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Properties

Label 18.0.27277478847...9616.1
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![1, 3, 18, 77, 165, 297, 767, 342, 711, 231, 162, 36, -16, -9, 18, -4, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 4*x^15 + 18*x^14 - 9*x^13 - 16*x^12 + 36*x^11 + 162*x^10 + 231*x^9 + 711*x^8 + 342*x^7 + 767*x^6 + 297*x^5 + 165*x^4 + 77*x^3 + 18*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 - 4*x^15 + 18*x^14 - 9*x^13 - 16*x^12 + 36*x^11 + 162*x^10 + 231*x^9 + 711*x^8 + 342*x^7 + 767*x^6 + 297*x^5 + 165*x^4 + 77*x^3 + 18*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 4 x^{15} + 18 x^{14} - 9 x^{13} - 16 x^{12} + 36 x^{11} + 162 x^{10} + 231 x^{9} + 711 x^{8} + 342 x^{7} + 767 x^{6} + 297 x^{5} + 165 x^{4} + 77 x^{3} + 18 x^{2} + 3 x + 1 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} + \frac{7}{23} a^{14} + \frac{9}{23} a^{13} - \frac{9}{23} a^{12} - \frac{10}{23} a^{11} + \frac{8}{23} a^{10} + \frac{10}{23} a^{9} + \frac{9}{23} a^{7} + \frac{8}{23} a^{6} + \frac{4}{23} a^{5} - \frac{8}{23} a^{4} + \frac{1}{23} a^{3} + \frac{2}{23} a^{2} + \frac{1}{23}$, $\frac{1}{25479115007} a^{16} + \frac{481153253}{25479115007} a^{15} + \frac{5078800081}{25479115007} a^{14} + \frac{9711247145}{25479115007} a^{13} - \frac{8980887569}{25479115007} a^{12} - \frac{2866714920}{25479115007} a^{11} - \frac{2913037878}{25479115007} a^{10} - \frac{9884265376}{25479115007} a^{9} + \frac{12489711}{480738019} a^{8} + \frac{12175129476}{25479115007} a^{7} + \frac{3068454264}{25479115007} a^{6} + \frac{145776388}{1498771471} a^{5} - \frac{10931194170}{25479115007} a^{4} + \frac{5988032070}{25479115007} a^{3} - \frac{4663860910}{25479115007} a^{2} + \frac{522715}{1210927} a + \frac{3841220417}{25479115007}$, $\frac{1}{25479115007} a^{17} - \frac{22742410}{25479115007} a^{15} + \frac{470911981}{1107787609} a^{14} - \frac{4345005071}{25479115007} a^{13} - \frac{8858872242}{25479115007} a^{12} + \frac{325434803}{1341006053} a^{11} - \frac{5160852714}{25479115007} a^{10} - \frac{7302342923}{25479115007} a^{9} - \frac{10702130837}{25479115007} a^{8} + \frac{11737663519}{25479115007} a^{7} + \frac{10211709093}{25479115007} a^{6} - \frac{1319780648}{25479115007} a^{5} + \frac{10243933737}{25479115007} a^{4} - \frac{2024993262}{25479115007} a^{3} + \frac{29413}{483943} a^{2} + \frac{2516142401}{25479115007} a + \frac{281633788}{25479115007}$

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:  \( 14940.9232602 \)
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

\(\Q(\sqrt{-11}) \), 3.1.44.1 x3, \(\Q(\zeta_{9})^+\), 6.0.21296.1, 6.0.8732691.1, 6.0.139723056.8 x2, 9.3.45270270144.6 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_11.6t1.2c1$1$ $ 3^{2} \cdot 11 $ $x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2_11.6t1.2c2$1$ $ 3^{2} \cdot 11 $ $x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e2_11.3t2.1c1$2$ $ 2^{2} \cdot 11 $ $x^{3} - x^{2} + x + 1$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e2_3e4_11.6t5.5c1$2$ $ 2^{2} \cdot 3^{4} \cdot 11 $ $x^{18} + 9 x^{16} - 4 x^{15} + 18 x^{14} - 9 x^{13} - 16 x^{12} + 36 x^{11} + 162 x^{10} + 231 x^{9} + 711 x^{8} + 342 x^{7} + 767 x^{6} + 297 x^{5} + 165 x^{4} + 77 x^{3} + 18 x^{2} + 3 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e2_3e4_11.6t5.5c2$2$ $ 2^{2} \cdot 3^{4} \cdot 11 $ $x^{18} + 9 x^{16} - 4 x^{15} + 18 x^{14} - 9 x^{13} - 16 x^{12} + 36 x^{11} + 162 x^{10} + 231 x^{9} + 711 x^{8} + 342 x^{7} + 767 x^{6} + 297 x^{5} + 165 x^{4} + 77 x^{3} + 18 x^{2} + 3 x + 1$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.