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Properties

Label 19.19.2222621977...7669.1
Degree $19$
Signature $[19, 0]$
Discriminant $18229^{9}$
Root discriminant $104.29$
Ramified prime $18229$
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![-2048, 61952, -475264, 1618560, -2688288, 1637520, 1188184, -2232580, 587984, 727151, -443620, -60856, 96924, -9209, -8614, 1763, 264, -78, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 - 78*x^17 + 264*x^16 + 1763*x^15 - 8614*x^14 - 9209*x^13 + 96924*x^12 - 60856*x^11 - 443620*x^10 + 727151*x^9 + 587984*x^8 - 2232580*x^7 + 1188184*x^6 + 1637520*x^5 - 2688288*x^4 + 1618560*x^3 - 475264*x^2 + 61952*x - 2048)
 
gp: K = bnfinit(x^19 - 2*x^18 - 78*x^17 + 264*x^16 + 1763*x^15 - 8614*x^14 - 9209*x^13 + 96924*x^12 - 60856*x^11 - 443620*x^10 + 727151*x^9 + 587984*x^8 - 2232580*x^7 + 1188184*x^6 + 1637520*x^5 - 2688288*x^4 + 1618560*x^3 - 475264*x^2 + 61952*x - 2048, 1)
 

Normalized defining polynomial

\( x^{19} - 2 x^{18} - 78 x^{17} + 264 x^{16} + 1763 x^{15} - 8614 x^{14} - 9209 x^{13} + 96924 x^{12} - 60856 x^{11} - 443620 x^{10} + 727151 x^{9} + 587984 x^{8} - 2232580 x^{7} + 1188184 x^{6} + 1637520 x^{5} - 2688288 x^{4} + 1618560 x^{3} - 475264 x^{2} + 61952 x - 2048 \)

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:  \(222262197774010870252934365204894747669=18229^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.29$
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:  $18229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{32} a^{6} - \frac{3}{16} a^{5} + \frac{3}{32} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{3}{32} a^{7} - \frac{1}{16} a^{6} - \frac{3}{32} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} + \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{3}{32} a^{9} + \frac{1}{64} a^{8} + \frac{1}{64} a^{7} - \frac{9}{64} a^{6} - \frac{7}{64} a^{5} - \frac{7}{32} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} + \frac{1}{128} a^{13} + \frac{1}{128} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} - \frac{3}{128} a^{9} - \frac{7}{128} a^{8} - \frac{15}{128} a^{7} + \frac{5}{128} a^{6} + \frac{7}{32} a^{5} + \frac{5}{32} a^{4} - \frac{7}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{512} a^{16} + \frac{1}{512} a^{15} + \frac{3}{512} a^{14} - \frac{5}{512} a^{13} - \frac{3}{256} a^{12} - \frac{5}{256} a^{11} - \frac{23}{512} a^{10} + \frac{3}{512} a^{9} + \frac{55}{512} a^{8} + \frac{103}{512} a^{7} - \frac{19}{256} a^{6} + \frac{25}{128} a^{5} - \frac{9}{64} a^{4} + \frac{11}{32} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{191488} a^{17} - \frac{163}{191488} a^{16} + \frac{431}{191488} a^{15} - \frac{1201}{191488} a^{14} + \frac{1439}{95744} a^{13} + \frac{343}{95744} a^{12} + \frac{2289}{191488} a^{11} - \frac{4545}{191488} a^{10} - \frac{8421}{191488} a^{9} + \frac{8587}{191488} a^{8} + \frac{8519}{95744} a^{7} - \frac{6945}{47872} a^{6} + \frac{437}{23936} a^{5} + \frac{2337}{11968} a^{4} + \frac{1623}{5984} a^{3} + \frac{179}{374} a^{2} - \frac{75}{374} a + \frac{26}{187}$, $\frac{1}{1229959264581363712} a^{18} - \frac{691446653473}{1229959264581363712} a^{17} + \frac{245251580811513}{1229959264581363712} a^{16} + \frac{1232978155059901}{1229959264581363712} a^{15} + \frac{2119894724886083}{307489816145340928} a^{14} + \frac{8018032510655885}{614979632290681856} a^{13} + \frac{18022763555617325}{1229959264581363712} a^{12} + \frac{34107458580509185}{1229959264581363712} a^{11} + \frac{75210721233189801}{1229959264581363712} a^{10} + \frac{2556689669929569}{72350544975374336} a^{9} - \frac{33872058240276891}{307489816145340928} a^{8} - \frac{2275919848403623}{153744908072670464} a^{7} + \frac{6106758388031}{9609056754541904} a^{6} + \frac{1320995637802705}{38436227018167616} a^{5} + \frac{1224999943973537}{9609056754541904} a^{4} + \frac{4964417068302599}{19218113509083808} a^{3} + \frac{1044617635456181}{2402264188635476} a^{2} + \frac{36487226149956}{600566047158869} a - \frac{2502891846676}{600566047158869}$

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:  \( 253663561168000 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $19$ $19$ $19$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $19$ $19$ $19$ $19$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $19$

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
18229Data not computed