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Properties

Label 18.0.30384393663...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{6}\cdot 7^{15}$
Root discriminant $13.74$
Ramified primes $2, 5, 7$
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![8, -32, 80, -144, 216, -272, 290, -264, 210, -151, 96, -59, 39, -29, 20, -12, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 7*x^16 - 12*x^15 + 20*x^14 - 29*x^13 + 39*x^12 - 59*x^11 + 96*x^10 - 151*x^9 + 210*x^8 - 264*x^7 + 290*x^6 - 272*x^5 + 216*x^4 - 144*x^3 + 80*x^2 - 32*x + 8)
 
gp: K = bnfinit(x^18 - 3*x^17 + 7*x^16 - 12*x^15 + 20*x^14 - 29*x^13 + 39*x^12 - 59*x^11 + 96*x^10 - 151*x^9 + 210*x^8 - 264*x^7 + 290*x^6 - 272*x^5 + 216*x^4 - 144*x^3 + 80*x^2 - 32*x + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 7 x^{16} - 12 x^{15} + 20 x^{14} - 29 x^{13} + 39 x^{12} - 59 x^{11} + 96 x^{10} - 151 x^{9} + 210 x^{8} - 264 x^{7} + 290 x^{6} - 272 x^{5} + 216 x^{4} - 144 x^{3} + 80 x^{2} - 32 x + 8 \)

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:  \(-303843936636352000000=-\,2^{12}\cdot 5^{6}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.74$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{11644} a^{17} - \frac{129}{2911} a^{16} - \frac{93}{5822} a^{15} - \frac{657}{11644} a^{14} + \frac{14}{71} a^{13} - \frac{1833}{11644} a^{12} + \frac{1513}{5822} a^{11} - \frac{417}{5822} a^{10} + \frac{21}{11644} a^{9} - \frac{2191}{11644} a^{8} - \frac{5275}{11644} a^{7} + \frac{373}{2911} a^{6} - \frac{606}{2911} a^{5} - \frac{667}{2911} a^{4} - \frac{1273}{2911} a^{3} + \frac{949}{2911} a^{2} - \frac{680}{2911} a - \frac{488}{2911}$

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:  \( \frac{60415}{11644} a^{17} - \frac{180723}{11644} a^{16} + \frac{94425}{2911} a^{15} - \frac{638679}{11644} a^{14} + \frac{24869}{284} a^{13} - \frac{1479221}{11644} a^{12} + \frac{1888407}{11644} a^{11} - \frac{741480}{2911} a^{10} + \frac{5029727}{11644} a^{9} - \frac{3864489}{5822} a^{8} + \frac{5215723}{5822} a^{7} - \frac{12622031}{11644} a^{6} + \frac{6669415}{5822} a^{5} - \frac{2881722}{2911} a^{4} + \frac{4189579}{5822} a^{3} - \frac{1276239}{2911} a^{2} + \frac{603320}{2911} a - \frac{171661}{2911} \) (order $14$)
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:  \( 4077.3155873667893 \)
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

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.1.140.1, \(\Q(\zeta_{7})\), 6.0.137200.1, 9.3.6588344000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$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}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.5_7.2t1.1c1$1$ $ 5 \cdot 7 $ $x^{2} - x + 9$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.5_7.6t1.2c1$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.6t1.1c1$1$ $ 7 $ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
1.5_7.6t1.2c2$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$ $C_6$ (as 6T1) $0$ $-1$
1.5_7.6t1.1c1$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
1.5_7.6t1.1c2$1$ $ 5 \cdot 7 $ $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ $C_6$ (as 6T1) $0$ $1$
* 1.7.6t1.1c2$1$ $ 7 $ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 2.2e2_5_7.3t2.1c1$2$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{3} + 2 x - 2$ $S_3$ (as 3T2) $1$ $0$
* 2.2e2_5_7.6t3.1c1$2$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{6} - x^{5} - x^{4} - x^{3} + 2 x + 2$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_5_7e2.12t18.1c1$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ $x^{18} - 3 x^{17} + 7 x^{16} - 12 x^{15} + 20 x^{14} - 29 x^{13} + 39 x^{12} - 59 x^{11} + 96 x^{10} - 151 x^{9} + 210 x^{8} - 264 x^{7} + 290 x^{6} - 272 x^{5} + 216 x^{4} - 144 x^{3} + 80 x^{2} - 32 x + 8$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e2_5_7e2.6t5.1c1$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ $x^{6} - 2 x^{5} - 3 x^{4} + 6 x^{3} + 37 x^{2} - 4 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2e2_5_7e2.12t18.1c2$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ $x^{18} - 3 x^{17} + 7 x^{16} - 12 x^{15} + 20 x^{14} - 29 x^{13} + 39 x^{12} - 59 x^{11} + 96 x^{10} - 151 x^{9} + 210 x^{8} - 264 x^{7} + 290 x^{6} - 272 x^{5} + 216 x^{4} - 144 x^{3} + 80 x^{2} - 32 x + 8$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.2e2_5_7e2.6t5.1c2$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ $x^{6} - 2 x^{5} - 3 x^{4} + 6 x^{3} + 37 x^{2} - 4 x + 1$ $S_3\times C_3$ (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 *.