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Properties

Label 18.0.29543127065...8643.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}$
Root discriminant $15.59$
Ramified prime $3$
Class number $1$
Class group Trivial
Galois group $C_9\times S_3$ (as 18T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 27, 0, 0, -53, 0, 0, 36, 0, 0, -9, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^15 + 36*x^12 - 53*x^9 + 27*x^6 + 1)
 
gp: K = bnfinit(x^18 - 9*x^15 + 36*x^12 - 53*x^9 + 27*x^6 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1 \)

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:  \(-2954312706550833698643=-\,3^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.59$
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:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2071} a^{15} - \frac{393}{2071} a^{12} - \frac{235}{2071} a^{9} - \frac{937}{2071} a^{6} - \frac{519}{2071} a^{3} + \frac{480}{2071}$, $\frac{1}{2071} a^{16} - \frac{393}{2071} a^{13} - \frac{235}{2071} a^{10} - \frac{937}{2071} a^{7} - \frac{519}{2071} a^{4} + \frac{480}{2071} a$, $\frac{1}{2071} a^{17} - \frac{393}{2071} a^{14} - \frac{235}{2071} a^{11} - \frac{937}{2071} a^{8} - \frac{519}{2071} a^{5} + \frac{480}{2071} a^{2}$

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:  \( \frac{350}{2071} a^{15} - \frac{2935}{2071} a^{12} + \frac{10945}{2071} a^{9} - \frac{13158}{2071} a^{6} + \frac{6811}{2071} a^{3} - \frac{1822}{2071} \) (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:  \( 23143.3269777 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9\times S_3$ (as 18T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_9\times S_3$
Character table for $C_9\times S_3$ is not computed

Intermediate fields

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

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{9}$ $18$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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

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) $1$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 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) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e3.18t1.1c1$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.18t1.1c2$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.18t1.1c3$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.9t1.1c1$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.18t1.1c4$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.9t1.1c2$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.18t1.1c5$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
1.3e3.9t1.1c3$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.9t1.1c4$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.9t1.1c5$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.9t1.1c6$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
1.3e3.18t1.1c6$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
2.3e5.3t2.1c1$2$ $ 3^{5}$ $x^{3} - 3$ $S_3$ (as 3T2) $1$ $0$
2.3e5.6t5.1c1$2$ $ 3^{5}$ $x^{6} - 3 x^{3} + 3$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.3e5.6t5.1c2$2$ $ 3^{5}$ $x^{6} - 3 x^{3} + 3$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3e6.18t16.2c1$2$ $ 3^{6}$ $x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.3e6.18t16.2c2$2$ $ 3^{6}$ $x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.3e6.18t16.2c3$2$ $ 3^{6}$ $x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.3e6.18t16.2c4$2$ $ 3^{6}$ $x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.3e6.18t16.2c5$2$ $ 3^{6}$ $x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1$ $C_9\times S_3$ (as 18T16) $0$ $0$
* 2.3e6.18t16.2c6$2$ $ 3^{6}$ $x^{18} - 9 x^{15} + 36 x^{12} - 53 x^{9} + 27 x^{6} + 1$ $C_9\times S_3$ (as 18T16) $0$ $0$

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 *.