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Properties

Label 19.1.33600614943...0139.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,19^{27}$
Root discriminant $65.64$
Ramified prime $19$
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![-110592, 229824, -445968, 696844, -695400, 532855, -236892, -144818, 263112, -78147, -64068, 39159, 6384, -7239, -228, 703, 0, -38, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 38*x^17 + 703*x^15 - 228*x^14 - 7239*x^13 + 6384*x^12 + 39159*x^11 - 64068*x^10 - 78147*x^9 + 263112*x^8 - 144818*x^7 - 236892*x^6 + 532855*x^5 - 695400*x^4 + 696844*x^3 - 445968*x^2 + 229824*x - 110592)
 
gp: K = bnfinit(x^19 - 38*x^17 + 703*x^15 - 228*x^14 - 7239*x^13 + 6384*x^12 + 39159*x^11 - 64068*x^10 - 78147*x^9 + 263112*x^8 - 144818*x^7 - 236892*x^6 + 532855*x^5 - 695400*x^4 + 696844*x^3 - 445968*x^2 + 229824*x - 110592, 1)
 

Normalized defining polynomial

\( x^{19} - 38 x^{17} + 703 x^{15} - 228 x^{14} - 7239 x^{13} + 6384 x^{12} + 39159 x^{11} - 64068 x^{10} - 78147 x^{9} + 263112 x^{8} - 144818 x^{7} - 236892 x^{6} + 532855 x^{5} - 695400 x^{4} + 696844 x^{3} - 445968 x^{2} + 229824 x - 110592 \)

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:  \(-33600614943460448322716069311260139=-\,19^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.64$
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:  $19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{9} + \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{24} a^{4} + \frac{1}{24} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{144} a^{11} - \frac{1}{48} a^{10} - \frac{1}{144} a^{9} + \frac{1}{18} a^{7} - \frac{5}{144} a^{5} + \frac{1}{48} a^{4} + \frac{61}{144} a^{3} + \frac{1}{3} a^{2} + \frac{17}{36} a + \frac{1}{3}$, $\frac{1}{144} a^{12} + \frac{1}{72} a^{10} - \frac{1}{48} a^{9} - \frac{1}{36} a^{8} - \frac{29}{144} a^{6} + \frac{1}{12} a^{5} - \frac{7}{72} a^{4} + \frac{7}{16} a^{3} - \frac{5}{18} a^{2} + \frac{5}{12} a$, $\frac{1}{288} a^{13} - \frac{1}{288} a^{11} - \frac{7}{288} a^{9} + \frac{1}{48} a^{8} + \frac{7}{288} a^{7} - \frac{1}{24} a^{6} + \frac{49}{288} a^{5} - \frac{1}{24} a^{4} + \frac{71}{288} a^{3} - \frac{11}{48} a^{2} + \frac{5}{12} a$, $\frac{1}{288} a^{14} - \frac{1}{288} a^{12} + \frac{5}{288} a^{10} - \frac{1}{48} a^{9} - \frac{5}{288} a^{8} + \frac{1}{24} a^{7} - \frac{47}{288} a^{6} + \frac{1}{8} a^{5} + \frac{59}{288} a^{4} - \frac{1}{48} a^{3} - \frac{5}{24} a^{2} + \frac{1}{12} a$, $\frac{1}{288} a^{15} - \frac{1}{48} a^{10} + \frac{1}{72} a^{9} + \frac{1}{48} a^{8} - \frac{1}{4} a^{6} - \frac{1}{18} a^{5} + \frac{5}{48} a^{4} + \frac{91}{288} a^{3} - \frac{5}{48} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{1728} a^{16} + \frac{1}{1728} a^{15} + \frac{1}{1728} a^{13} - \frac{1}{432} a^{12} - \frac{1}{576} a^{11} - \frac{5}{864} a^{10} + \frac{47}{1728} a^{9} - \frac{1}{54} a^{8} - \frac{1}{192} a^{7} - \frac{1}{54} a^{6} + \frac{331}{1728} a^{5} - \frac{1}{64} a^{4} - \frac{85}{216} a^{3} - \frac{145}{432} a^{2} - \frac{13}{36} a - \frac{1}{3}$, $\frac{1}{44928} a^{17} - \frac{7}{44928} a^{16} + \frac{1}{864} a^{15} - \frac{53}{44928} a^{14} + \frac{5}{3744} a^{13} - \frac{1}{3456} a^{12} + \frac{61}{22464} a^{11} - \frac{515}{44928} a^{10} + \frac{25}{936} a^{9} + \frac{1837}{44928} a^{8} + \frac{23}{5616} a^{7} - \frac{10219}{44928} a^{6} - \frac{227}{3456} a^{5} - \frac{2071}{22464} a^{4} - \frac{7}{936} a^{3} + \frac{851}{2808} a^{2} - \frac{7}{36} a + \frac{3}{13}$, $\frac{1}{431059976754143919901056} a^{18} + \frac{373866811795322195}{35921664729511993325088} a^{17} - \frac{81016404862559107673}{431059976754143919901056} a^{16} - \frac{8031016140055928353}{5321728108075850862976} a^{15} + \frac{40510207904925414565}{33158459750318763069312} a^{14} + \frac{119186151865216585309}{143686658918047973300352} a^{13} - \frac{135061596995156953769}{47895552972682657766784} a^{12} - \frac{27797133245399049917}{47895552972682657766784} a^{11} - \frac{734181150814994075581}{47895552972682657766784} a^{10} + \frac{1034003639877070564589}{47895552972682657766784} a^{9} - \frac{2699142237496533274855}{143686658918047973300352} a^{8} + \frac{1359106788617060853493}{47895552972682657766784} a^{7} - \frac{24613953369902649414209}{107764994188535979975264} a^{6} - \frac{22876120399774580903257}{143686658918047973300352} a^{5} - \frac{21865467284824721147179}{215529988377071959950528} a^{4} + \frac{135904126339845164035}{997824020264222036808} a^{3} + \frac{2857439909628598537709}{13470624273566997496908} a^{2} - \frac{860680176964427119933}{4490208091188999165636} a + \frac{171942530252109252305}{374184007599083263803}$

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:  \( 215015238772 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $19$ $19$ $19$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $19$ R $19$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
19Data not computed