Properties

Label 18.0.295...643.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.954\times 10^{21}$
Root discriminant $15.59$
Ramified prime $3$
Class number $1$
Class group trivial
Galois group $C_{18}$ (as 18T1)

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Normalized defining polynomial

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

\( x^{18} - x^{9} + 1 \)

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-2954312706550833698643\)\(\medspace = -\,3^{45}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.59$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $18$
This field is Galois and abelian over $\Q$.
Conductor:  \(27=3^{3}\)
Dirichlet character group:    $\lbrace$$\chi_{27}(1,·)$, $\chi_{27}(2,·)$, $\chi_{27}(4,·)$, $\chi_{27}(5,·)$, $\chi_{27}(7,·)$, $\chi_{27}(8,·)$, $\chi_{27}(10,·)$, $\chi_{27}(11,·)$, $\chi_{27}(13,·)$, $\chi_{27}(14,·)$, $\chi_{27}(16,·)$, $\chi_{27}(17,·)$, $\chi_{27}(19,·)$, $\chi_{27}(20,·)$, $\chi_{27}(22,·)$, $\chi_{27}(23,·)$, $\chi_{27}(25,·)$$\chi_{27}(26,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a \) (order $54$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{13} + a \),  \( a^{15} - a^{9} + 1 \),  \( a^{14} + a^{9} - a^{5} \),  \( a^{13} + a^{9} \),  \( a^{8} - a^{7} \),  \( a^{16} + a^{9} - a^{7} \),  \( a^{4} + a^{2} \),  \( a^{10} + a^{9} - a \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 40934.0329443 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{9}\cdot 40934.0329443 \cdot 1}{54\sqrt{2954312706550833698643}}\approx 0.212853776024$

Galois group

$C_{18}$ (as 18T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), \(\Q(\zeta_{27})^+\)

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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

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]])
 
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];
 

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.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.27.9t1.a.a$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
* 1.27.18t1.a.a$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.b$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
* 1.27.18t1.a.b$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.9.6t1.a.a$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.27.9t1.a.c$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
* 1.27.18t1.a.c$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.d$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
* 1.27.18t1.a.d$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.9.6t1.a.b$1$ $ 3^{2}$ $x^{6} - x^{3} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.27.9t1.a.e$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
* 1.27.18t1.a.e$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.f$1$ $ 3^{3}$ $x^{9} - 9 x^{7} + 27 x^{5} - 30 x^{3} + 9 x - 1$ $C_9$ (as 9T1) $0$ $1$
* 1.27.18t1.a.f$1$ $ 3^{3}$ $x^{18} - x^{9} + 1$ $C_{18}$ (as 18T1) $0$ $-1$

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 *.