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Properties

Label 18.18.1923380668...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{44}\cdot 5^{9}$
Root discriminant $32.79$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 81, 120, -810, -297, 2601, 261, -3888, -76, 3042, 0, -1269, 0, 270, 0, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1)
 
gp: K = bnfinit(x^18 - 27*x^16 + 270*x^14 - 1269*x^12 + 3042*x^10 - 76*x^9 - 3888*x^8 + 261*x^7 + 2601*x^6 - 297*x^5 - 810*x^4 + 120*x^3 + 81*x^2 - 9*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1923380668327365689220703125=3^{44}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.79$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(135=3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{135}(64,·)$, $\chi_{135}(1,·)$, $\chi_{135}(4,·)$, $\chi_{135}(76,·)$, $\chi_{135}(79,·)$, $\chi_{135}(16,·)$, $\chi_{135}(19,·)$, $\chi_{135}(91,·)$, $\chi_{135}(94,·)$, $\chi_{135}(31,·)$, $\chi_{135}(34,·)$, $\chi_{135}(106,·)$, $\chi_{135}(109,·)$, $\chi_{135}(46,·)$, $\chi_{135}(49,·)$, $\chi_{135}(121,·)$, $\chi_{135}(124,·)$, $\chi_{135}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{252470006105599} a^{17} - \frac{125005862637351}{252470006105599} a^{16} - \frac{55802239894318}{252470006105599} a^{15} + \frac{20683531428349}{252470006105599} a^{14} + \frac{75527407604533}{252470006105599} a^{13} + \frac{3308342371889}{252470006105599} a^{12} + \frac{59644436246245}{252470006105599} a^{11} + \frac{3762488467568}{252470006105599} a^{10} - \frac{125164617995790}{252470006105599} a^{9} - \frac{30792598734353}{252470006105599} a^{8} + \frac{99294107283645}{252470006105599} a^{7} - \frac{103580502572143}{252470006105599} a^{6} - \frac{79925047077669}{252470006105599} a^{5} - \frac{114505757343394}{252470006105599} a^{4} - \frac{39216739194526}{252470006105599} a^{3} - \frac{87257331108262}{252470006105599} a^{2} + \frac{15847820505903}{252470006105599} a + \frac{97497629787280}{252470006105599}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 49974435.7673 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1, \(\Q(\zeta_{27})^+\)

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 $18$ R R $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
5Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (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) $0$ $1$
* 1.3e3_5.18t1.2c1$1$ $ 3^{3} \cdot 5 $ $x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1$ $C_{18}$ (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_5.18t1.2c2$1$ $ 3^{3} \cdot 5 $ $x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1$ $C_{18}$ (as 18T1) $0$ $1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5.6t1.1c1$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (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) $0$ $1$
* 1.3e3_5.18t1.2c3$1$ $ 3^{3} \cdot 5 $ $x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1$ $C_{18}$ (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) $0$ $1$
* 1.3e3_5.18t1.2c4$1$ $ 3^{3} \cdot 5 $ $x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1$ $C_{18}$ (as 18T1) $0$ $1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5.6t1.1c2$1$ $ 3^{2} \cdot 5 $ $x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$ $C_6$ (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) $0$ $1$
* 1.3e3_5.18t1.2c5$1$ $ 3^{3} \cdot 5 $ $x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1$ $C_{18}$ (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) $0$ $1$
* 1.3e3_5.18t1.2c6$1$ $ 3^{3} \cdot 5 $ $x^{18} - 27 x^{16} + 270 x^{14} - 1269 x^{12} + 3042 x^{10} - 76 x^{9} - 3888 x^{8} + 261 x^{7} + 2601 x^{6} - 297 x^{5} - 810 x^{4} + 120 x^{3} + 81 x^{2} - 9 x - 1$ $C_{18}$ (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 *.