↑

Properties

Label 18.18.141...064.2
Degree $18$
Signature $[18, 0]$
Discriminant $1.412\times 10^{29}$
Root discriminant \(41.63\)
Ramified primes $2,257,43237$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_4^3.S_4$ (as 18T883)

Related objects

Downloads

Learn more

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^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 1)
 
Copy content gp:K = bnfinit(y^18 - 20*y^16 + 153*y^14 - 573*y^12 + 1136*y^10 - 1219*y^8 + 707*y^6 - 211*y^4 + 28*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 1)
 

\( x^{18} - 20x^{16} + 153x^{14} - 573x^{12} + 1136x^{10} - 1219x^{8} + 707x^{6} - 211x^{4} + 28x^{2} - 1 \) 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:  $18$
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:  $[18, 0]$
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:   \(141204899984457152700528984064\) \(\medspace = 2^{18}\cdot 257^{6}\cdot 43237^{2}\) 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:  \(41.63\)
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:  $2^{63/32}257^{1/2}43237^{1/2}\approx 13048.095961157862$
Ramified primes:   \(2\), \(257\), \(43237\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{227}a^{16}-\frac{90}{227}a^{14}+\frac{97}{227}a^{12}-\frac{99}{227}a^{10}-\frac{106}{227}a^{8}+\frac{72}{227}a^{6}-\frac{20}{227}a^{4}+\frac{54}{227}a^{2}+\frac{107}{227}$, $\frac{1}{227}a^{17}-\frac{90}{227}a^{15}+\frac{97}{227}a^{13}-\frac{99}{227}a^{11}-\frac{106}{227}a^{9}+\frac{72}{227}a^{7}-\frac{20}{227}a^{5}+\frac{54}{227}a^{3}+\frac{107}{227}a$ 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:  Not computed
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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $17$
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:   $a$, $\frac{1545}{227}a^{17}-\frac{30090}{227}a^{15}+\frac{220689}{227}a^{13}-\frac{771076}{227}a^{11}+\frac{1361216}{227}a^{9}-\frac{1202636}{227}a^{7}+\frac{511176}{227}a^{5}-\frac{90452}{227}a^{3}+\frac{4372}{227}a$, $\frac{1186}{227}a^{17}-\frac{23204}{227}a^{15}+\frac{171338}{227}a^{13}-\frac{604556}{227}a^{11}+\frac{1080789}{227}a^{9}-\frac{963575}{227}a^{7}+\frac{399181}{227}a^{5}-\frac{60352}{227}a^{3}+\frac{690}{227}a$, $\frac{391}{227}a^{16}-\frac{7496}{227}a^{14}+\frac{53590}{227}a^{12}-\frac{179222}{227}a^{10}+\frac{292698}{227}a^{8}-\frac{224272}{227}a^{6}+\frac{75489}{227}a^{4}-\frac{10666}{227}a^{2}+\frac{750}{227}$, $a^{17}-19a^{15}+134a^{13}-439a^{11}+697a^{9}-522a^{7}+185a^{5}-26a^{3}+2a$, $\frac{132}{227}a^{17}-\frac{2346}{227}a^{15}+\frac{14620}{227}a^{13}-\frac{36449}{227}a^{11}+\frac{22555}{227}a^{9}+\frac{37652}{227}a^{7}-\frac{48494}{227}a^{5}+\frac{17797}{227}a^{3}-\frac{2447}{227}a$, $\frac{1007}{227}a^{17}-\frac{19352}{227}a^{15}+\frac{138993}{227}a^{13}-\frac{469476}{227}a^{11}+\frac{785141}{227}a^{9}-\frac{640730}{227}a^{7}+\frac{253395}{227}a^{5}-\frac{44821}{227}a^{3}+\frac{2421}{227}a$, $\frac{296}{227}a^{16}-\frac{5529}{227}a^{14}+\frac{37792}{227}a^{12}-\frac{116018}{227}a^{10}+\frac{157034}{227}a^{8}-\frac{68126}{227}a^{6}-\frac{15000}{227}a^{4}+\frac{12806}{227}a^{2}-\frac{1470}{227}$, $\frac{132}{227}a^{17}-\frac{640}{227}a^{16}-\frac{2573}{227}a^{15}+\frac{12427}{227}a^{14}+\frac{18933}{227}a^{13}-\frac{90682}{227}a^{12}-\frac{66867}{227}a^{11}+\frac{313968}{227}a^{10}+\frac{122208}{227}a^{9}-\frac{544833}{227}a^{8}-\frac{120567}{227}a^{7}+\frac{465351}{227}a^{6}+\frac{70000}{227}a^{5}-\frac{185144}{227}a^{4}-\frac{24198}{227}a^{3}+\frac{28092}{227}a^{2}+\frac{3455}{227}a-\frac{380}{227}$, $a+1$, $\frac{132}{227}a^{17}-\frac{2573}{227}a^{15}+\frac{18933}{227}a^{13}-\frac{66867}{227}a^{11}+\frac{122208}{227}a^{9}-\frac{120567}{227}a^{7}+\frac{70000}{227}a^{5}-\frac{24198}{227}a^{3}+\frac{3228}{227}a-1$, $\frac{1545}{227}a^{17}-\frac{810}{227}a^{16}-\frac{30090}{227}a^{15}+\frac{15696}{227}a^{14}+\frac{220689}{227}a^{13}-\frac{114209}{227}a^{12}-\frac{771076}{227}a^{11}+\frac{393904}{227}a^{10}+\frac{1361216}{227}a^{9}-\frac{680719}{227}a^{8}-\frac{1202636}{227}a^{7}+\frac{581139}{227}a^{6}+\frac{511176}{227}a^{5}-\frac{235543}{227}a^{4}-\frac{90452}{227}a^{3}+\frac{38888}{227}a^{2}+\frac{4145}{227}a-\frac{1545}{227}$, $\frac{523}{227}a^{16}-a^{15}-\frac{9842}{227}a^{14}+19a^{13}+\frac{68210}{227}a^{12}-134a^{11}-\frac{215671}{227}a^{10}+439a^{9}+\frac{315253}{227}a^{8}-697a^{7}-\frac{186620}{227}a^{6}+522a^{5}+\frac{26995}{227}a^{4}-185a^{3}+\frac{7131}{227}a^{2}+25a-\frac{1470}{227}$, $\frac{296}{227}a^{17}-\frac{5529}{227}a^{15}+\frac{37792}{227}a^{13}-\frac{116018}{227}a^{11}+\frac{157034}{227}a^{9}-\frac{68126}{227}a^{7}-\frac{15000}{227}a^{5}+\frac{12806}{227}a^{3}+a^{2}-\frac{1243}{227}a-1$, $\frac{1621}{227}a^{17}+\frac{1098}{227}a^{16}-\frac{31028}{227}a^{15}-\frac{21186}{227}a^{14}+\frac{221251}{227}a^{13}+\frac{153041}{227}a^{12}-\frac{736605}{227}a^{11}-\frac{520934}{227}a^{10}+\frac{1193125}{227}a^{9}+\frac{877872}{227}a^{8}-\frac{899567}{227}a^{7}-\frac{712947}{227}a^{6}+\frac{291282}{227}a^{5}+\frac{264287}{227}a^{4}-\frac{32095}{227}a^{3}-\frac{39226}{227}a^{2}-\frac{208}{227}a+\frac{1262}{227}$, $\frac{1623}{227}a^{17}+\frac{1889}{227}a^{16}-\frac{32116}{227}a^{15}-\frac{36988}{227}a^{14}+\frac{241421}{227}a^{13}+\frac{273579}{227}a^{12}-\frac{877089}{227}a^{11}-\frac{969026}{227}a^{10}+\frac{1645778}{227}a^{9}+\frac{1749015}{227}a^{8}-\frac{1590638}{227}a^{7}-\frac{1599861}{227}a^{6}+\frac{750236}{227}a^{5}+\frac{711320}{227}a^{4}-\frac{142309}{227}a^{3}-\frac{131123}{227}a^{2}+\frac{4546}{227}a+\frac{5314}{227}$, $\frac{2247}{227}a^{17}+\frac{1281}{227}a^{16}-\frac{43784}{227}a^{15}-\frac{25171}{227}a^{14}+\frac{321244}{227}a^{13}+\frac{187136}{227}a^{12}-\frac{1121827}{227}a^{11}-\frac{667760}{227}a^{10}+\frac{1972117}{227}a^{9}+\frac{1216453}{227}a^{8}-\frac{1712101}{227}a^{7}-\frac{1119721}{227}a^{6}+\frac{683957}{227}a^{5}+\frac{489670}{227}a^{4}-\frac{99079}{227}a^{3}-\frac{84051}{227}a^{2}+\frac{2079}{227}a+\frac{3591}{227}$ 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:  \( 587034867.509 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 587034867.509 \cdot 1}{2\cdot\sqrt{141204899984457152700528984064}}\cr\approx \mathstrut & 0.204761800581 \end{aligned}\] (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^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 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^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 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^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 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^18 - 20*x^16 + 153*x^14 - 573*x^12 + 1136*x^10 - 1219*x^8 + 707*x^6 - 211*x^4 + 28*x^2 - 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

$S_4^3.S_4$ (as 18T883):

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 solvable group of order 331776
The 165 conjugacy class representatives for $S_4^3.S_4$
Character table for $S_4^3.S_4$

Intermediate fields

3.3.257.1, 9.9.733930477541.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.18.103634579676721781559152395037403098906624.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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 2.3.2.6a1.1$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$$2$$3$$6$$C_6$$$[2]^{3}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(257\) Copy content Toggle raw display $\Q_{257}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{257}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
\(43237\) Copy content Toggle raw display $\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{43237}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
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)