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Properties

Label 18.0.32268972922...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{44}\cdot 5^{9}$
Root discriminant $82.63$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157464, 0, 0, 489888, 0, 0, 391392, 0, 0, -53956, 0, 0, 3000, 0, 0, -84, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 84*x^15 + 3000*x^12 - 53956*x^9 + 391392*x^6 + 489888*x^3 + 157464)
 
gp: K = bnfinit(x^18 - 84*x^15 + 3000*x^12 - 53956*x^9 + 391392*x^6 + 489888*x^3 + 157464, 1)
 

Normalized defining polynomial

\( x^{18} - 84 x^{15} + 3000 x^{12} - 53956 x^{9} + 391392 x^{6} + 489888 x^{3} + 157464 \)

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:  \(-32268972922752572879044608000000000=-\,2^{24}\cdot 3^{44}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.63$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{10} + \frac{1}{3} a$, $\frac{1}{18} a^{11} + \frac{1}{3} a^{5} + \frac{1}{9} a^{2}$, $\frac{1}{108} a^{12} + \frac{1}{18} a^{9} - \frac{1}{18} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{11}{27} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{324} a^{13} + \frac{2}{27} a^{10} - \frac{2}{27} a^{7} + \frac{11}{81} a^{4} + \frac{1}{3} a$, $\frac{1}{2916} a^{14} - \frac{1}{972} a^{13} + \frac{1}{324} a^{12} + \frac{2}{243} a^{11} - \frac{2}{81} a^{10} + \frac{2}{27} a^{9} - \frac{20}{243} a^{8} + \frac{13}{162} a^{7} - \frac{2}{27} a^{6} + \frac{200}{729} a^{5} + \frac{43}{243} a^{4} - \frac{16}{81} a^{3} + \frac{5}{27} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{239005778868} a^{15} - \frac{308328319}{79668592956} a^{12} + \frac{272301136}{19917148239} a^{9} - \frac{4689860801}{119502889434} a^{6} - \frac{961941677}{2213016471} a^{3} + \frac{4488389}{81963573}$, $\frac{1}{717017336604} a^{16} - \frac{308328319}{239005778868} a^{13} - \frac{6094447141}{119502889434} a^{10} - \frac{12303504520}{179254334151} a^{7} + \frac{1988746951}{6639049413} a^{4} - \frac{77475184}{245890719} a$, $\frac{1}{2151052009812} a^{17} - \frac{15609400}{179254334151} a^{14} - \frac{1}{972} a^{13} + \frac{1}{324} a^{12} - \frac{3143758513}{358508668302} a^{11} - \frac{2}{81} a^{10} + \frac{2}{27} a^{9} + \frac{3187610777}{537763002453} a^{8} + \frac{13}{162} a^{7} - \frac{2}{27} a^{6} + \frac{9666001622}{19917148239} a^{5} + \frac{43}{243} a^{4} - \frac{16}{81} a^{3} - \frac{116895698}{245890719} a^{2} + \frac{1}{9} a - \frac{1}{3}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 8882169519.65668 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-5}) \), 3.1.243.1, 6.0.472392000.1, 9.1.2008387814976.4

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$