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Properties

Label 18.0.27277478847...9616.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 11^{9}$
Root discriminant $22.78$
Ramified primes $2, 3, 11$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -162, 918, -2916, 7182, -13275, 20355, -25056, 26112, -22468, 16527, -10128, 5309, -2298, 840, -243, 57, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 57*x^16 - 243*x^15 + 840*x^14 - 2298*x^13 + 5309*x^12 - 10128*x^11 + 16527*x^10 - 22468*x^9 + 26112*x^8 - 25056*x^7 + 20355*x^6 - 13275*x^5 + 7182*x^4 - 2916*x^3 + 918*x^2 - 162*x + 27)
 
gp: K = bnfinit(x^18 - 9*x^17 + 57*x^16 - 243*x^15 + 840*x^14 - 2298*x^13 + 5309*x^12 - 10128*x^11 + 16527*x^10 - 22468*x^9 + 26112*x^8 - 25056*x^7 + 20355*x^6 - 13275*x^5 + 7182*x^4 - 2916*x^3 + 918*x^2 - 162*x + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 57 x^{16} - 243 x^{15} + 840 x^{14} - 2298 x^{13} + 5309 x^{12} - 10128 x^{11} + 16527 x^{10} - 22468 x^{9} + 26112 x^{8} - 25056 x^{7} + 20355 x^{6} - 13275 x^{5} + 7182 x^{4} - 2916 x^{3} + 918 x^{2} - 162 x + 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2727747884710191986159616=-\,2^{12}\cdot 3^{24}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.78$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{1665} a^{15} - \frac{13}{555} a^{14} - \frac{6}{185} a^{13} + \frac{29}{555} a^{12} - \frac{10}{111} a^{11} - \frac{268}{555} a^{10} + \frac{341}{1665} a^{9} + \frac{69}{185} a^{8} - \frac{196}{555} a^{7} + \frac{467}{1665} a^{6} + \frac{92}{555} a^{5} + \frac{238}{555} a^{4} - \frac{61}{185} a^{3} + \frac{48}{185} a^{2} - \frac{5}{37} a + \frac{47}{185}$, $\frac{1}{118215} a^{16} - \frac{13}{118215} a^{15} - \frac{5536}{39405} a^{14} - \frac{2104}{39405} a^{13} + \frac{913}{13135} a^{12} + \frac{652}{39405} a^{11} + \frac{4412}{118215} a^{10} - \frac{10493}{118215} a^{9} - \frac{11279}{39405} a^{8} - \frac{26476}{118215} a^{7} - \frac{8672}{118215} a^{6} + \frac{2672}{7881} a^{5} - \frac{908}{7881} a^{4} + \frac{9446}{39405} a^{3} + \frac{4183}{13135} a^{2} - \frac{48}{13135} a + \frac{2332}{13135}$, $\frac{1}{189944197335} a^{17} - \frac{20858}{21104910815} a^{16} + \frac{43191667}{189944197335} a^{15} + \frac{9400698256}{63314732445} a^{14} - \frac{4108483084}{63314732445} a^{13} - \frac{1013468992}{63314732445} a^{12} + \frac{74463821363}{189944197335} a^{11} - \frac{9813386426}{63314732445} a^{10} - \frac{14240102107}{189944197335} a^{9} - \frac{12745896365}{37988839467} a^{8} - \frac{10274644789}{63314732445} a^{7} - \frac{77775941701}{189944197335} a^{6} - \frac{83724376}{580869105} a^{5} - \frac{417468224}{4220982163} a^{4} + \frac{8067833261}{63314732445} a^{3} - \frac{1010857881}{21104910815} a^{2} + \frac{9237803589}{21104910815} a + \frac{462264008}{21104910815}$

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

Class group and class number

$C_{3}$, which has order $3$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17836.0340898 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.3564.1 x3, \(\Q(\zeta_{9})^+\), 6.0.139723056.1, 6.0.1724976.1 x2, 6.0.8732691.1, 9.3.45270270144.5 x3

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

Sibling fields

Degree 6 sibling: 6.0.1724976.1
Degree 9 sibling: data not computed

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$