Properties

Label 19.1.19729438357...7516.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,2^{2}\cdot 49323595893763542101396994379$
Root discriminant $34.82$
Ramified primes $2, 49323595893763542101396994379$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group $S_{19}$ (as 19T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 3, 2, -8, -4, 16, 2, -25, 8, 21, -16, -9, 14, 0, -8, 4, 2, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - x - 1)
gp: K = bnfinit(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - x - 1, 1)

Normalized defining polynomial

\(x^{19} \) \(\mathstrut -\mathstrut 3 x^{18} \) \(\mathstrut +\mathstrut 2 x^{17} \) \(\mathstrut +\mathstrut 4 x^{16} \) \(\mathstrut -\mathstrut 8 x^{15} \) \(\mathstrut +\mathstrut 14 x^{13} \) \(\mathstrut -\mathstrut 9 x^{12} \) \(\mathstrut -\mathstrut 16 x^{11} \) \(\mathstrut +\mathstrut 21 x^{10} \) \(\mathstrut +\mathstrut 8 x^{9} \) \(\mathstrut -\mathstrut 25 x^{8} \) \(\mathstrut +\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 16 x^{6} \) \(\mathstrut -\mathstrut 4 x^{5} \) \(\mathstrut -\mathstrut 8 x^{4} \) \(\mathstrut +\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut -\mathstrut 1 \)

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:  \(-197294383575054168405587977516=-\,2^{2}\cdot 49323595893763542101396994379\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $34.82$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 49323595893763542101396994379$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$

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:  \( 20856736.7017 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_{19}$ (as 19T8):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 121645100408832000
The 490 conjugacy class representatives for $S_{19}$ are not computed
Character table for $S_{19}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $15{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $17{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $19$ $19$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.13.0.1$x^{13} + x^{4} - x^{3} - x + 1$$1$$13$$0$$C_{13}$$[\ ]^{13}$
49323595893763542101396994379Data not computed