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Properties

Label 19.7.832...904.1
Degree $19$
Signature $(7, 6)$
Discriminant $8.322\times 10^{30}$
Root discriminant \(42.40\)
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 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - x^2 + 4*x - 1)
 
Copy content gp:K = bnfinit(y^19 - 3*y^18 - y^17 + 15*y^16 - 22*y^15 - 6*y^14 + 59*y^13 - 70*y^12 - 6*y^11 + 106*y^10 - 107*y^9 - y^8 + 92*y^7 - 72*y^6 - y^5 + 35*y^4 - 19*y^3 - y^2 + 4*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - x^2 + 4*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - x^2 + 4*x - 1)
 

\( x^{19} - 3 x^{18} - x^{17} + 15 x^{16} - 22 x^{15} - 6 x^{14} + 59 x^{13} - 70 x^{12} - 6 x^{11} + \cdots - 1 \) 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:  $(7, 6)$
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:   \(8322004363062341150231840635904\) \(\medspace = 2^{10}\cdot 2746819\cdot 2958679616604904629509\) 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:  \(42.40\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}2746819^{1/2}2958679616604904629509^{1/2}\approx 254981683825377.03$
Ramified primes:   \(2\), \(2746819\), \(2958679616604904629509\) 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:  $\Q(\sqrt{81269\!\cdots\!81871}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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:  $12$
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:   $a$, $6a^{18}-16a^{17}-11a^{16}+85a^{15}-103a^{14}-65a^{13}+320a^{12}-308a^{11}-117a^{10}+554a^{9}-436a^{8}-114a^{7}+443a^{6}-249a^{5}-58a^{4}+137a^{3}-46a^{2}-11a+6$, $a^{18}-2a^{17}-3a^{16}+12a^{15}-10a^{14}-16a^{13}+43a^{12}-27a^{11}-33a^{10}+73a^{9}-34a^{8}-35a^{7}+57a^{6}-15a^{5}-16a^{4}+19a^{3}-a+2$, $9a^{18}-22a^{17}-22a^{16}+125a^{15}-128a^{14}-135a^{13}+472a^{12}-369a^{11}-292a^{10}+843a^{9}-514a^{8}-340a^{7}+712a^{6}-287a^{5}-191a^{4}+246a^{3}-51a^{2}-40a+19$, $a^{18}-2a^{17}-4a^{16}+13a^{15}-4a^{14}-26a^{13}+37a^{12}+5a^{11}-60a^{10}+46a^{9}+38a^{8}-72a^{7}+7a^{6}+58a^{5}-28a^{4}-19a^{3}+17a^{2}-4$, $8a^{18}-22a^{17}-12a^{16}+112a^{15}-147a^{14}-65a^{13}+416a^{12}-442a^{11}-92a^{10}+702a^{9}-619a^{8}-74a^{7}+551a^{6}-351a^{5}-46a^{4}+176a^{3}-66a^{2}-13a+11$, $3a^{18}-11a^{17}+4a^{16}+43a^{15}-93a^{14}+38a^{13}+154a^{12}-297a^{11}+153a^{10}+218a^{9}-419a^{8}+217a^{7}+131a^{6}-236a^{5}+94a^{4}+35a^{3}-41a^{2}+6a+2$, $7a^{18}-19a^{17}-12a^{16}+100a^{15}-125a^{14}-71a^{13}+379a^{12}-378a^{11}-123a^{10}+660a^{9}-543a^{8}-115a^{7}+535a^{6}-321a^{5}-59a^{4}+172a^{3}-64a^{2}-13a+9$, $8a^{18}-20a^{17}-20a^{16}+117a^{15}-120a^{14}-133a^{13}+460a^{12}-358a^{11}-300a^{10}+855a^{9}-533a^{8}-341a^{7}+743a^{6}-326a^{5}-179a^{4}+251a^{3}-59a^{2}-39a+19$, $17a^{18}-43a^{17}-36a^{16}+234a^{15}-264a^{14}-208a^{13}+874a^{12}-777a^{11}-401a^{10}+1516a^{9}-1082a^{8}-425a^{7}+1227a^{6}-602a^{5}-240a^{4}+410a^{3}-105a^{2}-60a+32$, $3a^{18}-5a^{17}-9a^{16}+31a^{15}-28a^{14}-42a^{13}+116a^{12}-80a^{11}-75a^{10}+206a^{9}-115a^{8}-67a^{7}+172a^{6}-75a^{5}-32a^{4}+62a^{3}-21a^{2}-9a+7$, $4a^{18}-11a^{17}-5a^{16}+51a^{15}-71a^{14}-12a^{13}+168a^{12}-210a^{11}+27a^{10}+228a^{9}-266a^{8}+69a^{7}+111a^{6}-118a^{5}+35a^{4}+13a^{3}-10a^{2}+2a$ 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:  \( 409832342.529 \) (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)
 
Unit signature rank:  \( 7 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{6}\cdot 409832342.529 \cdot 1}{2\cdot\sqrt{8322004363062341150231840635904}}\cr\approx \mathstrut & 0.559437330606 \end{aligned}\] (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 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - 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)))]
 
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 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^19 - 3*x^18 - x^17 + 15*x^16 - 22*x^15 - 6*x^14 + 59*x^13 - 70*x^12 - 6*x^11 + 106*x^10 - 107*x^9 - x^8 + 92*x^7 - 72*x^6 - x^5 + 35*x^4 - 19*x^3 - 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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
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(L)]
 
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 ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ $18{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.7.0.1}{7} }$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ $17{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $18{,}\,{\href{/padicField/31.1.0.1}{1} }$ $19$ $16{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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 2.2.2.4a2.1$x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$$2$$2$$4$$D_{4}$$$[2, 2]^{2}$$
2.1.4.6a1.2$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$$[2, 2]^{2}$$
2.11.1.0a1.1$x^{11} + x^{2} + 1$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$
\(2746819\) Copy content Toggle raw display 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 $11$$1$$11$$0$$C_{11}$$$[\ ]^{11}$$
\(295\!\cdots\!509\) Copy content Toggle raw display $\Q_{29\!\cdots\!09}$$x$$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 $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
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)