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Properties

Label 18.0.31329007244...0839.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 13^{9}$
Root discriminant $56.20$
Ramified primes $3, 13$
Class number $2524$ (GRH)
Class group $[2524]$ (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![3335149, -1319490, 531441, -1466100, 1180980, -439830, 1010394, -48870, 433026, -1810, 104247, 0, 14742, 0, 1215, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 54*x^16 + 1215*x^14 + 14742*x^12 + 104247*x^10 - 1810*x^9 + 433026*x^8 - 48870*x^7 + 1010394*x^6 - 439830*x^5 + 1180980*x^4 - 1466100*x^3 + 531441*x^2 - 1319490*x + 3335149)
 
gp: K = bnfinit(x^18 + 54*x^16 + 1215*x^14 + 14742*x^12 + 104247*x^10 - 1810*x^9 + 433026*x^8 - 48870*x^7 + 1010394*x^6 - 439830*x^5 + 1180980*x^4 - 1466100*x^3 + 531441*x^2 - 1319490*x + 3335149, 1)
 

Normalized defining polynomial

\( x^{18} + 54 x^{16} + 1215 x^{14} + 14742 x^{12} + 104247 x^{10} - 1810 x^{9} + 433026 x^{8} - 48870 x^{7} + 1010394 x^{6} - 439830 x^{5} + 1180980 x^{4} - 1466100 x^{3} + 531441 x^{2} - 1319490 x + 3335149 \)

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:  \(-31329007244264248949886964450839=-\,3^{45}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.20$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(351=3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{351}(1,·)$, $\chi_{351}(194,·)$, $\chi_{351}(196,·)$, $\chi_{351}(77,·)$, $\chi_{351}(79,·)$, $\chi_{351}(272,·)$, $\chi_{351}(274,·)$, $\chi_{351}(155,·)$, $\chi_{351}(157,·)$, $\chi_{351}(350,·)$, $\chi_{351}(38,·)$, $\chi_{351}(40,·)$, $\chi_{351}(233,·)$, $\chi_{351}(235,·)$, $\chi_{351}(116,·)$, $\chi_{351}(118,·)$, $\chi_{351}(311,·)$, $\chi_{351}(313,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{508} a^{9} + \frac{27}{508} a^{7} + \frac{243}{508} a^{5} - \frac{103}{254} a^{3} + \frac{221}{508} a - \frac{143}{508}$, $\frac{1}{508} a^{10} + \frac{27}{508} a^{8} + \frac{243}{508} a^{6} - \frac{103}{254} a^{4} + \frac{221}{508} a^{2} - \frac{143}{508} a$, $\frac{1}{508} a^{11} + \frac{11}{254} a^{7} - \frac{163}{508} a^{5} + \frac{195}{508} a^{3} - \frac{143}{508} a^{2} + \frac{129}{508} a - \frac{203}{508}$, $\frac{1}{508} a^{12} + \frac{11}{254} a^{8} - \frac{163}{508} a^{6} + \frac{195}{508} a^{4} - \frac{143}{508} a^{3} + \frac{129}{508} a^{2} - \frac{203}{508} a$, $\frac{1}{508} a^{13} - \frac{249}{508} a^{7} - \frac{71}{508} a^{5} - \frac{143}{508} a^{4} + \frac{89}{508} a^{3} - \frac{203}{508} a^{2} + \frac{109}{254} a + \frac{49}{254}$, $\frac{1}{1674257764} a^{14} - \frac{264482}{418564441} a^{13} + \frac{21}{837128882} a^{12} + \frac{1585987}{1674257764} a^{11} + \frac{693}{1674257764} a^{10} + \frac{179831}{418564441} a^{9} + \frac{2835}{837128882} a^{8} - \frac{89033201}{837128882} a^{7} + \frac{11907}{837128882} a^{6} + \frac{52650173}{418564441} a^{5} + \frac{234048221}{1674257764} a^{4} + \frac{11411708}{418564441} a^{3} + \frac{7424442}{418564441} a^{2} - \frac{67616494}{418564441} a + \frac{669048323}{1674257764}$, $\frac{1}{1674257764} a^{15} + \frac{45}{1674257764} a^{13} - \frac{121999}{1674257764} a^{12} + \frac{405}{837128882} a^{11} - \frac{1096181}{1674257764} a^{10} + \frac{7425}{1674257764} a^{9} + \frac{29694627}{1674257764} a^{8} + \frac{18225}{837128882} a^{7} - \frac{9684629}{1674257764} a^{6} + \frac{45927}{837128882} a^{5} + \frac{379585195}{1674257764} a^{4} - \frac{115301375}{837128882} a^{3} + \frac{722362917}{1674257764} a^{2} + \frac{400454037}{837128882} a - \frac{293275817}{837128882}$, $\frac{1}{1674257764} a^{16} + \frac{1343799}{1674257764} a^{13} - \frac{270}{418564441} a^{12} + \frac{20815}{837128882} a^{11} - \frac{5940}{418564441} a^{10} + \frac{310415}{837128882} a^{9} - \frac{54675}{418564441} a^{8} - \frac{147200705}{418564441} a^{7} - \frac{244944}{418564441} a^{6} - \frac{24165643}{418564441} a^{5} - \frac{816099601}{1674257764} a^{4} - \frac{349601189}{1674257764} a^{3} + \frac{68424123}{837128882} a^{2} - \frac{112316581}{1674257764} a + \frac{204240131}{837128882}$, $\frac{1}{1674257764} a^{17} - \frac{306}{418564441} a^{13} - \frac{369617}{1674257764} a^{12} - \frac{7344}{418564441} a^{11} - \frac{1221071}{1674257764} a^{10} - \frac{75735}{418564441} a^{9} - \frac{149131975}{837128882} a^{8} - \frac{396576}{418564441} a^{7} + \frac{447965167}{1674257764} a^{6} - \frac{1041012}{418564441} a^{5} + \frac{330977197}{1674257764} a^{4} - \frac{172570444}{418564441} a^{3} - \frac{315419811}{1674257764} a^{2} - \frac{21812969}{418564441} a - \frac{76975889}{418564441}$

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

Class group and class number

$C_{2524}$, which has order $2524$ (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

\(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), 6.0.43243551.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
13Data not computed