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Properties

Label 18.0.21458931758...9363.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 31^{12}$
Root discriminant $62.55$
Ramified primes $3, 7, 31$
Class number $324$ (GRH)
Class group $[18, 18]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19683, 0, -54837, 0, 34545, 0, 15760, 0, -9045, 0, -690, 0, 601, 0, 54, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 + 54*x^14 + 601*x^12 - 690*x^10 - 9045*x^8 + 15760*x^6 + 34545*x^4 - 54837*x^2 + 19683)
 
gp: K = bnfinit(x^18 + 3*x^16 + 54*x^14 + 601*x^12 - 690*x^10 - 9045*x^8 + 15760*x^6 + 34545*x^4 - 54837*x^2 + 19683, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} + 54 x^{14} + 601 x^{12} - 690 x^{10} - 9045 x^{8} + 15760 x^{6} + 34545 x^{4} - 54837 x^{2} + 19683 \)

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:  \(-214589317581007476486780131079363=-\,3^{9}\cdot 7^{12}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.55$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{2}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{90} a^{10} + \frac{1}{30} a^{8} - \frac{1}{6} a^{7} + \frac{1}{30} a^{6} - \frac{1}{6} a^{5} - \frac{43}{90} a^{4} - \frac{1}{6} a^{3} - \frac{1}{5} a^{2} + \frac{1}{10}$, $\frac{1}{90} a^{11} - \frac{1}{45} a^{9} + \frac{1}{30} a^{7} + \frac{17}{90} a^{5} + \frac{1}{45} a^{3} - \frac{1}{2} a^{2} + \frac{4}{15} a - \frac{1}{2}$, $\frac{1}{270} a^{12} - \frac{1}{270} a^{10} + \frac{1}{45} a^{8} - \frac{1}{6} a^{7} - \frac{1}{27} a^{6} - \frac{1}{6} a^{5} + \frac{19}{270} a^{4} + \frac{1}{3} a^{3} + \frac{11}{45} a^{2} - \frac{1}{2} a - \frac{3}{10}$, $\frac{1}{270} a^{13} - \frac{1}{270} a^{11} + \frac{1}{45} a^{9} - \frac{1}{6} a^{8} - \frac{1}{27} a^{7} - \frac{1}{6} a^{6} + \frac{19}{270} a^{5} + \frac{1}{3} a^{4} + \frac{11}{45} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a$, $\frac{1}{6750} a^{14} - \frac{1}{1125} a^{12} - \frac{31}{6750} a^{10} - \frac{1111}{6750} a^{8} + \frac{103}{1125} a^{6} - \frac{1}{2} a^{5} + \frac{11}{3375} a^{4} - \frac{1}{2} a^{3} + \frac{52}{225} a^{2} - \frac{7}{125}$, $\frac{1}{6750} a^{15} - \frac{1}{1125} a^{13} - \frac{31}{6750} a^{11} + \frac{7}{3375} a^{9} - \frac{1}{6} a^{8} + \frac{103}{1125} a^{7} + \frac{11}{3375} a^{5} - \frac{98}{225} a^{3} + \frac{1}{6} a^{2} + \frac{111}{250} a - \frac{1}{2}$, $\frac{1}{78659606250} a^{16} - \frac{722513}{26219868750} a^{14} - \frac{25776311}{26219868750} a^{12} + \frac{142895081}{39329803125} a^{10} - \frac{391395623}{26219868750} a^{8} - \frac{1}{6} a^{7} - \frac{1859459987}{13109934375} a^{6} + \frac{1}{3} a^{5} + \frac{5802411242}{39329803125} a^{4} - \frac{1}{6} a^{3} + \frac{5398504307}{13109934375} a^{2} - \frac{100009573}{485553125}$, $\frac{1}{707936456250} a^{17} + \frac{5465381}{117989409375} a^{15} - \frac{10711477}{13109934375} a^{13} + \frac{367363087}{707936456250} a^{11} - \frac{4306896023}{235978818750} a^{9} - \frac{1}{6} a^{8} - \frac{4263541711}{26219868750} a^{7} - \frac{1}{6} a^{6} - \frac{2261655058}{353968228125} a^{5} - \frac{1}{6} a^{4} - \frac{30668381818}{117989409375} a^{3} - \frac{1}{2} a^{2} + \frac{983745002}{4369978125} a$

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:  \( -\frac{4096153}{28317458250} a^{17} - \frac{2427098}{4719576375} a^{15} - \frac{4264649}{524397375} a^{13} - \frac{2586070681}{28317458250} a^{11} + \frac{222978877}{4719576375} a^{9} + \frac{1393217413}{1048794750} a^{7} - \frac{20063675606}{14158729125} a^{5} - \frac{26996786471}{4719576375} a^{3} + \frac{1340863403}{349598250} a + \frac{1}{2} \) (order $6$)
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:  \( 107864181.516 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.141267.1 x3, 3.3.47089.2, 6.0.59869095867.1, 6.0.1271403.2 x2, 6.0.59869095867.4, 9.3.2819175855281163.4 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: 6.0.1271403.2
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$31$31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31.9.6.1$x^{9} + 837 x^{6} + 232562 x^{3} + 21717639$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$