Properties

Label 19.1.207...120.1
Degree $19$
Signature $[1, 9]$
Discriminant $-2.075\times 10^{30}$
Root discriminant \(39.41\)
Ramified primes $2,5,928238261,426273972622159$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{19}$ (as 19T8)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 4*x - 8)
 
gp: K = bnfinit(y^19 - 4*y - 8, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 4*x - 8);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 4*x - 8)
 

\( x^{19} - 4x - 8 \) 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:   \(-2074522739311139777284664197120\) \(\medspace = -\,2^{20}\cdot 5\cdot 928238261\cdot 426273972622159\) 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:  \(39.41\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(928238261\), \(426273972622159\) 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:  $\Q(\sqrt{-19784\!\cdots\!27495}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

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

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

Monogenic:  Not computed
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:   $\frac{1}{2}a^{10}-a-1$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{2}a^{7}+\frac{1}{2}a^{5}-\frac{1}{2}a^{3}+a+1$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}+\frac{1}{4}a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+a^{7}-\frac{1}{2}a^{5}+\frac{1}{2}a^{3}+a^{2}-1$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-a^{2}+1$, $\frac{1}{4}a^{16}+\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-a^{9}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+2a^{2}+2a+1$, $\frac{1}{4}a^{14}-\frac{1}{2}a^{13}+\frac{1}{4}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+a^{4}-\frac{1}{2}a^{3}-1$, $\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{12}+\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+a^{6}+\frac{1}{2}a^{5}+\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+a^{2}+a+1$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{15}+\frac{1}{4}a^{14}+\frac{1}{4}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-a^{10}+a^{8}+\frac{3}{2}a^{7}-a^{6}-\frac{3}{2}a^{5}-\frac{3}{2}a^{4}+2a^{3}+a^{2}+a-3$, $\frac{1}{4}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{15}+\frac{1}{2}a^{14}-\frac{3}{2}a^{13}+2a^{12}-\frac{1}{2}a^{11}-a^{10}+\frac{5}{2}a^{9}-a^{8}+\frac{1}{2}a^{6}+2a^{5}-3a^{4}+a^{3}+3a^{2}-6a+3$ 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:  \( 110775507.502 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 110775507.502 \cdot 1}{2\cdot\sqrt{2074522739311139777284664197120}}\cr\approx \mathstrut & 1.17382529450 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^19 - 4*x - 8)
 
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 - 4*x - 8, 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 - 4*x - 8);
 
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 - 4*x - 8);
 
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

$S_{19}$ (as 19T8):

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 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 $\Q$.
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 $16{,}\,{\href{/padicField/3.3.0.1}{3} }$ R $18{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $19$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $19$ $19$ ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ $15{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $19$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.8.0.1}{8} }$ $19$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.18.20.1$x^{18} + 2 x^{6} + 2 x^{5} + 2 x^{3} + 2$$18$$1$$20$18T588$[10/9, 10/9, 10/9, 10/9, 10/9, 10/9, 4/3, 4/3]_{9}^{6}$
\(5\) Copy content Toggle raw display 5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.17.0.1$x^{17} + 3 x^{2} + 2 x + 3$$1$$17$$0$$C_{17}$$[\ ]^{17}$
\(928238261\) Copy content Toggle raw display $\Q_{928238261}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{928238261}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(426273972622159\) Copy content Toggle raw display $\Q_{426273972622159}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{426273972622159}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{426273972622159}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$