Properties

Label 19.1.23131036755...4871.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,1831^{9}$
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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Show commands for: Magma / SageMath / Pari/GP

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]);
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)

Normalized defining polynomial

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

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:  \(-231310367559550740879663744871=-\,1831^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $35.11$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $1831$
magma: PrimeDivisors(Discriminant(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 \(a\))

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

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:  \( 34462435.5384 \) (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 $19$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19$ $19$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $19$ $19$ $19$

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

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
1831Data not computed