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Properties

Label 18.6.17980759982...5017.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{27}\cdot 11^{9}$
Root discriminant $17.23$
Ramified primes $3, 11$
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, 48, -114, 169, -195, 189, -145, 54, -69, 48, 18, 75, 12, 3, -12, -10, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 10*x^15 - 12*x^14 + 3*x^13 + 12*x^12 + 75*x^11 + 18*x^10 + 48*x^9 - 69*x^8 + 54*x^7 - 145*x^6 + 189*x^5 - 195*x^4 + 169*x^3 - 114*x^2 + 48*x - 8)
 
gp: K = bnfinit(x^18 - 10*x^15 - 12*x^14 + 3*x^13 + 12*x^12 + 75*x^11 + 18*x^10 + 48*x^9 - 69*x^8 + 54*x^7 - 145*x^6 + 189*x^5 - 195*x^4 + 169*x^3 - 114*x^2 + 48*x - 8, 1)
 

Normalized defining polynomial

\( x^{18} - 10 x^{15} - 12 x^{14} + 3 x^{13} + 12 x^{12} + 75 x^{11} + 18 x^{10} + 48 x^{9} - 69 x^{8} + 54 x^{7} - 145 x^{6} + 189 x^{5} - 195 x^{4} + 169 x^{3} - 114 x^{2} + 48 x - 8 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17980759982220503815017=3^{27}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.23$
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:  $3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} + \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{3}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{16} + \frac{1}{16} a^{14} - \frac{1}{8} a^{13} - \frac{3}{16} a^{12} + \frac{1}{16} a^{11} - \frac{7}{16} a^{10} + \frac{1}{4} a^{9} - \frac{5}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{1932657330848} a^{17} - \frac{8962241417}{1932657330848} a^{16} + \frac{99452639265}{1932657330848} a^{15} + \frac{197281402997}{1932657330848} a^{14} + \frac{80347758791}{1932657330848} a^{13} + \frac{81616998075}{483164332712} a^{12} - \frac{18582308407}{120791083178} a^{11} + \frac{671582406739}{1932657330848} a^{10} - \frac{953203131553}{1932657330848} a^{9} - \frac{374577224991}{1932657330848} a^{8} + \frac{365637040105}{966328665424} a^{7} - \frac{619676017}{2073666664} a^{6} - \frac{802278826157}{1932657330848} a^{5} - \frac{372226310279}{966328665424} a^{4} - \frac{5883827413}{1932657330848} a^{3} - \frac{276535566665}{966328665424} a^{2} - \frac{72619725933}{241582166356} a + \frac{62673989391}{241582166356}$

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:  $11$
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:  \( 16013.863210903846 \)
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

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), 3.1.891.1, 6.2.26198073.1, 6.6.26198073.1, 9.3.707347971.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$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_11.2t1.1c1$1$ $ 3 \cdot 11 $ $x^{2} - x - 8$ $C_2$ (as 2T1) $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3e2.6t1.1c1$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.3e2_11.6t1.1c1$1$ $ 3^{2} \cdot 11 $ $x^{6} - 18 x^{4} - 8 x^{3} + 81 x^{2} + 72 x - 17$ $C_6$ (as 6T1) $0$ $1$
1.3e2.6t1.1c2$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.3e2_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_11.6t1.1c2$1$ $ 3^{2} \cdot 11 $ $x^{6} - 18 x^{4} - 8 x^{3} + 81 x^{2} + 72 x - 17$ $C_6$ (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) $0$ $1$
* 2.3e4_11.3t2.1c1$2$ $ 3^{4} \cdot 11 $ $x^{3} + 6 x - 1$ $S_3$ (as 3T2) $1$ $0$
* 2.3e4_11.6t3.1c1$2$ $ 3^{4} \cdot 11 $ $x^{6} - 3 x^{4} + 9 x^{2} - 36 x + 48$ $D_{6}$ (as 6T3) $1$ $0$
* 2.3e2_11.6t5.1c1$2$ $ 3^{2} \cdot 11 $ $x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3e3_11.12t18.1c1$2$ $ 3^{3} \cdot 11 $ $x^{18} - 10 x^{15} - 12 x^{14} + 3 x^{13} + 12 x^{12} + 75 x^{11} + 18 x^{10} + 48 x^{9} - 69 x^{8} + 54 x^{7} - 145 x^{6} + 189 x^{5} - 195 x^{4} + 169 x^{3} - 114 x^{2} + 48 x - 8$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.3e3_11.12t18.1c2$2$ $ 3^{3} \cdot 11 $ $x^{18} - 10 x^{15} - 12 x^{14} + 3 x^{13} + 12 x^{12} + 75 x^{11} + 18 x^{10} + 48 x^{9} - 69 x^{8} + 54 x^{7} - 145 x^{6} + 189 x^{5} - 195 x^{4} + 169 x^{3} - 114 x^{2} + 48 x - 8$ $S_3 \times C_6$ (as 18T6) $0$ $0$
* 2.3e2_11.6t5.1c2$2$ $ 3^{2} \cdot 11 $ $x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$ $S_3\times C_3$ (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 *.