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Properties

Label 18.0.22501648801...5616.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 11^{9}\cdot 13^{12}$
Root discriminant $29.11$
Ramified primes $2, 11, 13$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19683, -13122, 48114, -20412, 43497, -23733, 21735, -10863, 9291, -1429, 2504, 308, 406, 169, 56, 14, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 13*x^16 + 14*x^15 + 56*x^14 + 169*x^13 + 406*x^12 + 308*x^11 + 2504*x^10 - 1429*x^9 + 9291*x^8 - 10863*x^7 + 21735*x^6 - 23733*x^5 + 43497*x^4 - 20412*x^3 + 48114*x^2 - 13122*x + 19683)
 
gp: K = bnfinit(x^18 - x^17 + 13*x^16 + 14*x^15 + 56*x^14 + 169*x^13 + 406*x^12 + 308*x^11 + 2504*x^10 - 1429*x^9 + 9291*x^8 - 10863*x^7 + 21735*x^6 - 23733*x^5 + 43497*x^4 - 20412*x^3 + 48114*x^2 - 13122*x + 19683, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 13 x^{16} + 14 x^{15} + 56 x^{14} + 169 x^{13} + 406 x^{12} + 308 x^{11} + 2504 x^{10} - 1429 x^{9} + 9291 x^{8} - 10863 x^{7} + 21735 x^{6} - 23733 x^{5} + 43497 x^{4} - 20412 x^{3} + 48114 x^{2} - 13122 x + 19683 \)

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:  \(-225016488014952555075055616=-\,2^{12}\cdot 11^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.11$
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:  $2, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{4}{9} a^{9} - \frac{4}{9} a^{8} + \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{4}{27} a^{10} - \frac{4}{27} a^{9} - \frac{7}{27} a^{8} - \frac{11}{27} a^{7} + \frac{10}{27} a^{6} + \frac{2}{27} a^{5} + \frac{11}{27} a^{4} + \frac{11}{27} a^{3} + \frac{4}{9} a^{2}$, $\frac{1}{81} a^{13} - \frac{1}{81} a^{12} + \frac{4}{81} a^{11} - \frac{4}{81} a^{10} - \frac{34}{81} a^{9} + \frac{16}{81} a^{8} + \frac{10}{81} a^{7} + \frac{29}{81} a^{6} + \frac{38}{81} a^{5} - \frac{16}{81} a^{4} - \frac{5}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{243} a^{14} - \frac{1}{243} a^{13} + \frac{4}{243} a^{12} - \frac{4}{243} a^{11} - \frac{34}{243} a^{10} - \frac{65}{243} a^{9} + \frac{10}{243} a^{8} - \frac{52}{243} a^{7} + \frac{119}{243} a^{6} - \frac{97}{243} a^{5} + \frac{22}{81} a^{4} + \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{481869} a^{15} - \frac{487}{481869} a^{14} - \frac{2066}{481869} a^{13} + \frac{8708}{481869} a^{12} - \frac{25324}{481869} a^{11} - \frac{51563}{481869} a^{10} - \frac{174068}{481869} a^{9} + \frac{170543}{481869} a^{8} - \frac{197728}{481869} a^{7} + \frac{22898}{481869} a^{6} - \frac{54629}{160623} a^{5} - \frac{4600}{53541} a^{4} - \frac{3643}{17847} a^{3} - \frac{275}{661} a^{2} - \frac{145}{661} a - \frac{303}{661}$, $\frac{1}{1445607} a^{16} - \frac{1}{1445607} a^{15} - \frac{788}{1445607} a^{14} - \frac{1885}{1445607} a^{13} + \frac{18668}{1445607} a^{12} + \frac{20842}{1445607} a^{11} - \frac{75365}{1445607} a^{10} - \frac{682432}{1445607} a^{9} - \frac{64420}{1445607} a^{8} - \frac{294010}{1445607} a^{7} - \frac{184241}{481869} a^{6} - \frac{3047}{160623} a^{5} + \frac{125}{1983} a^{4} + \frac{8869}{17847} a^{3} - \frac{1480}{5949} a^{2} + \frac{205}{661} a - \frac{172}{661}$, $\frac{1}{33016396388940309483} a^{17} + \frac{1136931048269}{33016396388940309483} a^{16} - \frac{19464430344503}{33016396388940309483} a^{15} + \frac{60249443262676853}{33016396388940309483} a^{14} + \frac{44117041913306210}{33016396388940309483} a^{13} + \frac{261924484375896685}{33016396388940309483} a^{12} + \frac{100925983693253386}{33016396388940309483} a^{11} + \frac{2586993236700810572}{33016396388940309483} a^{10} - \frac{8261907683370627220}{33016396388940309483} a^{9} + \frac{8352539571008457497}{33016396388940309483} a^{8} - \frac{4105318126695127982}{11005465462980103161} a^{7} + \frac{173479033452836848}{407609831962226043} a^{6} - \frac{158283092060942728}{1222829495886678129} a^{5} - \frac{185932915851585140}{407609831962226043} a^{4} + \frac{53799330454901}{45289981329136227} a^{3} + \frac{3565056336923860}{45289981329136227} a^{2} - \frac{2277525405481518}{5032220147681803} a + \frac{1515558032722676}{5032220147681803}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( -1 \) (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:  \( 25638.894138274067 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.169.1, 3.1.676.1, 6.0.38014691.1, 6.0.608235056.2, 9.3.308915776.1

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 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
13Data not computed