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Properties

Label 19.1.49578494467...0343.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,1543^{9}$
Root discriminant $32.38$
Ramified prime $1543$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{19}$ (as 19T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 121, -8, 106, -275, -193, 116, 195, 181, -49, -238, 46, 74, 34, 4, -42, -6, 24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*x - 1)
 
gp: K = bnfinit(x^19 - 9*x^18 + 24*x^17 - 6*x^16 - 42*x^15 + 4*x^14 + 34*x^13 + 74*x^12 + 46*x^11 - 238*x^10 - 49*x^9 + 181*x^8 + 195*x^7 + 116*x^6 - 193*x^5 - 275*x^4 + 106*x^3 - 8*x^2 + 121*x - 1, 1)
 

Normalized defining polynomial

\( x^{19} - 9 x^{18} + 24 x^{17} - 6 x^{16} - 42 x^{15} + 4 x^{14} + 34 x^{13} + 74 x^{12} + 46 x^{11} - 238 x^{10} - 49 x^{9} + 181 x^{8} + 195 x^{7} + 116 x^{6} - 193 x^{5} - 275 x^{4} + 106 x^{3} - 8 x^{2} + 121 x - 1 \)

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

Invariants

Degree:  $19$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-49578494467761916312526550343=-\,1543^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.38$
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:  $1543$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{2} - \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a - \frac{2}{9}$, $\frac{1}{855} a^{16} - \frac{11}{285} a^{15} + \frac{41}{855} a^{14} + \frac{89}{855} a^{13} + \frac{14}{171} a^{12} + \frac{11}{285} a^{11} + \frac{103}{855} a^{10} + \frac{82}{855} a^{9} - \frac{4}{57} a^{8} - \frac{67}{285} a^{7} - \frac{17}{855} a^{6} - \frac{371}{855} a^{5} + \frac{73}{171} a^{4} - \frac{29}{285} a^{3} + \frac{251}{855} a^{2} - \frac{169}{855} a - \frac{61}{855}$, $\frac{1}{2565} a^{17} - \frac{98}{2565} a^{15} - \frac{26}{855} a^{14} - \frac{413}{2565} a^{13} - \frac{32}{2565} a^{12} - \frac{43}{2565} a^{11} - \frac{43}{855} a^{10} + \frac{122}{855} a^{9} + \frac{194}{2565} a^{8} - \frac{5}{27} a^{7} - \frac{184}{855} a^{6} + \frac{92}{2565} a^{5} - \frac{487}{2565} a^{4} + \frac{179}{513} a^{3} + \frac{298}{855} a^{2} + \frac{917}{2565} a + \frac{1027}{2565}$, $\frac{1}{4772442701295} a^{18} + \frac{662887919}{4772442701295} a^{17} + \frac{1752360862}{4772442701295} a^{16} + \frac{33487901806}{954488540259} a^{15} + \frac{13895980616}{954488540259} a^{14} + \frac{58546782047}{1590814233765} a^{13} + \frac{736040120899}{4772442701295} a^{12} - \frac{723266576861}{4772442701295} a^{11} - \frac{43584455216}{318162846753} a^{10} + \frac{474576464078}{4772442701295} a^{9} + \frac{935269937}{38800347165} a^{8} + \frac{647391070363}{4772442701295} a^{7} - \frac{1487278745026}{4772442701295} a^{6} - \frac{660860588378}{1590814233765} a^{5} - \frac{1990918487548}{4772442701295} a^{4} + \frac{326430003014}{4772442701295} a^{3} - \frac{26896025713}{60410667105} a^{2} + \frac{185213667625}{954488540259} a - \frac{51519525688}{251181194805}$

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:  $9$
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:  \( 17317158.8896 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{19}$ (as 19T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38
The 11 conjugacy class representatives for $D_{19}$
Character table for $D_{19}$

Intermediate fields

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

Sibling fields

Galois closure: 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 $19$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $19$ $19$ $19$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1543Data not computed