# Properties

 Label 19.7.10549160171...1804.1 Degree $19$ Signature $[7, 6]$ Discriminant $2^{2}\cdot 131\cdot 149\cdot 397\cdot 20117\cdot 218819\cdot 773146117559$ Root discriminant $33.69$ Ramified primes $2, 131, 149, 397, 20117, 218819, 773146117559$ Class number $1$ (GRH) Class group Trivial (GRH) Galois Group $S_{19}$ (as 19T8)

# Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, -1, -19, 35, -1, -73, 92, 0, -107, 106, -6, -70, 59, -6, -22, 15, -1, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 + 92*x^7 - 73*x^6 - x^5 + 35*x^4 - 19*x^3 - x^2 + 4*x - 1)
gp: K = bnfinit(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 + 92*x^7 - 73*x^6 - x^5 + 35*x^4 - 19*x^3 - x^2 + 4*x - 1, 1)

## Normalizeddefining polynomial

$$x^{19}$$ $$\mathstrut -\mathstrut 3 x^{18}$$ $$\mathstrut -\mathstrut x^{17}$$ $$\mathstrut +\mathstrut 15 x^{16}$$ $$\mathstrut -\mathstrut 22 x^{15}$$ $$\mathstrut -\mathstrut 6 x^{14}$$ $$\mathstrut +\mathstrut 59 x^{13}$$ $$\mathstrut -\mathstrut 70 x^{12}$$ $$\mathstrut -\mathstrut 6 x^{11}$$ $$\mathstrut +\mathstrut 106 x^{10}$$ $$\mathstrut -\mathstrut 107 x^{9}$$ $$\mathstrut +\mathstrut 92 x^{7}$$ $$\mathstrut -\mathstrut 73 x^{6}$$ $$\mathstrut -\mathstrut x^{5}$$ $$\mathstrut +\mathstrut 35 x^{4}$$ $$\mathstrut -\mathstrut 19 x^{3}$$ $$\mathstrut -\mathstrut x^{2}$$ $$\mathstrut +\mathstrut 4 x$$ $$\mathstrut -\mathstrut 1$$

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

## Invariants

 Degree: $19$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[7, 6]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$105491601716452677126377141804=2^{2}\cdot 131\cdot 149\cdot 397\cdot 20117\cdot 218819\cdot 773146117559$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $33.69$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 131, 149, 397, 20117, 218819, 773146117559$ 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: $12$ 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: $$52861133.6694$$ (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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R $17{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 2.7.0.1x^{7} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10} 131131.2.1.2x^{2} + 393$$2$$1$$1$$C_2$$[\ ]_{2}$
131.7.0.1$x^{7} - x + 11$$1$$7$$0$$C_7$$[\ ]^{7} 131.10.0.1x^{10} - x + 14$$1$$10$$0$$C_{10}$$[\ ]^{10}$
149Data not computed
397Data not computed
20117Data not computed
218819Data not computed
773146117559Data not computed