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)

Related objects

Show commands for: SageMath / Pari/GP / Magma

Normalizeddefining 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

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 *.