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Properties

Label 18.2.74757002560...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{9}\cdot 5^{9}\cdot 7^{15}$
Root discriminant $31.12$
Ramified primes $2, 3, 5, 7$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-216, 0, 0, 540, 0, 0, -1574, 0, 0, -1161, 0, 0, -209, 0, 0, -27, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^15 - 209*x^12 - 1161*x^9 - 1574*x^6 + 540*x^3 - 216)
 
gp: K = bnfinit(x^18 - 27*x^15 - 209*x^12 - 1161*x^9 - 1574*x^6 + 540*x^3 - 216, 1)
 

Normalized defining polynomial

\( x^{18} - 27 x^{15} - 209 x^{12} - 1161 x^{9} - 1574 x^{6} + 540 x^{3} - 216 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(747570025601664552000000000=2^{12}\cdot 3^{9}\cdot 5^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.12$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{21} a^{9} - \frac{1}{7} a^{6} - \frac{4}{21} a^{3} + \frac{2}{7}$, $\frac{1}{21} a^{10} - \frac{1}{7} a^{7} - \frac{4}{21} a^{4} + \frac{2}{7} a$, $\frac{1}{63} a^{11} - \frac{1}{63} a^{9} - \frac{1}{21} a^{8} + \frac{1}{21} a^{6} - \frac{25}{63} a^{5} + \frac{1}{3} a^{4} + \frac{4}{63} a^{3} + \frac{2}{21} a^{2} + \frac{1}{3} a - \frac{3}{7}$, $\frac{1}{630} a^{12} - \frac{1}{63} a^{10} - \frac{1}{70} a^{9} + \frac{1}{21} a^{7} - \frac{1}{90} a^{6} + \frac{1}{3} a^{5} + \frac{4}{63} a^{4} - \frac{11}{210} a^{3} + \frac{1}{3} a^{2} - \frac{3}{7} a - \frac{2}{35}$, $\frac{1}{630} a^{13} - \frac{1}{70} a^{10} - \frac{1}{63} a^{9} - \frac{1}{90} a^{7} + \frac{1}{21} a^{6} - \frac{1}{3} a^{5} - \frac{11}{210} a^{4} + \frac{25}{63} a^{3} + \frac{1}{3} a^{2} + \frac{29}{105} a - \frac{3}{7}$, $\frac{1}{1260} a^{14} + \frac{1}{1260} a^{11} + \frac{1}{63} a^{10} + \frac{1}{63} a^{9} - \frac{37}{1260} a^{8} - \frac{1}{21} a^{7} - \frac{1}{21} a^{6} + \frac{347}{1260} a^{5} - \frac{4}{63} a^{4} - \frac{25}{63} a^{3} - \frac{31}{210} a^{2} + \frac{2}{21} a + \frac{3}{7}$, $\frac{1}{522900} a^{15} + \frac{187}{522900} a^{12} + \frac{1}{63} a^{10} + \frac{3289}{522900} a^{9} - \frac{1}{21} a^{7} + \frac{15373}{104580} a^{6} + \frac{1}{3} a^{5} - \frac{4}{63} a^{4} + \frac{54844}{130725} a^{3} - \frac{1}{3} a^{2} + \frac{2}{21} a + \frac{799}{14525}$, $\frac{1}{3137400} a^{16} - \frac{491}{1045800} a^{13} + \frac{18229}{3137400} a^{10} - \frac{1}{63} a^{9} + \frac{1}{9} a^{8} - \frac{28961}{209160} a^{7} + \frac{1}{21} a^{6} + \frac{1}{3} a^{5} - \frac{280061}{784350} a^{4} + \frac{4}{63} a^{3} - \frac{4}{9} a^{2} - \frac{26591}{87150} a - \frac{2}{21}$, $\frac{1}{3137400} a^{17} + \frac{113}{348600} a^{14} + \frac{20719}{3137400} a^{11} - \frac{1}{63} a^{9} + \frac{11539}{69720} a^{8} + \frac{1}{21} a^{6} - \frac{18301}{224100} a^{5} + \frac{1}{3} a^{4} - \frac{17}{63} a^{3} - \frac{6576}{14525} a^{2} - \frac{3}{7}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $9$
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:  \( 2565657.342468663 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{105}) \), 3.1.588.1, 3.1.140.1, 6.2.226894500.1 x2, 6.2.907578000.1, 6.2.18522000.1, 9.1.177885288000.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$