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Properties

Label 18.0.48803823903...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 5^{6}$
Root discriminant $14.10$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -9, -19, 72, -12, -164, 171, 72, -257, 216, -123, 73, -33, 21, -15, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 15*x^15 + 21*x^14 - 33*x^13 + 73*x^12 - 123*x^11 + 216*x^10 - 257*x^9 + 72*x^8 + 171*x^7 - 164*x^6 - 12*x^5 + 72*x^4 - 19*x^3 - 9*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 15*x^15 + 21*x^14 - 33*x^13 + 73*x^12 - 123*x^11 + 216*x^10 - 257*x^9 + 72*x^8 + 171*x^7 - 164*x^6 - 12*x^5 + 72*x^4 - 19*x^3 - 9*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} - 15 x^{15} + 21 x^{14} - 33 x^{13} + 73 x^{12} - 123 x^{11} + 216 x^{10} - 257 x^{9} + 72 x^{8} + 171 x^{7} - 164 x^{6} - 12 x^{5} + 72 x^{4} - 19 x^{3} - 9 x^{2} + 3 x + 1 \)

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:  \(-488038239039168000000=-\,2^{12}\cdot 3^{27}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.10$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{485894187390167} a^{17} - \frac{6010941889730}{485894187390167} a^{16} + \frac{174308707278399}{485894187390167} a^{15} + \frac{18303240417875}{485894187390167} a^{14} + \frac{62901941517744}{485894187390167} a^{13} - \frac{50695902563032}{485894187390167} a^{12} + \frac{159380415856012}{485894187390167} a^{11} + \frac{175499251033698}{485894187390167} a^{10} - \frac{8745796077339}{485894187390167} a^{9} + \frac{58584764422042}{485894187390167} a^{8} + \frac{241974895650339}{485894187390167} a^{7} - \frac{183486570149081}{485894187390167} a^{6} + \frac{150069802678172}{485894187390167} a^{5} + \frac{3068046737857}{28582011022951} a^{4} - \frac{77363729952926}{485894187390167} a^{3} + \frac{71236179402893}{485894187390167} a^{2} - \frac{48457161056918}{485894187390167} a + \frac{20982018562327}{485894187390167}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{100198155075152}{485894187390167} a^{17} + \frac{222429353688448}{485894187390167} a^{16} - \frac{498069809457761}{485894187390167} a^{15} + \frac{1330233602192341}{485894187390167} a^{14} - \frac{1496621996011416}{485894187390167} a^{13} + \frac{3209042116075505}{485894187390167} a^{12} - \frac{6335638540269269}{485894187390167} a^{11} + \frac{9724188181759193}{485894187390167} a^{10} - \frac{19267835403418622}{485894187390167} a^{9} + \frac{19556677805338141}{485894187390167} a^{8} - \frac{7316473367731777}{485894187390167} a^{7} - \frac{4287075300891228}{485894187390167} a^{6} + \frac{7885962209046788}{485894187390167} a^{5} - \frac{323265079841958}{28582011022951} a^{4} + \frac{1075954927550176}{485894187390167} a^{3} + \frac{2585866452466049}{485894187390167} a^{2} - \frac{1079651289132205}{485894187390167} a + \frac{30508976631484}{485894187390167} \) (order $18$)
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:  \( 5780.39296417851 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.1.1620.1, 6.0.7873200.3, \(\Q(\zeta_{9})\), 9.3.4251528000.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_5.2t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{2} + 5$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_3_5.2t1.1c1$1$ $ 2^{2} \cdot 3 \cdot 5 $ $x^{2} - 15$ $C_2$ (as 2T1) $1$ $1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.2e2_3e2_5.6t1.2c1$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} + 9 x^{4} - 2 x^{3} + 84 x^{2} + 36 x + 321$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2_5.6t1.1c1$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} - 30 x^{4} + 225 x^{2} - 375$ $C_6$ (as 6T1) $0$ $1$
1.2e2_3e2_5.6t1.2c2$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} + 9 x^{4} - 2 x^{3} + 84 x^{2} + 36 x + 321$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.6t1.1c1$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.6t1.1c2$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_3e2_5.6t1.1c2$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} - 30 x^{4} + 225 x^{2} - 375$ $C_6$ (as 6T1) $0$ $1$
* 2.2e2_3e4_5.3t2.2c1$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ $x^{3} - 3 x - 8$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_3e4_5.6t3.4c1$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ $x^{6} - 8 x^{3} + 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_3e3_5.12t18.1c1$2$ $ 2^{2} \cdot 3^{3} \cdot 5 $ $x^{18} - 3 x^{17} + 6 x^{16} - 15 x^{15} + 21 x^{14} - 33 x^{13} + 73 x^{12} - 123 x^{11} + 216 x^{10} - 257 x^{9} + 72 x^{8} + 171 x^{7} - 164 x^{6} - 12 x^{5} + 72 x^{4} - 19 x^{3} - 9 x^{2} + 3 x + 1$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e2_3e2_5.6t5.1c1$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} + 7 x^{2} - 4 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_3e3_5.12t18.1c2$2$ $ 2^{2} \cdot 3^{3} \cdot 5 $ $x^{18} - 3 x^{17} + 6 x^{16} - 15 x^{15} + 21 x^{14} - 33 x^{13} + 73 x^{12} - 123 x^{11} + 216 x^{10} - 257 x^{9} + 72 x^{8} + 171 x^{7} - 164 x^{6} - 12 x^{5} + 72 x^{4} - 19 x^{3} - 9 x^{2} + 3 x + 1$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e2_3e2_5.6t5.1c2$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ $x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} + 7 x^{2} - 4 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.