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Properties

Label 18.0.54783205459...2864.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 11^{13}$
Root discriminant $26.91$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, -834, 2691, -5308, 7290, -7317, 5316, -2184, -378, 1613, -1461, 720, -63, -219, 219, -119, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125)
 
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 219*x^14 - 219*x^13 - 63*x^12 + 720*x^11 - 1461*x^10 + 1613*x^9 - 378*x^8 - 2184*x^7 + 5316*x^6 - 7317*x^5 + 7290*x^4 - 5308*x^3 + 2691*x^2 - 834*x + 125, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 42 x^{16} - 119 x^{15} + 219 x^{14} - 219 x^{13} - 63 x^{12} + 720 x^{11} - 1461 x^{10} + 1613 x^{9} - 378 x^{8} - 2184 x^{7} + 5316 x^{6} - 7317 x^{5} + 7290 x^{4} - 5308 x^{3} + 2691 x^{2} - 834 x + 125 \)

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:  \(-54783205459591112303652864=-\,2^{12}\cdot 3^{18}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.91$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{15} + \frac{2}{25} a^{14} - \frac{2}{25} a^{13} - \frac{2}{25} a^{12} - \frac{4}{25} a^{11} + \frac{2}{25} a^{10} + \frac{6}{25} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{25} a^{6} + \frac{1}{25} a^{5} - \frac{2}{25} a^{4} + \frac{7}{25} a^{3} - \frac{12}{25} a^{2} + \frac{4}{25} a$, $\frac{1}{25} a^{16} - \frac{1}{25} a^{14} + \frac{2}{25} a^{13} + \frac{12}{25} a^{10} + \frac{8}{25} a^{9} - \frac{2}{5} a^{8} - \frac{1}{25} a^{7} + \frac{8}{25} a^{6} + \frac{11}{25} a^{5} + \frac{1}{25} a^{4} + \frac{9}{25} a^{3} - \frac{2}{25} a^{2} + \frac{12}{25} a$, $\frac{1}{787448472669102625} a^{17} - \frac{2388168634565091}{787448472669102625} a^{16} + \frac{10961719205656944}{787448472669102625} a^{15} + \frac{21183715988952788}{787448472669102625} a^{14} + \frac{16089029361480568}{787448472669102625} a^{13} - \frac{222415676958069}{157489694533820525} a^{12} + \frac{56860902875836272}{787448472669102625} a^{11} + \frac{110604212955537956}{787448472669102625} a^{10} - \frac{26440919080244193}{787448472669102625} a^{9} - \frac{15786318452494201}{787448472669102625} a^{8} + \frac{249433800807017454}{787448472669102625} a^{7} - \frac{82820188577942177}{787448472669102625} a^{6} + \frac{53872351764675837}{157489694533820525} a^{5} + \frac{272420889378662143}{787448472669102625} a^{4} + \frac{164564538052086824}{787448472669102625} a^{3} + \frac{21485804053970814}{787448472669102625} a^{2} - \frac{264973052837483192}{787448472669102625} a + \frac{2567599315703162}{6299587781352821}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 311633.43352525914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.108.1, 6.0.15524784.3, 9.3.18443443392.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.12.10.2$x^{12} + 143 x^{6} + 5929$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$