Properties

Label 18.0.998...728.1
Degree $18$
Signature $[0, 9]$
Discriminant $-9.982\times 10^{26}$
Root discriminant \(31.62\)
Ramified primes $2,3,13$
Class number $3$ (GRH)
Class group [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 - 95*x^12 + 3667*x^6 + 27)
 
gp: K = bnfinit(y^18 - 95*y^12 + 3667*y^6 + 27, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 95*x^12 + 3667*x^6 + 27);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 95*x^12 + 3667*x^6 + 27)
 

\( x^{18} - 95x^{12} + 3667x^{6} + 27 \) 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:   \(-998220592474975337089609728\) \(\medspace = -\,2^{12}\cdot 3^{21}\cdot 13^{12}\) 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:  \(31.62\)
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:  $2^{2/3}3^{7/6}13^{2/3}\approx 31.619647871494593$
Ramified primes:   \(2\), \(3\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{6}-\frac{1}{12}a^{3}+\frac{1}{4}$, $\frac{1}{36}a^{10}-\frac{1}{4}a^{7}+\frac{11}{36}a^{4}+\frac{1}{4}a$, $\frac{1}{36}a^{11}-\frac{1}{12}a^{8}+\frac{11}{36}a^{5}+\frac{1}{12}a^{2}$, $\frac{1}{6120}a^{12}+\frac{613}{3060}a^{6}-\frac{1}{2}a^{3}-\frac{183}{680}$, $\frac{1}{6120}a^{13}+\frac{613}{3060}a^{7}-\frac{1}{2}a^{4}-\frac{183}{680}a$, $\frac{1}{6120}a^{14}+\frac{103}{3060}a^{8}-\frac{1}{2}a^{5}-\frac{209}{2040}a^{2}$, $\frac{1}{18360}a^{15}+\frac{1}{18360}a^{13}-\frac{1}{108}a^{11}+\frac{103}{9180}a^{9}-\frac{1}{12}a^{8}-\frac{917}{9180}a^{7}-\frac{47}{108}a^{5}-\frac{1}{2}a^{4}-\frac{209}{6120}a^{3}+\frac{1}{12}a^{2}-\frac{523}{2040}a-\frac{1}{2}$, $\frac{1}{18360}a^{16}+\frac{1}{18360}a^{14}+\frac{1}{18360}a^{12}+\frac{103}{9180}a^{10}+\frac{613}{9180}a^{8}+\frac{2143}{9180}a^{6}-\frac{1}{2}a^{5}-\frac{209}{6120}a^{4}-\frac{1}{2}a^{3}-\frac{863}{2040}a^{2}-\frac{1}{2}a+\frac{279}{680}$, $\frac{1}{18360}a^{17}-\frac{67}{9180}a^{11}-\frac{1}{36}a^{9}-\frac{1}{6}a^{7}-\frac{1}{4}a^{6}+\frac{1753}{18360}a^{5}-\frac{11}{36}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a+\frac{1}{4}$ 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

$C_{3}$, which has order $3$ (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:   \( \frac{1}{360} a^{15} - \frac{47}{180} a^{9} + \frac{1211}{120} a^{3} + \frac{1}{2} \)  (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:   $\frac{109}{9180}a^{17}+\frac{7}{1224}a^{16}+\frac{1}{540}a^{15}+\frac{7}{18360}a^{13}-\frac{863}{765}a^{11}-\frac{37}{68}a^{10}-\frac{47}{270}a^{9}-\frac{299}{9180}a^{7}+\frac{400007}{9180}a^{5}+\frac{25667}{1224}a^{4}+\frac{1211}{180}a^{3}+\frac{2119}{2040}a$, $\frac{109}{18360}a^{17}-\frac{7}{1224}a^{16}+\frac{1}{1080}a^{15}+\frac{1}{1224}a^{14}-\frac{7}{18360}a^{13}-\frac{863}{1530}a^{11}+\frac{37}{68}a^{10}-\frac{47}{540}a^{9}-\frac{25}{306}a^{8}+\frac{299}{9180}a^{7}+\frac{400007}{18360}a^{5}-\frac{25667}{1224}a^{4}+\frac{1211}{360}a^{3}+\frac{1321}{408}a^{2}-\frac{2119}{2040}a-\frac{1}{2}$, $\frac{19}{1080}a^{17}-\frac{61}{9180}a^{16}-\frac{1}{270}a^{15}+\frac{37}{18360}a^{14}+\frac{7}{18360}a^{13}-\frac{1}{9180}a^{12}-\frac{301}{180}a^{11}+\frac{2897}{4590}a^{10}+\frac{47}{135}a^{9}-\frac{1799}{9180}a^{8}-\frac{299}{9180}a^{7}+\frac{76}{2295}a^{6}+\frac{69707}{1080}a^{5}-\frac{74461}{3060}a^{4}-\frac{1211}{90}a^{3}+\frac{15329}{2040}a^{2}+\frac{1099}{2040}a-\frac{109}{340}$, $\frac{7}{1224}a^{17}-\frac{61}{9180}a^{16}+\frac{7}{18360}a^{14}-\frac{7}{6120}a^{13}-\frac{1}{9180}a^{12}-\frac{37}{68}a^{11}+\frac{2897}{4590}a^{10}-\frac{299}{9180}a^{8}+\frac{299}{3060}a^{7}+\frac{76}{2295}a^{6}+\frac{25667}{1224}a^{5}-\frac{74461}{3060}a^{4}+\frac{2119}{2040}a^{2}-\frac{2459}{680}a-\frac{789}{340}$, $\frac{1}{4590}a^{17}+\frac{131}{3060}a^{16}+\frac{1}{918}a^{15}-\frac{43}{6120}a^{14}+\frac{41}{18360}a^{13}-\frac{5}{1224}a^{12}-\frac{61}{3060}a^{11}-\frac{4123}{1020}a^{10}-\frac{251}{1836}a^{9}+\frac{973}{1530}a^{8}-\frac{28}{2295}a^{7}+\frac{37}{153}a^{6}+\frac{2911}{9180}a^{5}+\frac{119482}{765}a^{4}+\frac{4189}{612}a^{3}-\frac{18821}{680}a^{2}-\frac{1213}{2040}a+\frac{745}{136}$, $\frac{317}{18360}a^{17}-\frac{1}{1530}a^{16}-\frac{71}{18360}a^{15}+\frac{1}{765}a^{14}-\frac{1}{9180}a^{13}-\frac{1}{306}a^{12}-\frac{1661}{1020}a^{11}+\frac{61}{1020}a^{10}+\frac{2887}{9180}a^{9}-\frac{49}{765}a^{8}+\frac{1069}{9180}a^{7}-\frac{1}{153}a^{6}+\frac{1146961}{18360}a^{5}-\frac{4441}{3060}a^{4}-\frac{75601}{6120}a^{3}+\frac{7}{510}a^{2}+\frac{662}{255}a-\frac{2}{17}$, $\frac{157}{18360}a^{17}-\frac{1}{1530}a^{16}-\frac{13}{4590}a^{15}-\frac{7}{3060}a^{14}+\frac{1}{9180}a^{13}+\frac{4}{765}a^{12}-\frac{1229}{1530}a^{11}+\frac{61}{1020}a^{10}+\frac{2039}{9180}a^{9}+\frac{343}{3060}a^{8}-\frac{1069}{9180}a^{7}-\frac{1039}{3060}a^{6}+\frac{570851}{18360}a^{5}-\frac{4441}{3060}a^{4}-\frac{22531}{3060}a^{3}-\frac{2371}{510}a^{2}-\frac{662}{255}a-\frac{293}{340}$, $\frac{1}{1020}a^{15}-\frac{7}{1224}a^{12}-\frac{49}{1020}a^{9}+\frac{73}{306}a^{6}+\frac{193}{170}a^{3}+\frac{23}{136}$ 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:  \( 6280321.20776 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 6280321.20776 \cdot 3}{6\cdot\sqrt{998220592474975337089609728}}\cr\approx \mathstrut & 1.51690161434 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 95*x^12 + 3667*x^6 + 27)
 
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 - 95*x^12 + 3667*x^6 + 27, 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 - 95*x^12 + 3667*x^6 + 27);
 
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 - 95*x^12 + 3667*x^6 + 27);
 
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

\(\Q(\sqrt{-3}) \), 3.1.2028.1 x3, 3.1.18252.1 x3, 3.1.108.1 x3, 3.1.4563.1 x3, 6.0.12338352.2, 6.0.999406512.1, 6.0.34992.1, 6.0.62462907.1, 9.1.18241167657024.4 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: 9.1.18241167657024.4
Minimal sibling: 9.1.18241167657024.4

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 ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ R ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
\(13\) Copy content Toggle raw display 13.9.6.1$x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$