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Properties

Label 18.0.40830016558...0672.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 7^{6}$
Root discriminant $13.97$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 6, -6, 6, -24, 15, 12, 84, 68, 84, 12, 15, -24, 6, -6, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 6*x^15 + 6*x^14 - 24*x^13 + 15*x^12 + 12*x^11 + 84*x^10 + 68*x^9 + 84*x^8 + 12*x^7 + 15*x^6 - 24*x^5 + 6*x^4 - 6*x^3 + 6*x^2 + 1)
 
gp: K = bnfinit(x^18 + 6*x^16 - 6*x^15 + 6*x^14 - 24*x^13 + 15*x^12 + 12*x^11 + 84*x^10 + 68*x^9 + 84*x^8 + 12*x^7 + 15*x^6 - 24*x^5 + 6*x^4 - 6*x^3 + 6*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 6 x^{15} + 6 x^{14} - 24 x^{13} + 15 x^{12} + 12 x^{11} + 84 x^{10} + 68 x^{9} + 84 x^{8} + 12 x^{7} + 15 x^{6} - 24 x^{5} + 6 x^{4} - 6 x^{3} + 6 x^{2} + 1 \)

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:  \(-408300165580224540672=-\,2^{12}\cdot 3^{25}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.97$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{8} a^{13} - \frac{1}{16} a^{12} - \frac{1}{32} a^{11} + \frac{3}{32} a^{10} - \frac{3}{32} a^{9} + \frac{1}{32} a^{8} - \frac{5}{32} a^{7} + \frac{7}{32} a^{6} + \frac{1}{32} a^{5} - \frac{3}{32} a^{4} - \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{11}{32} a + \frac{11}{32}$, $\frac{1}{704} a^{16} + \frac{1}{88} a^{15} - \frac{85}{704} a^{14} - \frac{39}{352} a^{13} - \frac{27}{704} a^{12} + \frac{29}{352} a^{11} + \frac{1}{8} a^{10} + \frac{43}{352} a^{9} + \frac{17}{176} a^{8} + \frac{21}{352} a^{7} - \frac{1}{4} a^{6} - \frac{125}{352} a^{5} + \frac{215}{704} a^{4} - \frac{149}{352} a^{3} - \frac{63}{704} a^{2} + \frac{3}{22} a + \frac{155}{704}$, $\frac{1}{1408} a^{17} - \frac{1}{1408} a^{16} + \frac{19}{1408} a^{15} + \frac{159}{1408} a^{14} + \frac{147}{1408} a^{13} - \frac{51}{1408} a^{12} + \frac{135}{704} a^{11} + \frac{175}{704} a^{10} + \frac{175}{704} a^{9} - \frac{109}{704} a^{8} - \frac{101}{704} a^{7} - \frac{213}{704} a^{6} - \frac{351}{1408} a^{5} + \frac{21}{128} a^{4} - \frac{21}{1408} a^{3} + \frac{135}{1408} a^{2} - \frac{533}{1408} a + \frac{13}{1408}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1049}{704} a^{17} + \frac{53}{352} a^{16} + \frac{5939}{704} a^{15} - \frac{685}{88} a^{14} + \frac{3569}{704} a^{13} - \frac{5375}{176} a^{12} + \frac{2591}{176} a^{11} + \frac{11315}{352} a^{10} + \frac{2549}{22} a^{9} + \frac{38185}{352} a^{8} + \frac{17893}{176} a^{7} + \frac{4483}{352} a^{6} + \frac{763}{704} a^{5} - \frac{413}{16} a^{4} + \frac{2149}{704} a^{3} + \frac{985}{352} a^{2} + \frac{2851}{704} a + \frac{763}{352} \) (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:  \( 4190.38862513 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

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

\(\Q(\sqrt{-3}) \), 3.3.756.1, 3.1.108.1 x3, 6.0.1714608.1, 6.0.34992.1, 9.3.11666192832.1 x3

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: 6.0.1000188.1
Degree 9 sibling: 9.3.11666192832.1
Degree 12 sibling: 12.0.9003384318096.1
Degree 18 siblings: Deg 18, Deg 18

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 ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.1$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$$[\ ]_{3}^{2}$
3Data not computed
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$