}{4327223165521294948263316675859741} a^{7} + \frac{918270185567052916882506946968152}{4327223165521294948263316675859741} a^{6} - \frac{1385689641390177072865363263127800}{4327223165521294948263316675859741} a^{5} + \frac{397955530380109356011822791433465}{4327223165521294948263316675859741} a^{4} - \frac{149836593829282597436809093608200}{618174737931613564037616667979963} a^{3} + \frac{32856040302725271266545661721554}{88310676847373366291088095425709} a^{2} - \frac{6275928590812692883642438349052}{12615810978196195184441156489387} a - \frac{115612633198487845282252934344}{257465530167269289478390948763}$

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:  \( 2998086774930000 \) (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 $19$ $19$ $19$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{19}$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$ $19$

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
191Data not computed