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Properties

Label 19.1.512...799.1
Degree $19$
Signature $[1, 9]$
Discriminant $-5.121\times 10^{27}$
Root discriminant $28.73$
Ramified primes $11, 109$
Class number $1$
Class group trivial
Galois group $D_{19}$ (as 19T2)

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Normalized defining polynomial

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)
 
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]);
 

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

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

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-5121210743359411191500170799\)\(\medspace = -\,11^{9}\cdot 109^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $28.73$
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:  $11, 109$
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 \(a\))

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

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

Class group and class number

Trivial group, which has order $1$

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

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{9}\cdot 2481320.5796 \cdot 1}{2\sqrt{5121210743359411191500170799}}\approx 0.52919455720$

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $19$ $19$ $19$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ 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.

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