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Properties

Label 18.0.762...984.1
Degree $18$
Signature $[0, 9]$
Discriminant $-7.626\times 10^{22}$
Root discriminant \(18.67\)
Ramified primes $2,11,37$
Class number $1$
Class group trivial
Galois group $S_3\times D_6$ (as 18T29)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*x^2 + 1)
 
gp: K = bnfinit(y^18 - 20*y^14 + 39*y^12 + 40*y^10 - 112*y^8 + 55*y^6 + 12*y^4 - 8*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*x^2 + 1)
 

\( x^{18} - 20x^{14} + 39x^{12} + 40x^{10} - 112x^{8} + 55x^{6} + 12x^{4} - 8x^{2} + 1 \) 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:   \(-76258165114191147433984\) \(\medspace = -\,2^{24}\cdot 11^{6}\cdot 37^{6}\) 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:  \(18.67\)
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:  $2^{4/3}11^{1/2}37^{1/2}\approx 50.83590180772078$
Ramified primes:   \(2\), \(11\), \(37\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{12}+\frac{1}{6}a^{10}-\frac{1}{6}a^{8}+\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{13}+\frac{1}{6}a^{11}-\frac{1}{6}a^{9}+\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{3}a$, $\frac{1}{12}a^{14}+\frac{1}{12}a^{10}-\frac{1}{12}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{12}$, $\frac{1}{12}a^{15}+\frac{1}{12}a^{11}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}+\frac{5}{12}a-\frac{1}{2}$, $\frac{1}{20688}a^{16}+\frac{455}{20688}a^{14}+\frac{125}{20688}a^{12}-\frac{851}{10344}a^{10}+\frac{813}{3448}a^{8}-\frac{4015}{10344}a^{6}+\frac{1285}{20688}a^{4}-\frac{491}{6896}a^{2}+\frac{5585}{20688}$, $\frac{1}{41376}a^{17}-\frac{1}{41376}a^{16}+\frac{455}{41376}a^{15}-\frac{455}{41376}a^{14}+\frac{125}{41376}a^{13}-\frac{125}{41376}a^{12}+\frac{4321}{20688}a^{11}-\frac{4321}{20688}a^{10}-\frac{911}{6896}a^{9}+\frac{911}{6896}a^{8}-\frac{9187}{20688}a^{7}-\frac{1157}{20688}a^{6}-\frac{9059}{41376}a^{5}+\frac{9059}{41376}a^{4}+\frac{2957}{13792}a^{3}-\frac{2957}{13792}a^{2}-\frac{4759}{41376}a-\frac{15929}{41376}$ 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:   \( \frac{3995}{3448} a^{17} + \frac{629}{3448} a^{15} - \frac{79889}{3448} a^{13} + \frac{71529}{1724} a^{11} + \frac{91991}{1724} a^{9} - \frac{209295}{1724} a^{7} + \frac{146043}{3448} a^{5} + \frac{71785}{3448} a^{3} - \frac{15449}{3448} a \)  (order $4$) 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:   $\frac{2705}{6896}a^{16}+\frac{1241}{20688}a^{14}-\frac{161393}{20688}a^{12}+\frac{48719}{3448}a^{10}+\frac{59225}{3448}a^{8}-\frac{417919}{10344}a^{6}+\frac{361339}{20688}a^{4}+\frac{90469}{20688}a^{2}-\frac{34441}{20688}$, $\frac{13589}{41376}a^{17}-\frac{3665}{41376}a^{16}-\frac{2717}{41376}a^{15}-\frac{2191}{41376}a^{14}-\frac{273631}{41376}a^{13}+\frac{24289}{13792}a^{12}+\frac{291709}{20688}a^{11}-\frac{16599}{6896}a^{10}+\frac{237583}{20688}a^{9}-\frac{112055}{20688}a^{8}-\frac{280573}{6896}a^{7}+\frac{157519}{20688}a^{6}+\frac{311867}{13792}a^{5}+\frac{21131}{41376}a^{4}+\frac{178339}{41376}a^{3}-\frac{163079}{41376}a^{2}-\frac{130387}{41376}a+\frac{1733}{13792}$, $\frac{629}{3448}a^{16}+\frac{11}{3448}a^{14}-\frac{12747}{3448}a^{12}+\frac{12091}{1724}a^{10}+\frac{14425}{1724}a^{8}-\frac{36841}{1724}a^{6}+\frac{23845}{3448}a^{4}+\frac{16511}{3448}a^{2}-\frac{3995}{3448}$, $\frac{1979}{10344}a^{17}+\frac{1379}{10344}a^{15}-\frac{38809}{10344}a^{13}+\frac{4131}{862}a^{11}+\frac{14657}{1293}a^{9}-\frac{11383}{862}a^{7}-\frac{3697}{3448}a^{5}+\frac{15733}{10344}a^{3}+\frac{13081}{10344}a$, $\frac{1467}{862}a^{17}+\frac{911}{1293}a^{16}+\frac{2213}{5172}a^{15}+\frac{275}{1724}a^{14}-\frac{87755}{2586}a^{13}-\frac{12151}{862}a^{12}+\frac{299239}{5172}a^{11}+\frac{125423}{5172}a^{10}+\frac{427919}{5172}a^{9}+\frac{178133}{5172}a^{8}-\frac{880903}{5172}a^{7}-\frac{372683}{5172}a^{6}+\frac{263609}{5172}a^{5}+\frac{105751}{5172}a^{4}+\frac{89215}{2586}a^{3}+\frac{20057}{1293}a^{2}-\frac{11867}{1724}a-\frac{16889}{5172}$, $\frac{1837}{5172}a^{17}-\frac{323}{10344}a^{16}+\frac{21}{862}a^{15}+\frac{437}{10344}a^{14}-\frac{36733}{5172}a^{13}+\frac{6173}{10344}a^{12}+\frac{23097}{1724}a^{11}-\frac{2720}{1293}a^{10}+\frac{78407}{5172}a^{9}+\frac{401}{431}a^{8}-\frac{203567}{5172}a^{7}+\frac{12913}{2586}a^{6}+\frac{45665}{2586}a^{5}-\frac{86633}{10344}a^{4}+\frac{12615}{1724}a^{3}+\frac{6881}{3448}a^{2}-\frac{12223}{2586}a+\frac{14003}{10344}$, $\frac{16531}{20688}a^{17}+\frac{1517}{20688}a^{15}-\frac{109995}{6896}a^{13}+\frac{102277}{3448}a^{11}+\frac{353341}{10344}a^{9}-\frac{877205}{10344}a^{7}+\frac{754319}{20688}a^{5}+\frac{189253}{20688}a^{3}-\frac{21175}{6896}a$, $\frac{1}{4}a^{17}+\frac{1}{4}a^{16}+\frac{1}{4}a^{15}+\frac{1}{4}a^{14}-\frac{19}{4}a^{13}-\frac{19}{4}a^{12}+5a^{11}+5a^{10}+15a^{9}+15a^{8}-13a^{7}-13a^{6}+\frac{3}{4}a^{5}+\frac{3}{4}a^{4}+\frac{15}{4}a^{3}+\frac{15}{4}a^{2}+\frac{7}{4}a+\frac{3}{4}$ 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:  \( 36250.2959789 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 36250.2959789 \cdot 1}{4\cdot\sqrt{76258165114191147433984}}\cr\approx \mathstrut & 0.500872520038 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*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

$S_3\times D_6$ (as 18T29):

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 72
The 18 conjugacy class representatives for $S_3\times D_6$
Character table for $S_3\times D_6$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.148.1, 3.1.44.1, 6.0.350464.1, 6.0.30976.1, 9.3.4314825152.1

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 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 sibling: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.0.74470864369327292416.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ R ${\href{/padicField/13.2.0.1}{2} }^{8}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{9}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ R ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 10 x^{11} + 47 x^{10} + 144 x^{9} + 330 x^{8} + 578 x^{7} + 785 x^{6} + 830 x^{5} + 530 x^{4} - 64 x^{3} - 189 x^{2} - 30 x + 25$$6$$2$$16$$D_6$$[2]_{3}^{2}$
\(11\) Copy content Toggle raw display 11.6.0.1$x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.6.1$x^{12} + 72 x^{10} + 8 x^{9} + 2034 x^{8} + 38 x^{7} + 27996 x^{6} - 6312 x^{5} + 196025 x^{4} - 84710 x^{3} + 695581 x^{2} - 235284 x + 1080083$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
\(37\) Copy content Toggle raw display 37.3.0.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$[\ ]^{3}$
37.6.3.1$x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} + 2775 x^{5} + 2566998 x^{4} + 791680745 x^{3} + 110476893 x^{2} + 4842384300 x + 27887445532$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$