# Properties

 Label 19.1.51212107433...0799.1 Degree $19$ Signature $[1, 9]$ Discriminant $-\,11^{9}\cdot 109^{9}$ Root discriminant $28.73$ Ramified primes $11, 109$ Class number $1$ Class group Trivial Galois Group $D_{19}$ (as 19T2)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -210, 408, 482, -1490, 4212, -5699, 7142, -6561, 5497, -3757, 2423, -1319, 709, -310, 135, -47, 17, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49)
gp: K = bnfinit(x^19 - 3*x^18 + 17*x^17 - 47*x^16 + 135*x^15 - 310*x^14 + 709*x^13 - 1319*x^12 + 2423*x^11 - 3757*x^10 + 5497*x^9 - 6561*x^8 + 7142*x^7 - 5699*x^6 + 4212*x^5 - 1490*x^4 + 482*x^3 + 408*x^2 - 210*x + 49, 1)

## Normalizeddefining polynomial

$$x^{19}$$ $$\mathstrut -\mathstrut 3 x^{18}$$ $$\mathstrut +\mathstrut 17 x^{17}$$ $$\mathstrut -\mathstrut 47 x^{16}$$ $$\mathstrut +\mathstrut 135 x^{15}$$ $$\mathstrut -\mathstrut 310 x^{14}$$ $$\mathstrut +\mathstrut 709 x^{13}$$ $$\mathstrut -\mathstrut 1319 x^{12}$$ $$\mathstrut +\mathstrut 2423 x^{11}$$ $$\mathstrut -\mathstrut 3757 x^{10}$$ $$\mathstrut +\mathstrut 5497 x^{9}$$ $$\mathstrut -\mathstrut 6561 x^{8}$$ $$\mathstrut +\mathstrut 7142 x^{7}$$ $$\mathstrut -\mathstrut 5699 x^{6}$$ $$\mathstrut +\mathstrut 4212 x^{5}$$ $$\mathstrut -\mathstrut 1490 x^{4}$$ $$\mathstrut +\mathstrut 482 x^{3}$$ $$\mathstrut +\mathstrut 408 x^{2}$$ $$\mathstrut -\mathstrut 210 x$$ $$\mathstrut +\mathstrut 49$$

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: $$-5121210743359411191500170799=-\,11^{9}\cdot 109^{9}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $28.73$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $11, 109$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11} a^{11} - \frac{3}{11} a^{10} + \frac{3}{11} a^{8} - \frac{4}{11} a^{7} - \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{3}{11} a^{2} - \frac{2}{11}$, $\frac{1}{11} a^{12} + \frac{2}{11} a^{10} + \frac{3}{11} a^{9} + \frac{5}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{1}{11} a^{5} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} - \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{10} + \frac{5}{11} a^{9} + \frac{3}{11} a^{7} - \frac{4}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{11} a^{4} + \frac{2}{11} a^{3} + \frac{4}{11} a^{2} + \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{10} - \frac{2}{11} a^{8} - \frac{1}{11} a^{7} + \frac{2}{11} a^{4} + \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{15} - \frac{3}{11} a^{10} - \frac{2}{11} a^{9} + \frac{2}{11} a^{8} - \frac{4}{11} a^{7} - \frac{4}{11} a^{6} - \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{4}{11} a - \frac{2}{11}$, $\frac{1}{77} a^{16} - \frac{2}{77} a^{15} - \frac{2}{77} a^{14} - \frac{1}{77} a^{13} + \frac{1}{77} a^{11} - \frac{4}{77} a^{10} - \frac{10}{77} a^{9} - \frac{25}{77} a^{8} - \frac{5}{11} a^{7} - \frac{4}{11} a^{6} - \frac{16}{77} a^{5} - \frac{3}{11} a^{4} + \frac{26}{77} a^{3} + \frac{24}{77} a^{2} - \frac{18}{77} a$, $\frac{1}{847} a^{17} + \frac{3}{847} a^{16} + \frac{37}{847} a^{15} + \frac{38}{847} a^{14} + \frac{30}{847} a^{13} + \frac{1}{847} a^{12} - \frac{6}{847} a^{11} + \frac{17}{77} a^{10} + \frac{387}{847} a^{9} - \frac{335}{847} a^{8} + \frac{43}{121} a^{7} + \frac{152}{847} a^{6} + \frac{417}{847} a^{5} + \frac{3}{77} a^{4} + \frac{20}{121} a^{3} + \frac{263}{847} a^{2} + \frac{393}{847} a + \frac{24}{121}$, $\frac{1}{51727543335603443} a^{18} - \frac{29701844258160}{51727543335603443} a^{17} - \frac{172604384349723}{51727543335603443} a^{16} + \frac{842653333060995}{51727543335603443} a^{15} - \frac{268650353179353}{7389649047943349} a^{14} - \frac{66104475505676}{7389649047943349} a^{13} + \frac{1898574407964536}{51727543335603443} a^{12} + \frac{94415154718519}{4702503939600313} a^{11} - \frac{5289571219058366}{51727543335603443} a^{10} + \frac{146979307371132}{1202966124083801} a^{9} - \frac{8703660410832559}{51727543335603443} a^{8} - \frac{11008289411013897}{51727543335603443} a^{7} + \frac{18133680031107288}{51727543335603443} a^{6} + \frac{2003985898343274}{4702503939600313} a^{5} + \frac{800344130348002}{7389649047943349} a^{4} - \frac{18476274851913583}{51727543335603443} a^{3} - \frac{3933526836525905}{51727543335603443} a^{2} + \frac{2729455584365821}{51727543335603443} a + \frac{209747834143810}{671786277085759}$

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

## Class group and class number

Trivial group, which has order $1$

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 magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$2481320.5796$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_{19}$ (as 19T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 38 The 11 conjugacy class representatives for $D_{19}$ Character table for $D_{19}$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling fields

 Galois closure: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 $19$ $19$ $19$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R $19$ $19$ $19$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $19$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0Trivial[\ ] 11.2.1.1x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.1x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.1x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.1x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.1x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0Trivial[\ ] 109.2.1.2x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2} 109.2.1.2x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2} 109.2.1.2x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2} 109.2.1.2x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2} 109.2.1.2x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$