Properties

Label 19.1.27217203547...6391.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,311^{9}$
Root discriminant $15.16$
Ramified prime $311$
Class number $1$
Class group Trivial
Galois group $D_{19}$ (as 19T2)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^19 - x^18 + 2*x^17 - 5*x^16 + 8*x^15 - 14*x^14 + 13*x^13 - 10*x^12 - x^11 + 9*x^10 - 18*x^9 + 25*x^8 - 10*x^7 - 4*x^6 + 38*x^5 - 42*x^4 + 37*x^3 - 16*x^2 + 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -16, 37, -42, 38, -4, -10, 25, -18, 9, -1, -10, 13, -14, 8, -5, 2, -1, 1]);
 

Normalized defining polynomial

\( x^{19} - x^{18} + 2 x^{17} - 5 x^{16} + 8 x^{15} - 14 x^{14} + 13 x^{13} - 10 x^{12} - x^{11} + 9 x^{10} - 18 x^{9} + 25 x^{8} - 10 x^{7} - 4 x^{6} + 38 x^{5} - 42 x^{4} + 37 x^{3} - 16 x^{2} + 4 x + 1 \)

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

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-27217203547650508966391=-\,311^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.16$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $311$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{391} a^{17} - \frac{2}{17} a^{16} - \frac{128}{391} a^{15} - \frac{179}{391} a^{14} - \frac{4}{23} a^{13} - \frac{19}{391} a^{12} - \frac{67}{391} a^{11} - \frac{160}{391} a^{10} + \frac{154}{391} a^{9} - \frac{174}{391} a^{8} + \frac{189}{391} a^{7} + \frac{133}{391} a^{6} + \frac{94}{391} a^{5} - \frac{65}{391} a^{4} - \frac{126}{391} a^{3} + \frac{48}{391} a^{2} - \frac{11}{23} a + \frac{158}{391}$, $\frac{1}{161454457} a^{18} + \frac{135026}{161454457} a^{17} + \frac{69148682}{161454457} a^{16} - \frac{655864}{161454457} a^{15} - \frac{133411}{161454457} a^{14} + \frac{38668402}{161454457} a^{13} - \frac{39486610}{161454457} a^{12} + \frac{1235520}{9497321} a^{11} + \frac{15740082}{161454457} a^{10} + \frac{11932252}{161454457} a^{9} - \frac{43903937}{161454457} a^{8} - \frac{39923095}{161454457} a^{7} - \frac{18000683}{161454457} a^{6} - \frac{2167161}{8497603} a^{5} + \frac{1272916}{8497603} a^{4} - \frac{16289027}{161454457} a^{3} - \frac{42424184}{161454457} a^{2} + \frac{23572120}{161454457} a - \frac{75616849}{161454457}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2166.17030371 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
311Data not computed