Properties

Label 19.1.23974020934...2368.1
Degree $19$
Signature $[1, 9]$
Discriminant $-\,2^{7}\cdot 7\cdot 60259\cdot 4440286064038502416537$
Root discriminant $35.18$
Ramified primes $2, 7, 60259, 4440286064038502416537$
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, -9, -4, 17, 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 + 17*x^6 - 4*x^5 - 9*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 + 17*x^6 - 4*x^5 - 9*x^4 + 2*x^3 + 3*x^2 - x - 1, 1)

Normalized defining polynomial

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

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:  \(-239740209347874920937820362368=-\,2^{7}\cdot 7\cdot 60259\cdot 4440286064038502416537\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $35.18$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 60259, 4440286064038502416537$
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:  \( 21087865.9745 \) (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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ $17{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $16{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $19$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.9.0.1$x^{9} + x^{4} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
60259Data not computed
4440286064038502416537Data not computed