Properties

Label 18.0.96511468169...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 5^{6}$
Root discriminant $11.34$
Ramified primes $3, 5$
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![3, 0, -9, 0, 36, -27, -57, 81, 45, -156, 72, 90, -111, 0, 84, -78, 36, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3)
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 9 x^{17} \) \(\mathstrut +\mathstrut 36 x^{16} \) \(\mathstrut -\mathstrut 78 x^{15} \) \(\mathstrut +\mathstrut 84 x^{14} \) \(\mathstrut -\mathstrut 111 x^{12} \) \(\mathstrut +\mathstrut 90 x^{11} \) \(\mathstrut +\mathstrut 72 x^{10} \) \(\mathstrut -\mathstrut 156 x^{9} \) \(\mathstrut +\mathstrut 45 x^{8} \) \(\mathstrut +\mathstrut 81 x^{7} \) \(\mathstrut -\mathstrut 57 x^{6} \) \(\mathstrut -\mathstrut 27 x^{5} \) \(\mathstrut +\mathstrut 36 x^{4} \) \(\mathstrut -\mathstrut 9 x^{2} \) \(\mathstrut +\mathstrut 3 \)

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:  \(-9651146816936671875=-\,3^{31}\cdot 5^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.34$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{78986} a^{17} - \frac{5056}{39493} a^{16} - \frac{6819}{78986} a^{15} + \frac{16487}{78986} a^{14} + \frac{13397}{78986} a^{13} - \frac{3690}{39493} a^{12} - \frac{2755}{78986} a^{11} - \frac{4355}{39493} a^{10} + \frac{3399}{39493} a^{9} - \frac{2023}{78986} a^{8} - \frac{9480}{39493} a^{7} - \frac{13791}{39493} a^{6} - \frac{1719}{78986} a^{5} + \frac{29603}{78986} a^{4} - \frac{38077}{78986} a^{3} + \frac{30111}{78986} a^{2} + \frac{3137}{78986} a - \frac{19725}{78986}$

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:  \( -\frac{40878}{39493} a^{17} + \frac{301049}{39493} a^{16} - \frac{1924181}{78986} a^{15} + \frac{1493193}{39493} a^{14} - \frac{625023}{39493} a^{13} - \frac{1508121}{39493} a^{12} + \frac{4037487}{78986} a^{11} + \frac{1418225}{78986} a^{10} - \frac{3056857}{39493} a^{9} + \frac{2562749}{78986} a^{8} + \frac{3350415}{78986} a^{7} - \frac{1489395}{39493} a^{6} - \frac{1122827}{78986} a^{5} + \frac{2080287}{78986} a^{4} - \frac{26003}{39493} a^{3} - \frac{236085}{39493} a^{2} + \frac{78471}{39493} a + \frac{177403}{78986} \) (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:  \( 868.676171318 \)
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

\(\Q(\sqrt{-3}) \), 3.1.135.1, \(\Q(\zeta_{9})^+\), 6.0.54675.1, \(\Q(\zeta_{9})\), 9.3.1793613375.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
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) $1$ $-1$
1.3_5.2t1.1c1$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3e2_5.6t1.1c1$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 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) $0$ $1$
1.3e2_5.6t1.2c1$1$ $ 3^{2} \cdot 5 $ $x^{6} + 6 x^{4} - 4 x^{3} + 9 x^{2} - 12 x + 19$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2.6t1.1c1$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_5.6t1.2c2$1$ $ 3^{2} \cdot 5 $ $x^{6} + 6 x^{4} - 4 x^{3} + 9 x^{2} - 12 x + 19$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_5.6t1.1c2$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 1.3e2.6t1.1c2$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 2.3e3_5.3t2.1c1$2$ $ 3^{3} \cdot 5 $ $x^{3} + 3 x - 1$ $S_3$ (as 3T2) $1$ $0$
* 2.3e3_5.6t3.1c1$2$ $ 3^{3} \cdot 5 $ $x^{6} - x^{3} - 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.3e4_5.12t18.1c1$2$ $ 3^{4} \cdot 5 $ $x^{18} - 9 x^{17} + 36 x^{16} - 78 x^{15} + 84 x^{14} - 111 x^{12} + 90 x^{11} + 72 x^{10} - 156 x^{9} + 45 x^{8} + 81 x^{7} - 57 x^{6} - 27 x^{5} + 36 x^{4} - 9 x^{2} + 3$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.3e4_5.12t18.1c2$2$ $ 3^{4} \cdot 5 $ $x^{18} - 9 x^{17} + 36 x^{16} - 78 x^{15} + 84 x^{14} - 111 x^{12} + 90 x^{11} + 72 x^{10} - 156 x^{9} + 45 x^{8} + 81 x^{7} - 57 x^{6} - 27 x^{5} + 36 x^{4} - 9 x^{2} + 3$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.3e4_5.6t5.1c1$2$ $ 3^{4} \cdot 5 $ $x^{6} + 6 x^{4} - x^{3} + 9 x^{2} - 3 x + 4$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3e4_5.6t5.1c2$2$ $ 3^{4} \cdot 5 $ $x^{6} + 6 x^{4} - x^{3} + 9 x^{2} - 3 x + 4$ $S_3\times C_3$ (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 *.