Properties

Label 19.1.38641008899...4375.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,5^{9}\cdot 19^{19}$
Root discriminant $40.72$
Ramified primes $5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois Group $F_{19}$ (as 19T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 19, 0, 285, 0, 1254, 0, 2508, 0, 2717, 0, 1729, 0, 665, 0, 152, 0, 19, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*x - 1)
gp: K = bnfinit(x^19 + 19*x^17 + 152*x^15 + 665*x^13 + 1729*x^11 + 2717*x^9 + 2508*x^7 + 1254*x^5 + 285*x^3 + 19*x - 1, 1)

Normalized defining polynomial

\(x^{19} \) \(\mathstrut +\mathstrut 19 x^{17} \) \(\mathstrut +\mathstrut 152 x^{15} \) \(\mathstrut +\mathstrut 665 x^{13} \) \(\mathstrut +\mathstrut 1729 x^{11} \) \(\mathstrut +\mathstrut 2717 x^{9} \) \(\mathstrut +\mathstrut 2508 x^{7} \) \(\mathstrut +\mathstrut 1254 x^{5} \) \(\mathstrut +\mathstrut 285 x^{3} \) \(\mathstrut +\mathstrut 19 x \) \(\mathstrut -\mathstrut 1 \)

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:  \(-3864100889961549978757771484375=-\,5^{9}\cdot 19^{19}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $40.72$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 19$
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:  \( 123041861.807 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$F_{19}$ (as 19T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 342
The 19 conjugacy class representatives for $F_{19}$
Character table for $F_{19}$

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R $18{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $19$ $18{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $18{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
5Data not computed
19Data not computed