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Properties

Label 18.0.43232495562...1152.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 7^{12}$
Root discriminant $30.18$
Ramified primes $2, 3, 7$
Class number $9$
Class group $[3, 3]$
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![1, 0, 0, -8, 0, 0, 155, 0, 0, -260, 0, 0, 155, 0, 0, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^15 + 155*x^12 - 260*x^9 + 155*x^6 - 8*x^3 + 1)
 
gp: K = bnfinit(x^18 - 8*x^15 + 155*x^12 - 260*x^9 + 155*x^6 - 8*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{15} + 155 x^{12} - 260 x^{9} + 155 x^{6} - 8 x^{3} + 1 \)

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:  \(-432324955623130532869681152=-\,2^{12}\cdot 3^{27}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.18$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{14} a^{9} + \frac{3}{14} a^{6} + \frac{3}{14} a^{3} + \frac{1}{14}$, $\frac{1}{14} a^{10} + \frac{3}{14} a^{7} + \frac{3}{14} a^{4} + \frac{1}{14} a$, $\frac{1}{14} a^{11} + \frac{3}{14} a^{8} + \frac{3}{14} a^{5} + \frac{1}{14} a^{2}$, $\frac{1}{126} a^{12} - \frac{31}{63} a^{6} - \frac{2}{7} a^{3} - \frac{17}{126}$, $\frac{1}{126} a^{13} - \frac{31}{63} a^{7} - \frac{2}{7} a^{4} - \frac{17}{126} a$, $\frac{1}{126} a^{14} - \frac{31}{63} a^{8} - \frac{2}{7} a^{5} - \frac{17}{126} a^{2}$, $\frac{1}{756} a^{15} - \frac{1}{756} a^{12} - \frac{2}{189} a^{9} + \frac{31}{378} a^{6} + \frac{55}{756} a^{3} + \frac{323}{756}$, $\frac{1}{756} a^{16} - \frac{1}{756} a^{13} - \frac{2}{189} a^{10} + \frac{31}{378} a^{7} + \frac{55}{756} a^{4} + \frac{323}{756} a$, $\frac{1}{756} a^{17} - \frac{1}{756} a^{14} - \frac{2}{189} a^{11} + \frac{31}{378} a^{8} + \frac{55}{756} a^{5} + \frac{323}{756} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:  \( \frac{1}{12} a^{15} - \frac{55}{84} a^{12} + \frac{77}{6} a^{9} - \frac{418}{21} a^{6} + \frac{979}{84} a^{3} + \frac{11}{84} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2670473.74348 \)
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

\(\Q(\sqrt{-3}) \), 3.1.588.1 x3, 3.3.3969.1, 6.0.1037232.1, 6.0.47258883.2, 6.0.15431472.2 x2, 9.3.12004512424128.4 x3

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

Sibling fields

Degree 6 sibling: 6.0.15431472.2
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$