Properties

Label 19.13.6102326046...2484.1
Degree $19$
Signature $[13, 3]$
Discriminant $-\,2^{2}\cdot 90313\cdot 16881409121807\cdot 1000636759045331$
Root discriminant $60.00$
Ramified primes $2, 90313, 16881409121807, 1000636759045331$
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, -18, -14, 128, 72, -461, -163, 904, 143, -987, 8, 614, -88, -216, 52, 40, -12, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 - 12*x^17 + 40*x^16 + 52*x^15 - 216*x^14 - 88*x^13 + 614*x^12 + 8*x^11 - 987*x^10 + 143*x^9 + 904*x^8 - 163*x^7 - 461*x^6 + 72*x^5 + 128*x^4 - 14*x^3 - 18*x^2 + x + 1)
gp: K = bnfinit(x^19 - 3*x^18 - 12*x^17 + 40*x^16 + 52*x^15 - 216*x^14 - 88*x^13 + 614*x^12 + 8*x^11 - 987*x^10 + 143*x^9 + 904*x^8 - 163*x^7 - 461*x^6 + 72*x^5 + 128*x^4 - 14*x^3 - 18*x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{19} \) \(\mathstrut -\mathstrut 3 x^{18} \) \(\mathstrut -\mathstrut 12 x^{17} \) \(\mathstrut +\mathstrut 40 x^{16} \) \(\mathstrut +\mathstrut 52 x^{15} \) \(\mathstrut -\mathstrut 216 x^{14} \) \(\mathstrut -\mathstrut 88 x^{13} \) \(\mathstrut +\mathstrut 614 x^{12} \) \(\mathstrut +\mathstrut 8 x^{11} \) \(\mathstrut -\mathstrut 987 x^{10} \) \(\mathstrut +\mathstrut 143 x^{9} \) \(\mathstrut +\mathstrut 904 x^{8} \) \(\mathstrut -\mathstrut 163 x^{7} \) \(\mathstrut -\mathstrut 461 x^{6} \) \(\mathstrut +\mathstrut 72 x^{5} \) \(\mathstrut +\mathstrut 128 x^{4} \) \(\mathstrut -\mathstrut 14 x^{3} \) \(\mathstrut -\mathstrut 18 x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $19$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[13, 3]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-6102326046691495371062145790782484=-\,2^{2}\cdot 90313\cdot 16881409121807\cdot 1000636759045331\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $60.00$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 90313, 16881409121807, 1000636759045331$
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:  $15$
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:  \( 61782008553.5 \) (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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $17{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
90313Data not computed
16881409121807Data not computed
1000636759045331Data not computed