Properties

Label 19.1.200...264.1
Degree $19$
Signature $[1, 9]$
Discriminant $-2.009\times 10^{38}$
Root discriminant \(103.74\)
Ramified primes $2,3,19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $F_{19}$ (as 19T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 6)
 
gp: K = bnfinit(y^19 - 6, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 6);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 6)
 

\( x^{19} - 6 \) Copy content Toggle raw display

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

Invariants

Degree:  $19$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-200928214500803951615477553772817547264\) \(\medspace = -\,2^{18}\cdot 3^{18}\cdot 19^{19}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(103.74\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{18/19}3^{18/19}19^{359/342}\approx 120.09178438864606$
Ramified primes:   \(2\), \(3\), \(19\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-19}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a^{10}-a^{9}+a^{8}-a^{6}+a^{5}-a^{3}+a^{2}-a-1$, $3a^{18}-4a^{17}-3a^{16}+4a^{15}-8a^{14}-a^{13}+a^{12}-12a^{11}+5a^{10}-3a^{9}-12a^{8}+10a^{7}-16a^{6}-7a^{5}+10a^{4}-27a^{3}+9a^{2}-5a-31$, $4a^{18}-3a^{17}-a^{15}+3a^{14}+2a^{13}-5a^{12}+2a^{11}-5a^{10}+13a^{9}-9a^{8}+7a^{7}-17a^{6}+19a^{5}-11a^{4}+20a^{3}-28a^{2}+17a-11$, $11a^{18}+16a^{17}+12a^{16}-3a^{15}-16a^{14}-18a^{13}-14a^{12}-8a^{11}-2a^{10}+16a^{9}+34a^{8}+40a^{7}+10a^{6}-23a^{5}-48a^{4}-38a^{3}-29a^{2}-12a+11$, $2a^{18}-3a^{17}+8a^{16}+5a^{15}+a^{14}-2a^{13}+22a^{12}-11a^{11}+5a^{10}+24a^{9}-11a^{8}+9a^{7}+17a^{6}+11a^{5}-12a^{4}+31a^{3}+25a^{2}-36a+61$, $5a^{18}-5a^{16}-2a^{15}+6a^{14}+7a^{13}-2a^{12}-12a^{11}+2a^{10}+9a^{9}+7a^{8}-5a^{7}-21a^{6}+9a^{5}+15a^{4}+a^{3}-15a^{2}-26a+19$, $3a^{18}-a^{17}-a^{16}-2a^{15}+8a^{13}-9a^{12}+9a^{11}-18a^{10}+22a^{9}-14a^{8}+13a^{7}-18a^{6}+6a^{5}+5a^{4}+5a^{3}-5a^{2}-8a-1$, $185a^{18}-126a^{17}+22a^{16}+110a^{15}-242a^{14}+335a^{13}-352a^{12}+274a^{11}-109a^{10}-119a^{9}+365a^{8}-564a^{7}+644a^{6}-561a^{5}+313a^{4}+68a^{3}-513a^{2}+918a-1151$, $204a^{18}+141a^{17}+23a^{16}-68a^{15}-122a^{14}-111a^{13}-152a^{12}-165a^{11}-118a^{10}+76a^{9}+403a^{8}+571a^{7}+469a^{6}-80a^{5}-709a^{4}-1028a^{3}-772a^{2}+81a+841$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 929704933317 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 929704933317 \cdot 1}{2\cdot\sqrt{200928214500803951615477553772817547264}}\cr\approx \mathstrut & 1.00102169086913 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^19 - 6)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^19 - 6, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^19 - 6);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 6);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_{19}$ (as 19T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 342
The 19 conjugacy class representatives for $F_{19}$
Character table for $F_{19}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.9.0.1}{9} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.3.0.1}{3} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ R ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $18{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{9}{,}\,{\href{/padicField/37.1.0.1}{1} }$ $18{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.9.0.1}{9} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $18{,}\,{\href{/padicField/53.1.0.1}{1} }$ $18{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.19.18.1$x^{19} + 2$$19$$1$$18$$F_{19}$$[\ ]_{19}^{18}$
\(3\) Copy content Toggle raw display 3.19.18.1$x^{19} + 3$$19$$1$$18$$F_{19}$$[\ ]_{19}^{18}$
\(19\) Copy content Toggle raw display 19.19.19.10$x^{19} + 19 x + 19$$19$$1$$19$$F_{19}$$[19/18]_{18}$