Properties

Label 18.0.315...808.6
Degree $18$
Signature $[0, 9]$
Discriminant $-3.152\times 10^{29}$
Root discriminant \(43.53\)
Ramified primes $2,3,7$
Class number $27$ (GRH)
Class group [3, 3, 3] (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 87*x^12 + 8991*x^6 + 19683)
 
gp: K = bnfinit(y^18 - 87*y^12 + 8991*y^6 + 19683, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 87*x^12 + 8991*x^6 + 19683);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 87*x^12 + 8991*x^6 + 19683)
 

\( x^{18} - 87x^{12} + 8991x^{6} + 19683 \) 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:   \(-315164892649262158461997559808\) \(\medspace = -\,2^{12}\cdot 3^{33}\cdot 7^{12}\) 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:  \(43.53\)
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:  $2^{2/3}3^{11/6}7^{2/3}\approx 43.531903466499294$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{54}a^{9}-\frac{1}{6}a^{6}+\frac{7}{18}a^{3}-\frac{1}{2}$, $\frac{1}{54}a^{10}-\frac{1}{6}a^{7}+\frac{7}{18}a^{4}-\frac{1}{2}a$, $\frac{1}{162}a^{11}-\frac{1}{18}a^{8}+\frac{25}{54}a^{5}-\frac{1}{6}a^{2}$, $\frac{1}{43740}a^{12}-\frac{227}{1458}a^{6}-\frac{1}{2}a^{3}+\frac{77}{180}$, $\frac{1}{131220}a^{13}+\frac{259}{4374}a^{7}-\frac{1}{2}a^{4}-\frac{103}{540}a$, $\frac{1}{131220}a^{14}-\frac{227}{4374}a^{8}-\frac{1}{2}a^{5}+\frac{257}{540}a^{2}$, $\frac{1}{393660}a^{15}+\frac{8}{6561}a^{9}+\frac{347}{1620}a^{3}-\frac{1}{2}$, $\frac{1}{1180980}a^{16}+\frac{259}{39366}a^{10}-\frac{1}{6}a^{7}+\frac{977}{4860}a^{4}$, $\frac{1}{1180980}a^{17}+\frac{8}{19683}a^{11}-\frac{1273}{4860}a^{5}-\frac{1}{2}a^{2}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (assuming GRH)

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:   \( -\frac{5}{78732} a^{15} + \frac{43}{6561} a^{9} - \frac{187}{324} a^{3} + \frac{1}{2} \)  (order $6$) 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:   $\frac{1}{393660}a^{15}-\frac{13}{43740}a^{12}+\frac{8}{6561}a^{9}+\frac{35}{1458}a^{6}-\frac{463}{1620}a^{3}-\frac{11}{180}$, $\frac{1}{21870}a^{12}+\frac{16}{729}a^{6}-\frac{13}{90}$, $\frac{19}{196830}a^{17}-\frac{5}{39366}a^{16}-\frac{1}{10935}a^{15}+\frac{19}{65610}a^{14}+\frac{11}{65610}a^{13}-\frac{1}{1620}a^{12}-\frac{121}{13122}a^{11}+\frac{86}{6561}a^{10}+\frac{17}{1458}a^{9}-\frac{121}{4374}a^{8}-\frac{67}{2187}a^{7}+\frac{1}{27}a^{6}+\frac{394}{405}a^{5}-\frac{187}{162}a^{4}-\frac{47}{45}a^{3}+\frac{259}{135}a^{2}+\frac{487}{270}a-\frac{41}{20}$, $\frac{19}{196830}a^{17}-\frac{1}{14580}a^{16}-\frac{7}{98415}a^{15}+\frac{19}{65610}a^{14}-\frac{11}{131220}a^{13}+\frac{1}{2916}a^{12}-\frac{121}{13122}a^{11}+\frac{1}{243}a^{10}+\frac{19}{6561}a^{9}-\frac{121}{4374}a^{8}+\frac{67}{4374}a^{7}-\frac{1}{486}a^{6}+\frac{394}{405}a^{5}-\frac{91}{180}a^{4}-\frac{583}{810}a^{3}+\frac{259}{135}a^{2}-\frac{1297}{540}a+\frac{29}{12}$, $\frac{1}{295245}a^{17}-\frac{83}{590490}a^{16}+\frac{1}{21870}a^{15}+\frac{47}{65610}a^{14}+\frac{127}{131220}a^{13}-\frac{2}{10935}a^{12}-\frac{179}{39366}a^{11}+\frac{17}{39366}a^{10}-\frac{11}{729}a^{9}-\frac{197}{4374}a^{8}+\frac{44}{2187}a^{7}+\frac{179}{729}a^{6}+\frac{1189}{2430}a^{5}-\frac{2251}{2430}a^{4}-\frac{83}{90}a^{3}+\frac{167}{135}a^{2}+\frac{689}{540}a-\frac{64}{45}$, $\frac{16}{295245}a^{17}-\frac{23}{295245}a^{16}-\frac{1}{131220}a^{14}-\frac{1}{3645}a^{13}+\frac{1}{21870}a^{12}-\frac{191}{39366}a^{11}+\frac{229}{39366}a^{10}-\frac{8}{2187}a^{8}+\frac{17}{486}a^{7}+\frac{16}{729}a^{6}+\frac{512}{1215}a^{5}-\frac{1607}{2430}a^{4}+\frac{193}{540}a^{2}-\frac{49}{30}a-\frac{283}{90}$, $\frac{103}{295245}a^{17}-\frac{19}{65610}a^{16}+\frac{5}{19683}a^{15}-\frac{7}{21870}a^{13}-\frac{592}{19683}a^{11}+\frac{121}{4374}a^{10}-\frac{172}{6561}a^{9}+\frac{19}{1458}a^{7}+\frac{3971}{1215}a^{5}-\frac{394}{135}a^{4}+\frac{187}{81}a^{3}-\frac{112}{45}a+5$, $\frac{103}{590490}a^{17}+\frac{19}{65610}a^{16}+\frac{35}{78732}a^{15}+\frac{13}{21870}a^{14}+\frac{7}{21870}a^{13}-\frac{296}{19683}a^{11}-\frac{121}{4374}a^{10}-\frac{301}{6561}a^{9}-\frac{35}{729}a^{8}-\frac{19}{1458}a^{7}+\frac{3971}{2430}a^{5}+\frac{394}{135}a^{4}+\frac{1309}{324}a^{3}+\frac{371}{90}a^{2}+\frac{112}{45}a-\frac{1}{2}$ Copy content Toggle raw display (assuming GRH)
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:  \( 30334220.019 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 30334220.019 \cdot 27}{6\cdot\sqrt{315164892649262158461997559808}}\cr\approx \mathstrut & 3.7110366591 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 87*x^12 + 8991*x^6 + 19683)
 
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 - 87*x^12 + 8991*x^6 + 19683, 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 - 87*x^12 + 8991*x^6 + 19683);
 
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 - 87*x^12 + 8991*x^6 + 19683);
 
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

$C_3:S_3$ (as 18T4):

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 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 3.1.47628.3 x3, 3.1.588.1 x3, 3.1.11907.2 x3, 6.0.2834352.2, 6.0.6805279152.2, 6.0.1037232.1, 6.0.425329947.5, 9.1.324121835451456.10 x9

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 9 sibling: 9.1.324121835451456.10
Minimal sibling: 9.1.324121835451456.10

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.2.0.1}{2} }^{9}$ R ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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
\(2\) Copy content Toggle raw display 2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.6.11.7$x^{6} + 9 x^{2} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.7$x^{6} + 9 x^{2} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.7$x^{6} + 9 x^{2} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
\(7\) Copy content Toggle raw display 7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$