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Properties

Label 18.0.44428426669...8107.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 17^{12}$
Root discriminant $34.35$
Ramified primes $3, 17$
Class number $9$ (GRH)
Class group $[3, 3]$ (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![1728, 0, 0, -1368, 0, 0, 793, 0, 0, -342, 0, 0, 127, 0, 0, -18, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^15 + 127*x^12 - 342*x^9 + 793*x^6 - 1368*x^3 + 1728)
 
gp: K = bnfinit(x^18 - 18*x^15 + 127*x^12 - 342*x^9 + 793*x^6 - 1368*x^3 + 1728, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{15} + 127 x^{12} - 342 x^{9} + 793 x^{6} - 1368 x^{3} + 1728 \)

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:  \(-4442842666916764759667098107=-\,3^{27}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.35$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} + \frac{3}{8} a^{6} + \frac{3}{8} a^{3}$, $\frac{1}{16} a^{10} + \frac{3}{16} a^{7} - \frac{5}{16} a^{4} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{8} - \frac{1}{16} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{6} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{13} + \frac{1}{8} a^{7} + \frac{7}{16} a^{4} - \frac{1}{2} a$, $\frac{1}{48} a^{14} - \frac{1}{24} a^{8} - \frac{7}{16} a^{5} - \frac{5}{12} a^{2}$, $\frac{1}{852816} a^{15} + \frac{2789}{142136} a^{12} + \frac{2267}{106602} a^{9} + \frac{52711}{284272} a^{6} - \frac{70349}{213204} a^{3} - \frac{140}{17767}$, $\frac{1}{2558448} a^{16} - \frac{4063}{284272} a^{13} + \frac{2267}{319806} a^{10} - \frac{41653}{284272} a^{7} - \frac{228095}{2558448} a^{4} - \frac{5969}{17767} a$, $\frac{1}{5116896} a^{17} + \frac{2789}{852816} a^{14} - \frac{141767}{5116896} a^{11} - \frac{13399}{213204} a^{8} - \frac{1187513}{5116896} a^{5} + \frac{53021}{213204} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{73}{159903} a^{16} + \frac{1795}{284272} a^{13} - \frac{18913}{639612} a^{10} + \frac{2329}{142136} a^{7} - \frac{1102759}{2558448} a^{4} + \frac{7128}{17767} a \) (order $18$)
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:  \( 5779497.98045 \) (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

\(\Q(\sqrt{-3}) \), 3.1.867.1 x3, \(\Q(\zeta_{9})^+\), 6.0.2255067.2, 6.0.1643943843.1 x2, \(\Q(\zeta_{9})\), 9.3.38483081420787.3 x3

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

Sibling fields

Degree 6 sibling: 6.0.1643943843.1
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$