Label 19.1.720...723.1
Degree $19$
Signature $[1, 9]$
Discriminant $-7.207\times 10^{32}$
Root discriminant $53.62$
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{19}$ (as 19T8)

Related objects


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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x - 3)
gp: K = bnfinit(x^19 - 3*x - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);

\(x^{19} - 3 x - 3\)  Toggle raw display

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


Degree:  $19$
gp: poldegree(K.pol)
magma: Degree(K);
Signature:  $[1, 9]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
Discriminant:  \(-720749496472355508369324104418723\)\(\medspace = -\,3^{18}\cdot 30011\cdot 461269\cdot 134390022739373\)
sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
Root discriminant:  $53.62$
sage: (K.disc().abs())^(1./
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
Ramified primes:  $3, 30011, 461269, 134390022739373$
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

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

sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);

Unit group

sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
Rank:  $9$
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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)
sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
Regulator:  \( 1413210922.08 \) (assuming GRH)
sage: K.regulator()
gp: K.reg
magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 1413210922.08 \cdot 1}{2\sqrt{720749496472355508369324104418723}}\approx 0.803403328425$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
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 ${\href{/LocalNumberField/}{7} }{,}\,{\href{/LocalNumberField/}{5} }{,}\,{\href{/LocalNumberField/}{4} }{,}\,{\href{/LocalNumberField/}{3} }$ R ${\href{/LocalNumberField/}{13} }{,}\,{\href{/LocalNumberField/}{4} }{,}\,{\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{9} }{,}\,{\href{/LocalNumberField/}{7} }{,}\,{\href{/LocalNumberField/}{2} }{,}\,{\href{/LocalNumberField/}{1} }$ ${\href{/LocalNumberField/}{8} }{,}\,{\href{/LocalNumberField/}{6} }{,}\,{\href{/LocalNumberField/}{5} }$ ${\href{/LocalNumberField/}{9} }{,}\,{\href{/LocalNumberField/}{8} }{,}\,{\href{/LocalNumberField/}{2} }$ ${\href{/LocalNumberField/}{10} }{,}\,{\href{/LocalNumberField/}{9} }$ $18{,}\,{\href{/LocalNumberField/}{1} }$ $15{,}\,{\href{/LocalNumberField/}{1} }^{4}$ ${\href{/LocalNumberField/}{10} }{,}\,{\href{/LocalNumberField/}{5} }{,}\,{\href{/LocalNumberField/}{2} }{,}\,{\href{/LocalNumberField/}{1} }^{2}$ ${\href{/LocalNumberField/}{9} }{,}\,{\href{/LocalNumberField/}{6} }{,}\,{\href{/LocalNumberField/}{3} }{,}\,{\href{/LocalNumberField/}{1} }$ ${\href{/LocalNumberField/}{7} }{,}\,{\href{/LocalNumberField/}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/}{2} }{,}\,{\href{/LocalNumberField/}{1} }$ $16{,}\,{\href{/LocalNumberField/}{2} }{,}\,{\href{/LocalNumberField/}{1} }$ ${\href{/LocalNumberField/}{8} }{,}\,{\href{/LocalNumberField/}{4} }^{2}{,}\,{\href{/LocalNumberField/}{1} }^{3}$ ${\href{/LocalNumberField/}{8} }{,}\,{\href{/LocalNumberField/}{6} }{,}\,{\href{/LocalNumberField/}{4} }{,}\,{\href{/LocalNumberField/}{1} }$ $17{,}\,{\href{/LocalNumberField/}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
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];

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
30011Data not computed
461269Data not computed
134390022739373Data not computed