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Properties

Label 19.1.197...516.1
Degree $19$
Signature $[1, 9]$
Discriminant $-1.973\times 10^{29}$
Root discriminant \(34.82\)
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 / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - x - 1)
 
Copy content gp:K = bnfinit(y^19 - 3*y^18 + 2*y^17 + 4*y^16 - 8*y^15 + 14*y^13 - 9*y^12 - 16*y^11 + 21*y^10 + 8*y^9 - 25*y^8 + 2*y^7 + 16*y^6 - 4*y^5 - 8*y^4 + 2*y^3 + 3*y^2 - y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^19 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - x - 1)
 

\( x^{19} - 3 x^{18} + 2 x^{17} + 4 x^{16} - 8 x^{15} + 14 x^{13} - 9 x^{12} - 16 x^{11} + 21 x^{10} + \cdots - 1 \) 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:  $[1, 9]$
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:   \(-197294383575054168405587977516\) \(\medspace = -\,2^{2}\cdot 49323595893763542101396994379\) 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:  \(34.82\)
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:  $2^{2/3}49323595893763542101396994379^{1/2}\approx 352544569446363.8$
Ramified primes:   \(2\), \(49323595893763542101396994379\) 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:  $\Q(\sqrt{-49323\!\cdots\!94379}$)
$\Aut(K/\Q)$:   $C_1$
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:  $9$
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:   $a$, $a^{18}-4a^{17}+6a^{16}-2a^{15}-6a^{14}+6a^{13}+8a^{12}-17a^{11}+a^{10}+20a^{9}-12a^{8}-13a^{7}+15a^{6}+a^{5}-5a^{4}-2a^{3}+3a^{2}-1$, $a^{18}-4a^{17}+6a^{16}-2a^{15}-7a^{14}+9a^{13}+5a^{12}-17a^{11}+4a^{10}+20a^{9}-20a^{8}-6a^{7}+19a^{6}-6a^{5}-5a^{4}+a^{3}+3a^{2}-a-1$, $a^{18}-2a^{17}-a^{16}+6a^{15}-4a^{14}-8a^{13}+14a^{12}+5a^{11}-25a^{10}+5a^{9}+29a^{8}-18a^{7}-21a^{6}+17a^{5}+11a^{4}-9a^{3}-7a^{2}+2a+2$, $a^{18}-4a^{17}+6a^{16}-3a^{15}-5a^{14}+11a^{13}-4a^{12}-10a^{11}+13a^{10}-21a^{8}+25a^{7}+5a^{6}-38a^{5}+24a^{4}+18a^{3}-21a^{2}-2a+6$, $a^{17}-2a^{16}+a^{15}+2a^{14}-5a^{13}+2a^{12}+4a^{11}-3a^{10}-3a^{9}+7a^{8}-5a^{7}-2a^{6}+7a^{5}-8a^{4}-3a^{3}+4a^{2}-2a-2$, $3a^{18}-11a^{17}+11a^{16}+12a^{15}-38a^{14}+17a^{13}+52a^{12}-67a^{11}-35a^{10}+116a^{9}-25a^{8}-116a^{7}+82a^{6}+53a^{5}-71a^{4}-3a^{3}+28a^{2}-5a-6$, $a^{18}+2a^{17}-13a^{16}+13a^{15}+10a^{14}-31a^{13}+11a^{12}+48a^{11}-44a^{10}-53a^{9}+74a^{8}+21a^{7}-66a^{6}+7a^{5}+31a^{4}-3a^{3}-16a^{2}+a+4$, $2a^{18}-7a^{17}+3a^{16}+16a^{15}-23a^{14}-7a^{13}+46a^{12}-20a^{11}-64a^{10}+60a^{9}+56a^{8}-85a^{7}-25a^{6}+69a^{5}+13a^{4}-33a^{3}-12a^{2}+12a+6$ 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:  \( 20856736.7017 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{9}\cdot 20856736.7017 \cdot 1}{2\cdot\sqrt{197294383575054168405587977516}}\cr\approx \mathstrut & 0.716651167422 \end{aligned}\] (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 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - 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 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - 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 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - 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 - 3*x^18 + 2*x^17 + 4*x^16 - 8*x^15 + 14*x^13 - 9*x^12 - 16*x^11 + 21*x^10 + 8*x^9 - 25*x^8 + 2*x^7 + 16*x^6 - 4*x^5 - 8*x^4 + 2*x^3 + 3*x^2 - 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 R $15{,}\,{\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $18{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ $17{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $19$ $19$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

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
\(2\) Copy content Toggle raw display 2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.13.1.0a1.1$x^{13} + x^{4} + x^{3} + x + 1$$1$$13$$0$$C_{13}$$$[\ ]^{13}$$
\(493\!\cdots\!379\) Copy content Toggle raw display $\Q_{49\!\cdots\!79}$$x$$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 $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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