Properties

Label 19.3.146...972.1
Degree $19$
Signature $[3, 8]$
Discriminant $1.465\times 10^{26}$
Root discriminant \(23.83\)
Ramified primes see page
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 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1)
 
gp: K = bnfinit(y^19 - 2*y^18 + 7*y^16 - 10*y^15 - y^14 + 19*y^13 - 19*y^12 - 9*y^11 + 33*y^10 - 19*y^9 - 20*y^8 + 32*y^7 - 9*y^6 - 17*y^5 + 17*y^4 - 3*y^3 - 5*y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1)
 

\( x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} - 19 x^{9} - 20 x^{8} + 32 x^{7} - 9 x^{6} - 17 x^{5} + 17 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 1 \) 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:  $[3, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(146525077331272123613812972\) \(\medspace = 2^{2}\cdot 139\cdot 126311\cdot 1021043\cdot 2043393285469\) 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:  \(23.83\)
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:   \(2\), \(139\), \(126311\), \(1021043\), \(2043393285469\) 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:  $\Q(\sqrt{36631\!\cdots\!53243}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $10$
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:   $a$, $a^{18}-2a^{17}-a^{16}+8a^{15}-8a^{14}-7a^{13}+21a^{12}-10a^{11}-21a^{10}+31a^{9}-32a^{7}+20a^{6}+11a^{5}-17a^{4}+6a^{3}+3a^{2}-5a+1$, $5a^{18}-13a^{17}+44a^{15}-64a^{14}-16a^{13}+133a^{12}-113a^{11}-89a^{10}+233a^{9}-85a^{8}-185a^{7}+210a^{6}+14a^{5}-145a^{4}+84a^{3}+20a^{2}-40a+12$, $3a^{17}-3a^{16}-5a^{15}+18a^{14}-8a^{13}-24a^{12}+38a^{11}+a^{10}-56a^{9}+41a^{8}+30a^{7}-65a^{6}+7a^{5}+34a^{4}-26a^{3}-4a^{2}+10a-2$, $a^{18}+a^{17}-3a^{16}+3a^{15}+8a^{14}-11a^{13}-a^{12}+21a^{11}-16a^{10}-18a^{9}+30a^{8}-5a^{7}-33a^{6}+12a^{5}+6a^{4}-18a^{3}+3a-3$, $5a^{18}-6a^{17}-8a^{16}+33a^{15}-20a^{14}-41a^{13}+78a^{12}-14a^{11}-101a^{10}+99a^{9}+35a^{8}-130a^{7}+45a^{6}+58a^{5}-67a^{4}+8a^{3}+22a^{2}-13a+2$, $7a^{18}-6a^{17}-13a^{16}+41a^{15}-13a^{14}-61a^{13}+84a^{12}+14a^{11}-135a^{10}+83a^{9}+84a^{8}-147a^{7}-a^{6}+85a^{5}-54a^{4}-19a^{3}+24a^{2}-3a-1$, $9a^{18}-9a^{17}-17a^{16}+57a^{15}-23a^{14}-84a^{13}+126a^{12}+9a^{11}-193a^{10}+138a^{9}+112a^{8}-226a^{7}+23a^{6}+133a^{5}-93a^{4}-21a^{3}+43a^{2}-9a-2$, $a^{18}-5a^{16}+8a^{15}+4a^{14}-24a^{13}+20a^{12}+17a^{11}-47a^{10}+17a^{9}+43a^{8}-53a^{7}-4a^{6}+42a^{5}-24a^{4}-6a^{3}+16a^{2}-6a+2$, $2a^{18}+a^{17}-6a^{16}+6a^{15}+14a^{14}-23a^{13}-4a^{12}+45a^{11}-35a^{10}-39a^{9}+70a^{8}-7a^{7}-76a^{6}+37a^{5}+26a^{4}-42a^{3}+4a^{2}+12a-6$ 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:  \( 737873.586636 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{8}\cdot 737873.586636 \cdot 1}{2\cdot\sqrt{146525077331272123613812972}}\cr\approx \mathstrut & 0.592277074992 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1)
 
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 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1, 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 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1);
 
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 - 2*x^18 + 7*x^16 - 10*x^15 - x^14 + 19*x^13 - 19*x^12 - 9*x^11 + 33*x^10 - 19*x^9 - 20*x^8 + 32*x^7 - 9*x^6 - 17*x^5 + 17*x^4 - 3*x^3 - 5*x^2 + 4*x - 1);
 
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 ${\href{/padicField/3.11.0.1}{11} }{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/17.1.0.1}{1} }$ $15{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ $17{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ $19$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ $19$

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 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.13.0.1$x^{13} + x^{4} + x^{3} + x + 1$$1$$13$$0$$C_{13}$$[\ ]^{13}$
\(139\) Copy content Toggle raw display $\Q_{139}$$x + 137$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 137$$1$$1$$0$Trivial$[\ ]$
139.2.1.1$x^{2} + 278$$2$$1$$1$$C_2$$[\ ]_{2}$
139.4.0.1$x^{4} + 7 x^{2} + 96 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
139.11.0.1$x^{11} + 7 x + 137$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(126311\) Copy content Toggle raw display $\Q_{126311}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(1021043\) Copy content Toggle raw display $\Q_{1021043}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(2043393285469\) Copy content Toggle raw display $\Q_{2043393285469}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$