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Properties

Label 18.0.33724510211...9488.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 11^{12}\cdot 17^{12}$
Root discriminant $496.59$
Ramified primes $2, 3, 11, 17$
Class number $1289945088$ (GRH)
Class group $[2, 2, 2, 6, 6, 6, 12, 12, 72, 72]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![747377296875, 0, 0, 0, 0, 0, 3634057251, 0, 0, 0, 0, 0, -46827, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 46827*x^12 + 3634057251*x^6 + 747377296875)
 
gp: K = bnfinit(x^18 - 46827*x^12 + 3634057251*x^6 + 747377296875, 1)
 

Normalized defining polynomial

\( x^{18} - 46827 x^{12} + 3634057251 x^{6} + 747377296875 \)

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:  \(-3372451021135065952308015612910739392504066879488=-\,2^{12}\cdot 3^{37}\cdot 11^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $496.59$
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:  $2, 3, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{11} a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{363} a^{6}$, $\frac{1}{1815} a^{7} - \frac{2}{5} a$, $\frac{1}{1815} a^{8} - \frac{2}{5} a^{2}$, $\frac{1}{39930} a^{9} - \frac{1}{726} a^{6} + \frac{3}{110} a^{3} - \frac{1}{2}$, $\frac{1}{119790} a^{10} - \frac{1}{3630} a^{7} + \frac{13}{330} a^{4} - \frac{3}{10} a$, $\frac{1}{119790} a^{11} - \frac{1}{3630} a^{8} + \frac{13}{330} a^{5} - \frac{3}{10} a^{2}$, $\frac{1}{12668271660} a^{12} - \frac{2526}{2908235} a^{6} - \frac{1}{22} a^{3} + \frac{3673}{19228}$, $\frac{1}{38004814980} a^{13} + \frac{1}{119790} a^{9} + \frac{2036}{26174115} a^{7} - \frac{1}{726} a^{6} + \frac{1}{33} a^{5} - \frac{1}{22} a^{4} + \frac{13}{330} a^{3} - \frac{58547}{288420} a - \frac{1}{2}$, $\frac{1}{190024074900} a^{14} - \frac{26806}{130870575} a^{8} - \frac{1}{22} a^{5} - \frac{116231}{1442100} a^{2}$, $\frac{1}{10451324119500} a^{15} + \frac{45299}{7197881625} a^{9} - \frac{1}{726} a^{6} - \frac{693071}{79315500} a^{3}$, $\frac{1}{52256620597500} a^{16} - \frac{74876}{35989408125} a^{10} - \frac{1}{3630} a^{7} + \frac{2492943}{132192500} a^{4} + \frac{1}{5} a$, $\frac{1}{261283102987500} a^{17} - \frac{74876}{179947040625} a^{11} - \frac{1}{3630} a^{8} - \frac{9524557}{660962500} a^{5} + \frac{1}{5} a^{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{12}\times C_{12}\times C_{72}\times C_{72}$, which has order $1289945088$ (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:  \( \frac{1}{61118854500} a^{15} - \frac{181}{252557250} a^{9} + \frac{83937}{1391500} a^{3} + \frac{1}{2} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 795492382.7560022 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.117612.4 x3, 3.1.8497467.4 x3, 3.1.33989868.7 x3, 3.1.31212.1 x3, 6.0.41497747632.3, 6.0.216620836248267.2, 6.0.3465933379972272.2, 6.0.2922566832.1, 9.1.1060259562738462320460864.1 x9

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

Sibling fields

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
3Data not computed
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{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}$