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Properties

Label 19.3.531...757.1
Degree $19$
Signature $[3, 8]$
Discriminant $5.315\times 10^{40}$
Root discriminant \(139.14\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{19}$ (as 19T8)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^19 - 9*x - 1)
 
Copy content gp:K = bnfinit(y^19 - 9*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 9*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^19 - 9*x - 1)
 

\( x^{19} - 9x - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $19$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[3, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(53151162932776812454737157860046003696757\) \(\medspace = 101\cdot 9221\cdot 1141909\cdot 263051581967\cdot 189994470465354439\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(139.14\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $101^{1/2}9221^{1/2}1141909^{1/2}263051581967^{1/2}189994470465354439^{1/2}\approx 2.3054535981619066e+20$
Ramified primes:   \(101\), \(9221\), \(1141909\), \(263051581967\), \(189994470465354439\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{53151\!\cdots\!96757}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $10$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $a$, $a^{9}-3$, $a^{6}-2$, $7a^{17}-5a^{16}-5a^{14}+16a^{13}-6a^{12}+5a^{11}-13a^{10}+19a^{9}+6a^{8}+10a^{7}-25a^{6}+23a^{5}+2a^{4}+40a^{3}-47a^{2}+12a-1$, $18a^{18}-73a^{17}+111a^{16}-85a^{15}-6a^{14}+105a^{13}-131a^{12}+52a^{11}+82a^{10}-166a^{9}+144a^{8}-63a^{7}+18a^{6}-45a^{5}+75a^{4}-28a^{3}-84a^{2}+90a+12$, $285a^{18}+598a^{17}+706a^{16}+491a^{15}-17a^{14}-641a^{13}-1081a^{12}-1061a^{11}-484a^{10}+483a^{9}+1396a^{8}+1740a^{7}+1155a^{6}-308a^{5}-2056a^{4}-3211a^{3}-2901a^{2}-793a-59$, $397a^{18}-448a^{17}+508a^{16}-565a^{15}+635a^{14}-721a^{13}+814a^{12}-903a^{11}+1020a^{10}-1160a^{9}+1293a^{8}-1440a^{7}+1626a^{6}-1841a^{5}+2035a^{4}-2282a^{3}+2582a^{2}-2893a-356$, $6610a^{18}+6265a^{17}-1480a^{16}-10253a^{15}-9922a^{14}+2558a^{13}+15981a^{12}+14299a^{11}-5021a^{10}-23411a^{9}-19136a^{8}+8012a^{7}+31931a^{6}+25969a^{5}-9437a^{4}-42744a^{3}-38951a^{2}+8154a+1331$, $10150a^{18}-11993a^{17}+7736a^{16}+2668a^{15}-12654a^{14}+20680a^{13}-17327a^{12}+4884a^{11}+13511a^{10}-30603a^{9}+34059a^{8}-19924a^{7}-6454a^{6}+41555a^{5}-56069a^{4}+50739a^{3}-11887a^{2}-43835a-4646$, $763a^{18}+546a^{17}-34a^{16}-411a^{15}-270a^{14}+29a^{13}-156a^{12}-880a^{11}-1253a^{10}-215a^{9}+2016a^{8}+3572a^{7}+2531a^{6}-1169a^{5}-5144a^{4}-5998a^{3}-2335a^{2}+3576a+418$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 11114856383000 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{8}\cdot 11114856383000 \cdot 1}{2\cdot\sqrt{53151162932776812454737157860046003696757}}\cr\approx \mathstrut & 0.468431841109576 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^19 - 9*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^19 - 9*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 9*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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^19 - 9*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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 121645100408832000
The 490 conjugacy class representatives for $S_{19}$
Character table for $S_{19}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ $18{,}\,{\href{/padicField/3.1.0.1}{1} }$ $16{,}\,{\href{/padicField/5.3.0.1}{3} }$ $16{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $19$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ $18{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ $17{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(101\) Copy content Toggle raw display $\Q_{101}$$x + 99$$1$$1$$0$Trivial$$[\ ]$$
101.2.1.0a1.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
101.1.2.1a1.1$x^{2} + 101$$2$$1$$1$$C_2$$$[\ ]_{2}$$
101.14.1.0a1.1$x^{14} - x + 15$$1$$14$$0$$C_{14}$$$[\ ]^{14}$$
\(9221\) Copy content Toggle raw display $\Q_{9221}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
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 $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$
\(1141909\) Copy content Toggle raw display $\Q_{1141909}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $14$$1$$14$$0$$C_{14}$$$[\ ]^{14}$$
\(263051581967\) Copy content Toggle raw display $\Q_{263051581967}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $14$$1$$14$$0$$C_{14}$$$[\ ]^{14}$$
\(189994470465354439\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)