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Properties

Label 19.3.531...861.1
Degree $19$
Signature $[3, 8]$
Discriminant $5.314\times 10^{40}$
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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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 - 9*x - 5)
 
Copy content gp:K = bnfinit(y^19 - 9*y - 5, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 9*x - 5);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^19 - 9*x - 5)
 

\( x^{19} - 9x - 5 \) 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:   \(53143615860726108280854552247929172898861\) \(\medspace = 6219297888079\cdot 8544954240347693873940241859\) 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:  $6219297888079^{1/2}8544954240347693873940241859^{1/2}\approx 2.305289913670862e+20$
Ramified primes:   \(6219297888079\), \(8544954240347693873940241859\) 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:  $\Q(\sqrt{53143\!\cdots\!98861}$)
$\Aut(K/\Q)$:   $C_1$
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:   $a^{10}+3a+1$, $6a^{18}-a^{17}-a^{16}+13a^{15}-13a^{14}+9a^{13}-9a^{12}-13a^{11}+11a^{10}-11a^{9}+14a^{8}+19a^{7}-12a^{6}+19a^{5}-16a^{4}-31a^{3}+23a^{2}-56a-21$, $2a^{17}+2a^{16}-3a^{15}-a^{14}-2a^{13}+3a^{12}+5a^{11}+2a^{10}-2a^{9}-4a^{8}-6a^{7}+3a^{6}+8a^{5}+3a^{4}+a^{3}-24a^{2}-10a+2$, $198a^{18}-114a^{17}+81a^{16}-33a^{15}-10a^{14}-20a^{13}+34a^{12}+5a^{11}-14a^{10}+12a^{9}+7a^{8}-56a^{7}-22a^{6}+87a^{5}+50a^{4}-82a^{3}-51a^{2}+41a-1811$, $293a^{18}-347a^{17}-539a^{16}+213a^{15}+796a^{14}+78a^{13}-999a^{12}-552a^{11}+1056a^{10}+1192a^{9}-845a^{8}-1925a^{7}+234a^{6}+2612a^{5}+868a^{4}-3019a^{3}-2474a^{2}+2821a+1841$, $323a^{18}-62a^{17}-529a^{16}-862a^{15}-827a^{14}-344a^{13}+484a^{12}+1308a^{11}+1707a^{10}+1309a^{9}+84a^{8}-1610a^{7}-2985a^{6}-3205a^{5}-1721a^{4}+1193a^{3}+4437a^{2}+6281a+2383$, $271a^{18}-140a^{17}-47a^{16}+222a^{15}-290a^{14}+199a^{13}+27a^{12}-295a^{11}+457a^{10}-392a^{9}+99a^{8}+318a^{7}-659a^{6}+695a^{5}-344a^{4}-273a^{3}+896a^{2}-1147a-1657$, $109a^{18}+213a^{17}+282a^{16}+285a^{15}+214a^{14}+84a^{13}-58a^{12}-161a^{11}-196a^{10}-168a^{9}-145a^{8}-198a^{7}-402a^{6}-732a^{5}-1082a^{4}-1242a^{3}-992a^{2}-213a+64$, $142a^{18}+199a^{17}+169a^{16}+59a^{15}-162a^{14}-443a^{13}-539a^{12}-369a^{11}-52a^{10}+321a^{9}+721a^{8}+790a^{7}+256a^{6}-620a^{5}-1342a^{4}-1807a^{3}-1708a^{2}-570a-4$, $678a^{18}+3646a^{17}-4740a^{16}+467a^{15}+5482a^{14}-6405a^{13}-42a^{12}+8145a^{11}-8584a^{10}-1066a^{9}+11968a^{8}-11400a^{7}-2931a^{6}+17425a^{5}-14958a^{4}-6108a^{3}+25192a^{2}-19309a-17384$ 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:  \( 27546004978300 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{8}\cdot 27546004978300 \cdot 1}{2\cdot\sqrt{53143615860726108280854552247929172898861}}\cr\approx \mathstrut & 1.16099943855555 \end{aligned}\] (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 - 5) 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 - 5, 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 - 5); 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 - 5); 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 ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ $18{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ $16{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.8.0.1}{8} }$ ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $16{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $18{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ $15{,}\,{\href{/padicField/47.4.0.1}{4} }$ $15{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(6219297888079\) Copy content Toggle raw display $\Q_{6219297888079}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $10$$1$$10$$0$$C_{10}$$$[\ ]^{10}$$
\(854\!\cdots\!859\) Copy content Toggle raw display $\Q_{85\!\cdots\!59}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{85\!\cdots\!59}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $15$$1$$15$$0$$C_{15}$$$[\ ]^{15}$$

Spectrum of ring of integers

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