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Properties

Label 18.0.119...867.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.198\times 10^{18}$
Root discriminant \(10.10\)
Ramified primes $31,67$
Class number $1$
Class group trivial
Galois group $C_6\wr S_3$ (as 18T284)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1)
 
gp: K = bnfinit(y^18 - 4*y^17 + 9*y^16 - 14*y^15 + 17*y^14 - 16*y^13 + 15*y^12 - 23*y^11 + 45*y^10 - 65*y^9 + 62*y^8 - 40*y^7 + 29*y^6 - 39*y^5 + 49*y^4 - 41*y^3 + 22*y^2 - 7*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1)
 

\( x^{18} - 4 x^{17} + 9 x^{16} - 14 x^{15} + 17 x^{14} - 16 x^{13} + 15 x^{12} - 23 x^{11} + 45 x^{10} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1198241002273018867\) \(\medspace = -\,31^{6}\cdot 67^{5}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(10.10\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $31^{1/2}67^{5/6}\approx 185.1014620777746$
Ramified primes:   \(31\), \(67\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-67}) \)
$\card{ \Aut(K/\Q) }$:  $6$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6457}a^{17}+\frac{2237}{6457}a^{16}+\frac{2494}{6457}a^{15}-\frac{2722}{6457}a^{14}+\frac{1880}{6457}a^{13}+\frac{3100}{6457}a^{12}-\frac{617}{6457}a^{11}-\frac{922}{6457}a^{10}+\frac{83}{6457}a^{9}-\frac{1315}{6457}a^{8}-\frac{2461}{6457}a^{7}-\frac{863}{6457}a^{6}+\frac{286}{587}a^{5}-\frac{897}{6457}a^{4}-\frac{2001}{6457}a^{3}-\frac{284}{587}a^{2}-\frac{134}{587}a+\frac{2743}{6457}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{80980}{6457}a^{17}-\frac{270069}{6457}a^{16}+\frac{544462}{6457}a^{15}-\frac{760420}{6457}a^{14}+\frac{851578}{6457}a^{13}-\frac{707516}{6457}a^{12}+\frac{722790}{6457}a^{11}-\frac{1370153}{6457}a^{10}+\frac{2711543}{6457}a^{9}-\frac{3409152}{6457}a^{8}+\frac{2657352}{6457}a^{7}-\frac{1389884}{6457}a^{6}+\frac{126400}{587}a^{5}-\frac{2225475}{6457}a^{4}+\frac{2438181}{6457}a^{3}-\frac{148171}{587}a^{2}+\frac{59249}{587}a-\frac{121800}{6457}$, $a$, $\frac{86349}{6457}a^{17}-\frac{282550}{6457}a^{16}+\frac{562301}{6457}a^{15}-\frac{775561}{6457}a^{14}+\frac{859464}{6457}a^{13}-\frac{703305}{6457}a^{12}+\frac{729015}{6457}a^{11}-\frac{1419508}{6457}a^{10}+\frac{2802035}{6457}a^{9}-\frac{3457085}{6457}a^{8}+\frac{2616523}{6457}a^{7}-\frac{1329092}{6457}a^{6}+\frac{128690}{587}a^{5}-\frac{2295573}{6457}a^{4}+\frac{2452174}{6457}a^{3}-\frac{143832}{587}a^{2}+\frac{54769}{587}a-\frac{103679}{6457}$, $\frac{23316}{6457}a^{17}-\frac{92152}{6457}a^{16}+\frac{204986}{6457}a^{15}-\frac{310235}{6457}a^{14}+\frac{365556}{6457}a^{13}-\frac{329365}{6457}a^{12}+\frac{303703}{6457}a^{11}-\frac{492731}{6457}a^{10}+\frac{1005420}{6457}a^{9}-\frac{1436158}{6457}a^{8}+\frac{1300540}{6457}a^{7}-\frac{737794}{6457}a^{6}+\frac{47603}{587}a^{5}-\frac{833182}{6457}a^{4}+\frac{1081285}{6457}a^{3}-\frac{75520}{587}a^{2}+\frac{33716}{587}a-\frac{78281}{6457}$, $\frac{63887}{6457}a^{17}-\frac{217100}{6457}a^{16}+\frac{446779}{6457}a^{15}-\frac{633276}{6457}a^{14}+\frac{717630}{6457}a^{13}-\frac{606362}{6457}a^{12}+\frac{608664}{6457}a^{11}-\frac{1113664}{6457}a^{10}+\frac{2216175}{6457}a^{9}-\frac{2853372}{6457}a^{8}+\frac{2307192}{6457}a^{7}-\frac{1231445}{6457}a^{6}+\frac{103445}{587}a^{5}-\frac{1815181}{6457}a^{4}+\frac{2064039}{6457}a^{3}-\frac{129465}{587}a^{2}+\frac{52780}{587}a-\frac{110708}{6457}$, $\frac{77169}{6457}a^{17}-\frac{278493}{6457}a^{16}+\frac{576817}{6457}a^{15}-\frac{827847}{6457}a^{14}+\frac{938109}{6457}a^{13}-\frac{802161}{6457}a^{12}+\frac{775485}{6457}a^{11}-\frac{1420675}{6457}a^{10}+\frac{2860134}{6457}a^{9}-\frac{3769911}{6457}a^{8}+\frac{3060993}{6457}a^{7}-\frac{1607142}{6457}a^{6}+\frac{130035}{587}a^{5}-\frac{2377729}{6457}a^{4}+\frac{2741754}{6457}a^{3}-\frac{172342}{587}a^{2}+\frac{69799}{587}a-\frac{140861}{6457}$, $\frac{172}{587}a^{17}+\frac{279}{587}a^{16}-\frac{129}{587}a^{15}+\frac{242}{587}a^{14}+\frac{510}{587}a^{13}-\frac{383}{587}a^{12}+\frac{1884}{587}a^{11}-\frac{94}{587}a^{10}-\frac{986}{587}a^{9}+\frac{2750}{587}a^{8}+\frac{4044}{587}a^{7}-\frac{6382}{587}a^{6}+\frac{3420}{587}a^{5}+\frac{4206}{587}a^{4}-\frac{190}{587}a^{3}-\frac{3745}{587}a^{2}+\frac{4165}{587}a-\frac{1326}{587}$, $\frac{74523}{6457}a^{17}-\frac{244241}{6457}a^{16}+\frac{486349}{6457}a^{15}-\frac{670022}{6457}a^{14}+\frac{741809}{6457}a^{13}-\frac{604204}{6457}a^{12}+\frac{625935}{6457}a^{11}-\frac{1221642}{6457}a^{10}+\frac{2420978}{6457}a^{9}-\frac{2989447}{6457}a^{8}+\frac{2257018}{6457}a^{7}-\frac{1131604}{6457}a^{6}+\frac{109377}{587}a^{5}-\frac{1973652}{6457}a^{4}+\frac{2121788}{6457}a^{3}-\frac{124104}{587}a^{2}+\frac{46335}{587}a-\frac{83058}{6457}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 22.3834819773 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 22.3834819773 \cdot 1}{2\cdot\sqrt{1198241002273018867}}\cr\approx \mathstrut & 0.156043031051 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*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

$C_6\wr S_3$ (as 18T284):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 1296
The 98 conjugacy class representatives for $C_6\wr S_3$
Character table for $C_6\wr S_3$

Intermediate fields

3.1.31.1, 6.0.64387.1, 9.3.133731799.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 18 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $18$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{3}$ $18$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ R ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{3}$ $18$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(31\) Copy content Toggle raw display 31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.6.0.1$x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.5.1$x^{6} + 201$$6$$1$$5$$C_6$$[\ ]_{6}$