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Properties

Label 19.1.231...871.1
Degree $19$
Signature $[1, 9]$
Discriminant $-2.313\times 10^{29}$
Root discriminant $35.11$
Ramified prime $1831$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{19}$ (as 19T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 6*x^18 + 18*x^17 - 39*x^16 + 73*x^15 - 200*x^14 + 265*x^13 + 305*x^12 - 931*x^11 + 1905*x^10 - 5214*x^9 + 10284*x^8 - 13343*x^7 + 12719*x^6 - 8662*x^5 + 4443*x^4 - 1732*x^3 + 614*x^2 - 152*x + 39)
 
gp: K = bnfinit(x^19 - 6*x^18 + 18*x^17 - 39*x^16 + 73*x^15 - 200*x^14 + 265*x^13 + 305*x^12 - 931*x^11 + 1905*x^10 - 5214*x^9 + 10284*x^8 - 13343*x^7 + 12719*x^6 - 8662*x^5 + 4443*x^4 - 1732*x^3 + 614*x^2 - 152*x + 39, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![39, -152, 614, -1732, 4443, -8662, 12719, -13343, 10284, -5214, 1905, -931, 305, 265, -200, 73, -39, 18, -6, 1]);
 

\( x^{19} - 6 x^{18} + 18 x^{17} - 39 x^{16} + 73 x^{15} - 200 x^{14} + 265 x^{13} + 305 x^{12} - 931 x^{11} + 1905 x^{10} - 5214 x^{9} + 10284 x^{8} - 13343 x^{7} + 12719 x^{6} - 8662 x^{5} + 4443 x^{4} - 1732 x^{3} + 614 x^{2} - 152 x + 39 \)

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:  \(-231310367559550740879663744871\)\(\medspace = -\,1831^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.11$
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:  $1831$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{5} - \frac{2}{9} a^{3} + \frac{4}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} + \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} - \frac{1}{27} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{27} a^{6} - \frac{1}{27} a^{5} + \frac{11}{27} a^{4} - \frac{4}{27} a^{3} - \frac{4}{27} a^{2} - \frac{7}{27} a - \frac{4}{9}$, $\frac{1}{189} a^{16} - \frac{1}{63} a^{15} + \frac{2}{63} a^{14} + \frac{4}{189} a^{13} - \frac{1}{21} a^{12} - \frac{1}{63} a^{11} + \frac{25}{189} a^{10} + \frac{1}{9} a^{9} + \frac{8}{63} a^{8} - \frac{10}{189} a^{7} - \frac{8}{63} a^{6} + \frac{31}{63} a^{4} + \frac{1}{9} a^{3} - \frac{31}{63} a^{2} - \frac{74}{189} a + \frac{25}{63}$, $\frac{1}{5677371} a^{17} + \frac{4}{9639} a^{16} - \frac{39061}{5677371} a^{15} + \frac{23431}{630819} a^{14} + \frac{44839}{1892457} a^{13} - \frac{108944}{5677371} a^{12} - \frac{115825}{5677371} a^{11} - \frac{83947}{811053} a^{10} - \frac{7723}{630819} a^{9} + \frac{683246}{5677371} a^{8} - \frac{322213}{5677371} a^{7} - \frac{128144}{811053} a^{6} + \frac{2283073}{5677371} a^{5} + \frac{195559}{811053} a^{4} - \frac{2635103}{5677371} a^{3} + \frac{177047}{5677371} a^{2} + \frac{1613687}{5677371} a - \frac{90832}{270351}$, $\frac{1}{6376620657827511} a^{18} - \frac{286725058}{6376620657827511} a^{17} + \frac{5114476798316}{2125540219275837} a^{16} - \frac{91393300682065}{6376620657827511} a^{15} + \frac{29099678211083}{2125540219275837} a^{14} - \frac{353851745036513}{6376620657827511} a^{13} + \frac{56230169410196}{2125540219275837} a^{12} - \frac{5525634038423}{205697440575081} a^{11} + \frac{10920697707662}{6376620657827511} a^{10} - \frac{408800903654779}{6376620657827511} a^{9} + \frac{827255474969296}{6376620657827511} a^{8} + \frac{4028471867507}{54501031263483} a^{7} - \frac{98232255326917}{910945808261073} a^{6} - \frac{11019316777724}{375095332813383} a^{5} - \frac{1126468650470110}{6376620657827511} a^{4} - \frac{15674151521204}{125031777604461} a^{3} + \frac{1758133891772611}{6376620657827511} a^{2} + \frac{638318059312385}{6376620657827511} a - \frac{10834185621670}{163503093790449}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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 (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 34462435.5384 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 34462435.5384 \cdot 1}{2\sqrt{231310367559550740879663744871}}\approx 1.09362198601$ (assuming GRH)

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 $19$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ $19$ $19$

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
1831Data not computed