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Properties

Label 18.0.35034567037...3571.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 17^{9}$
Root discriminant $64.27$
Ramified primes $3, 17$
Class number $11096$ (GRH)
Class group $[2, 2, 2774]$ (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![22810681, -10812672, 5308416, -9010560, 8847360, -2027376, 5677056, -168948, 1824768, -4693, 329472, 0, 34944, 0, 2160, 0, 72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 72*x^16 + 2160*x^14 + 34944*x^12 + 329472*x^10 - 4693*x^9 + 1824768*x^8 - 168948*x^7 + 5677056*x^6 - 2027376*x^5 + 8847360*x^4 - 9010560*x^3 + 5308416*x^2 - 10812672*x + 22810681)
 
gp: K = bnfinit(x^18 + 72*x^16 + 2160*x^14 + 34944*x^12 + 329472*x^10 - 4693*x^9 + 1824768*x^8 - 168948*x^7 + 5677056*x^6 - 2027376*x^5 + 8847360*x^4 - 9010560*x^3 + 5308416*x^2 - 10812672*x + 22810681, 1)
 

Normalized defining polynomial

\( x^{18} + 72 x^{16} + 2160 x^{14} + 34944 x^{12} + 329472 x^{10} - 4693 x^{9} + 1824768 x^{8} - 168948 x^{7} + 5677056 x^{6} - 2027376 x^{5} + 8847360 x^{4} - 9010560 x^{3} + 5308416 x^{2} - 10812672 x + 22810681 \)

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:  \(-350345670377968069507061800493571=-\,3^{45}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.27$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(459=3^{3}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{459}(256,·)$, $\chi_{459}(1,·)$, $\chi_{459}(458,·)$, $\chi_{459}(203,·)$, $\chi_{459}(205,·)$, $\chi_{459}(407,·)$, $\chi_{459}(152,·)$, $\chi_{459}(409,·)$, $\chi_{459}(154,·)$, $\chi_{459}(356,·)$, $\chi_{459}(101,·)$, $\chi_{459}(358,·)$, $\chi_{459}(103,·)$, $\chi_{459}(305,·)$, $\chi_{459}(50,·)$, $\chi_{459}(307,·)$, $\chi_{459}(52,·)$, $\chi_{459}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1165} a^{9} + \frac{36}{1165} a^{7} + \frac{432}{1165} a^{5} - \frac{82}{233} a^{3} - \frac{26}{1165} a + \frac{566}{1165}$, $\frac{1}{1165} a^{10} + \frac{36}{1165} a^{8} + \frac{432}{1165} a^{6} - \frac{82}{233} a^{4} - \frac{26}{1165} a^{2} + \frac{566}{1165} a$, $\frac{1}{1165} a^{11} + \frac{301}{1165} a^{7} + \frac{348}{1165} a^{5} - \frac{411}{1165} a^{3} + \frac{566}{1165} a^{2} - \frac{229}{1165} a - \frac{571}{1165}$, $\frac{1}{1165} a^{12} + \frac{301}{1165} a^{8} + \frac{348}{1165} a^{6} - \frac{411}{1165} a^{4} + \frac{566}{1165} a^{3} - \frac{229}{1165} a^{2} - \frac{571}{1165} a$, $\frac{1}{1165} a^{13} - \frac{3}{1165} a^{7} + \frac{37}{1165} a^{5} + \frac{566}{1165} a^{4} - \frac{309}{1165} a^{3} - \frac{571}{1165} a^{2} - \frac{329}{1165} a - \frac{276}{1165}$, $\frac{1}{25963647845} a^{14} - \frac{261187}{25963647845} a^{13} + \frac{56}{25963647845} a^{12} + \frac{8704669}{25963647845} a^{11} + \frac{1232}{25963647845} a^{10} - \frac{4197764}{25963647845} a^{9} + \frac{2688}{5192729569} a^{8} - \frac{12235412784}{25963647845} a^{7} + \frac{75264}{25963647845} a^{6} - \frac{3031803574}{25963647845} a^{5} + \frac{10095936733}{25963647845} a^{4} - \frac{11546059826}{25963647845} a^{3} - \frac{7599459309}{25963647845} a^{2} + \frac{7130669616}{25963647845} a - \frac{5638424661}{25963647845}$, $\frac{1}{25963647845} a^{15} + \frac{12}{5192729569} a^{13} + \frac{1044748}{25963647845} a^{12} + \frac{288}{5192729569} a^{11} + \frac{5575118}{25963647845} a^{10} + \frac{3520}{5192729569} a^{9} - \frac{701958024}{25963647845} a^{8} + \frac{23040}{5192729569} a^{7} + \frac{3900644391}{25963647845} a^{6} + \frac{387072}{25963647845} a^{5} + \frac{6955325676}{25963647845} a^{4} + \frac{11544925014}{25963647845} a^{3} + \frac{10967608524}{25963647845} a^{2} + \frac{9449676392}{25963647845} a - \frac{11187168582}{25963647845}$, $\frac{1}{25963647845} a^{16} - \frac{1114085}{5192729569} a^{13} - \frac{384}{5192729569} a^{12} - \frac{4117983}{25963647845} a^{11} - \frac{11264}{5192729569} a^{10} - \frac{4364324}{25963647845} a^{9} - \frac{138240}{5192729569} a^{8} - \frac{300502266}{25963647845} a^{7} - \frac{4128768}{25963647845} a^{6} - \frac{12226583923}{25963647845} a^{5} - \frac{9661476969}{25963647845} a^{4} + \frac{5632223752}{25963647845} a^{3} - \frac{10642405941}{25963647845} a^{2} + \frac{10533774054}{25963647845} a - \frac{7512480521}{25963647845}$, $\frac{1}{25963647845} a^{17} - \frac{2176}{25963647845} a^{13} - \frac{836737}{5192729569} a^{12} - \frac{69632}{25963647845} a^{11} - \frac{5809768}{25963647845} a^{10} - \frac{191488}{5192729569} a^{9} - \frac{6823897502}{25963647845} a^{8} - \frac{6684672}{25963647845} a^{7} + \frac{7363997608}{25963647845} a^{6} - \frac{23396352}{25963647845} a^{5} + \frac{6577926613}{25963647845} a^{4} + \frac{5736524203}{25963647845} a^{3} + \frac{1894740309}{5192729569} a^{2} - \frac{8618145266}{25963647845} a - \frac{11003350412}{25963647845}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2774}$, which has order $11096$ (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

\(\Q(\sqrt{-51}) \), \(\Q(\zeta_{9})^+\), 6.0.96702579.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$