↑

Properties

Label 18.0.25815178338...6864.7
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}$
Root discriminant $29.33$
Ramified primes $2, 3$
Class number $19$
Class group $[19]$
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 81, 0, 540, 0, 1386, 0, 1782, 0, 1287, 0, 546, 0, 135, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 + 1782*x^8 + 1386*x^6 + 540*x^4 + 81*x^2 + 1)
 
gp: K = bnfinit(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 + 1782*x^8 + 1386*x^6 + 540*x^4 + 81*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1 \)

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:  \(-258151783382020583032356864=-\,2^{18}\cdot 3^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.33$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(108=2^{2}\cdot 3^{3}\)
Dirichlet character group:    $\lbrace$$\chi_{108}(1,·)$, $\chi_{108}(67,·)$, $\chi_{108}(7,·)$, $\chi_{108}(73,·)$, $\chi_{108}(13,·)$, $\chi_{108}(79,·)$, $\chi_{108}(19,·)$, $\chi_{108}(85,·)$, $\chi_{108}(25,·)$, $\chi_{108}(91,·)$, $\chi_{108}(31,·)$, $\chi_{108}(97,·)$, $\chi_{108}(37,·)$, $\chi_{108}(103,·)$, $\chi_{108}(43,·)$, $\chi_{108}(49,·)$, $\chi_{108}(55,·)$, $\chi_{108}(61,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{19}$, which has order $19$

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:  \( a^{9} + 9 a^{7} + 27 a^{5} + 30 a^{3} + 9 a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a^{3} + 3 a \),  \( a^{9} + 9 a^{7} + 27 a^{5} + 29 a^{3} + 6 a \),  \( a^{17} + 16 a^{15} + 104 a^{13} + 353 a^{11} + 670 a^{9} + 707 a^{7} + 387 a^{5} + 93 a^{3} + 6 a \),  \( a^{11} + 11 a^{9} + 44 a^{7} + 77 a^{5} + 55 a^{3} + 11 a \),  \( a^{7} + 7 a^{5} + 14 a^{3} + 7 a \),  \( a^{8} + 9 a^{6} + 27 a^{4} + 30 a^{2} + 9 \),  \( a^{12} + 11 a^{10} + 45 a^{8} + 84 a^{6} + 70 a^{4} + 21 a^{2} + 1 \),  \( a^{5} + 5 a^{3} + 5 a \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 40934.0329443 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), 6.0.419904.1, \(\Q(\zeta_{27})^+\)

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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
2Data not computed
3Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 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) $0$ $1$
* 1.2e2_3e3.18t1.1c1$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (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) $0$ $1$
* 1.2e2_3e3.18t1.1c2$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2.6t1.2c1$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 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) $0$ $1$
* 1.2e2_3e3.18t1.1c3$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 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) $0$ $1$
* 1.2e2_3e3.18t1.1c4$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_3e2.6t1.2c2$1$ $ 2^{2} \cdot 3^{2}$ $x^{6} + 6 x^{4} + 9 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
* 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) $0$ $1$
* 1.2e2_3e3.18t1.1c5$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (as 18T1) $0$ $-1$
* 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) $0$ $1$
* 1.2e2_3e3.18t1.1c6$1$ $ 2^{2} \cdot 3^{3}$ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} + 1782 x^{8} + 1386 x^{6} + 540 x^{4} + 81 x^{2} + 1$ $C_{18}$ (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 *.