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Properties

Label 18.0.57701420049...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 5^{9}$
Root discriminant $34.86$
Ramified primes $3, 5$
Class number $74$ (GRH)
Class group $[74]$ (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![5779, -684, 81, -2280, 540, -2052, 1386, -684, 1782, -76, 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 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779)
 
gp: K = bnfinit(x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779 \)

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:  \(-5770142004982097067662109375=-\,3^{45}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.86$
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$$\chi_{135}(1,·)$, $\chi_{135}(134,·)$, $\chi_{135}(74,·)$, $\chi_{135}(76,·)$, $\chi_{135}(14,·)$, $\chi_{135}(16,·)$, $\chi_{135}(89,·)$, $\chi_{135}(91,·)$, $\chi_{135}(29,·)$, $\chi_{135}(31,·)$, $\chi_{135}(104,·)$, $\chi_{135}(106,·)$, $\chi_{135}(44,·)$, $\chi_{135}(46,·)$, $\chi_{135}(119,·)$, $\chi_{135}(121,·)$, $\chi_{135}(59,·)$, $\chi_{135}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{34} a^{9} + \frac{9}{34} a^{7} - \frac{7}{34} a^{5} - \frac{2}{17} a^{3} + \frac{9}{34} a + \frac{13}{34}$, $\frac{1}{34} a^{10} + \frac{9}{34} a^{8} - \frac{7}{34} a^{6} - \frac{2}{17} a^{4} + \frac{9}{34} a^{2} + \frac{13}{34} a$, $\frac{1}{34} a^{11} + \frac{7}{17} a^{7} - \frac{9}{34} a^{5} + \frac{11}{34} a^{3} + \frac{13}{34} a^{2} - \frac{13}{34} a - \frac{15}{34}$, $\frac{1}{34} a^{12} + \frac{7}{17} a^{8} - \frac{9}{34} a^{6} + \frac{11}{34} a^{4} + \frac{13}{34} a^{3} - \frac{13}{34} a^{2} - \frac{15}{34} a$, $\frac{1}{34} a^{13} + \frac{1}{34} a^{7} + \frac{7}{34} a^{5} + \frac{13}{34} a^{4} + \frac{9}{34} a^{3} - \frac{15}{34} a^{2} + \frac{5}{17} a - \frac{6}{17}$, $\frac{1}{196418} a^{14} - \frac{38}{98209} a^{13} + \frac{7}{98209} a^{12} - \frac{494}{98209} a^{11} + \frac{77}{196418} a^{10} + \frac{837}{196418} a^{9} + \frac{105}{98209} a^{8} + \frac{2361}{11554} a^{7} + \frac{147}{98209} a^{6} + \frac{33523}{98209} a^{5} + \frac{5973}{196418} a^{4} - \frac{12693}{196418} a^{3} + \frac{23157}{196418} a^{2} + \frac{34168}{98209} a + \frac{86657}{196418}$, $\frac{1}{196418} a^{15} + \frac{15}{196418} a^{13} + \frac{38}{98209} a^{12} + \frac{45}{98209} a^{11} + \frac{456}{98209} a^{10} + \frac{275}{196418} a^{9} + \frac{2052}{98209} a^{8} + \frac{225}{98209} a^{7} - \frac{3917}{11554} a^{6} + \frac{189}{98209} a^{5} - \frac{24895}{98209} a^{4} - \frac{5637}{196418} a^{3} - \frac{83919}{196418} a^{2} - \frac{8658}{98209} a + \frac{23184}{98209}$, $\frac{1}{196418} a^{16} + \frac{608}{98209} a^{13} - \frac{60}{98209} a^{12} - \frac{1599}{196418} a^{11} - \frac{440}{98209} a^{10} - \frac{1337}{98209} a^{9} - \frac{1350}{98209} a^{8} - \frac{73613}{196418} a^{7} - \frac{2016}{98209} a^{6} + \frac{21075}{98209} a^{5} - \frac{47616}{98209} a^{4} + \frac{89145}{196418} a^{3} - \frac{360}{98209} a^{2} + \frac{42148}{98209} a + \frac{17301}{196418}$, $\frac{1}{196418} a^{17} - \frac{4}{5777} a^{13} - \frac{38}{5777} a^{12} - \frac{32}{5777} a^{11} + \frac{1903}{196418} a^{10} + \frac{2037}{196418} a^{9} + \frac{5935}{98209} a^{8} + \frac{45465}{196418} a^{7} + \frac{9856}{98209} a^{6} - \frac{46151}{196418} a^{5} + \frac{32279}{98209} a^{4} + \frac{72925}{196418} a^{3} - \frac{4975}{98209} a^{2} - \frac{11673}{98209} a - \frac{77533}{196418}$

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

Class group and class number

$C_{74}$, which has order $74$ (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:  $8$
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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), 6.0.2460375.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R R $18$ $18$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
3Data not computed
5Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_5.2t1.1c1$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (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_5.18t1.1c1$1$ $ 3^{3} \cdot 5 $ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$ $C_{18}$ (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_5.18t1.1c2$1$ $ 3^{3} \cdot 5 $ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5.6t1.2c1$1$ $ 3^{2} \cdot 5 $ $x^{6} + 6 x^{4} - 4 x^{3} + 9 x^{2} - 12 x + 19$ $C_6$ (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_5.18t1.1c3$1$ $ 3^{3} \cdot 5 $ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$ $C_{18}$ (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_5.18t1.1c4$1$ $ 3^{3} \cdot 5 $ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$ $C_{18}$ (as 18T1) $0$ $-1$
* 1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.3e2_5.6t1.2c2$1$ $ 3^{2} \cdot 5 $ $x^{6} + 6 x^{4} - 4 x^{3} + 9 x^{2} - 12 x + 19$ $C_6$ (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_5.18t1.1c5$1$ $ 3^{3} \cdot 5 $ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$ $C_{18}$ (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_5.18t1.1c6$1$ $ 3^{3} \cdot 5 $ $x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$ $C_{18}$ (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 *.