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Properties

Label 18.0.17249251837...9984.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 13^{12}$
Root discriminant $47.84$
Ramified primes $2, 3, 13$
Class number $324$ (GRH)
Class group $[18, 18]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2809, 0, 24555, 0, 74337, 0, 99645, 0, 69201, 0, 26937, 0, 5996, 0, 738, 0, 45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 45*x^16 + 738*x^14 + 5996*x^12 + 26937*x^10 + 69201*x^8 + 99645*x^6 + 74337*x^4 + 24555*x^2 + 2809)
 
gp: K = bnfinit(x^18 + 45*x^16 + 738*x^14 + 5996*x^12 + 26937*x^10 + 69201*x^8 + 99645*x^6 + 74337*x^4 + 24555*x^2 + 2809, 1)
 

Normalized defining polynomial

\( x^{18} + 45 x^{16} + 738 x^{14} + 5996 x^{12} + 26937 x^{10} + 69201 x^{8} + 99645 x^{6} + 74337 x^{4} + 24555 x^{2} + 2809 \)

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:  \(-1724925183796757382490845609984=-\,2^{18}\cdot 3^{24}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.84$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(468=2^{2}\cdot 3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(451,·)$, $\chi_{468}(133,·)$, $\chi_{468}(391,·)$, $\chi_{468}(139,·)$, $\chi_{468}(79,·)$, $\chi_{468}(211,·)$, $\chi_{468}(217,·)$, $\chi_{468}(157,·)$, $\chi_{468}(445,·)$, $\chi_{468}(289,·)$, $\chi_{468}(295,·)$, $\chi_{468}(235,·)$, $\chi_{468}(367,·)$, $\chi_{468}(373,·)$, $\chi_{468}(55,·)$, $\chi_{468}(313,·)$, $\chi_{468}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{6904} a^{14} + \frac{785}{6904} a^{12} + \frac{293}{6904} a^{10} + \frac{463}{6904} a^{8} - \frac{232}{863} a^{6} - \frac{131}{863} a^{4} + \frac{2657}{6904} a^{2} + \frac{1287}{6904}$, $\frac{1}{6904} a^{15} + \frac{785}{6904} a^{13} + \frac{293}{6904} a^{11} + \frac{463}{6904} a^{9} - \frac{232}{863} a^{7} - \frac{131}{863} a^{5} + \frac{2657}{6904} a^{3} + \frac{1287}{6904} a$, $\frac{1}{47665216} a^{16} + \frac{751}{11916304} a^{14} - \frac{1513365}{23832608} a^{12} + \frac{2486267}{23832608} a^{10} - \frac{2880397}{47665216} a^{8} + \frac{4524071}{23832608} a^{6} + \frac{21449343}{47665216} a^{4} - \frac{4735567}{23832608} a^{2} - \frac{9830247}{47665216}$, $\frac{1}{2526256448} a^{17} + \frac{21463}{631564112} a^{15} - \frac{106033021}{1263128224} a^{13} - \frac{62832477}{1263128224} a^{11} - \frac{238596765}{2526256448} a^{9} - \frac{560927337}{1263128224} a^{7} + \frac{649602879}{2526256448} a^{5} - \frac{55542103}{1263128224} a^{3} - \frac{236861383}{2526256448} a$

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

Class group and class number

$C_{18}\times C_{18}$, which has order $324$ (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:  \( -\frac{74385}{315782056} a^{17} - \frac{597337}{39472757} a^{15} - \frac{57456135}{157891028} a^{13} - \frac{658084131}{157891028} a^{11} - \frac{7835651195}{315782056} a^{9} - \frac{12340599279}{157891028} a^{7} - \frac{39206115879}{315782056} a^{5} - \frac{13433248857}{157891028} a^{3} - \frac{5276234421}{315782056} a \) (order $4$)
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:  \( 400417.136445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.3.13689.2, 3.3.169.1, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 6.0.11992878144.4, 6.0.1827904.1, 6.0.419904.1, 6.0.11992878144.3, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$