# Properties

 Label 19.5.23920741637...4200.1 Degree $19$ Signature $[5, 7]$ Discriminant $-\,2^{3}\cdot 5^{2}\cdot 430709\cdot 19005923\cdot 14610723851226353$ Root discriminant $44.82$ Ramified primes $2, 5, 430709, 19005923, 14610723851226353$ Class number $1$ (GRH) Class group Trivial (GRH) Galois group $S_{19}$ (as 19T8)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - x^18 - 6*x^17 + 7*x^16 + 13*x^15 - 21*x^14 - 16*x^13 + 41*x^12 + 10*x^11 - 51*x^10 + 5*x^9 + 43*x^8 - 19*x^7 - 24*x^6 + 18*x^5 + 5*x^4 - 11*x^3 + 2*x^2 + 3*x - 1)

gp: K = bnfinit(x^19 - x^18 - 6*x^17 + 7*x^16 + 13*x^15 - 21*x^14 - 16*x^13 + 41*x^12 + 10*x^11 - 51*x^10 + 5*x^9 + 43*x^8 - 19*x^7 - 24*x^6 + 18*x^5 + 5*x^4 - 11*x^3 + 2*x^2 + 3*x - 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -11, 5, 18, -24, -19, 43, 5, -51, 10, 41, -16, -21, 13, 7, -6, -1, 1]);

## Normalizeddefining polynomial

$$x^{19} - x^{18} - 6 x^{17} + 7 x^{16} + 13 x^{15} - 21 x^{14} - 16 x^{13} + 41 x^{12} + 10 x^{11} - 51 x^{10} + 5 x^{9} + 43 x^{8} - 19 x^{7} - 24 x^{6} + 18 x^{5} + 5 x^{4} - 11 x^{3} + 2 x^{2} + 3 x - 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $19$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[5, 7]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-23920741637672928000872108534200=-\,2^{3}\cdot 5^{2}\cdot 430709\cdot 19005923\cdot 14610723851226353$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $44.82$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 5, 430709, 19005923, 14610723851226353$ 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 $$a$$)

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

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: $11$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) 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: $$840306267.085$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## 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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R $15{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.9.0.1}{9} }$ $15{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $17{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

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
2Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2} 5.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2} 5.3.0.1x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 5.6.0.1x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
430709Data not computed
19005923Data not computed
14610723851226353Data not computed