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Properties

Label 18.6.31773322854...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{26}\cdot 5^{15}$
Root discriminant $29.67$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (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![36, 108, 0, -36, 990, 2268, 1539, -90, -333, -60, -180, -270, -125, 0, 0, -6, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 6*x^15 - 125*x^12 - 270*x^11 - 180*x^10 - 60*x^9 - 333*x^8 - 90*x^7 + 1539*x^6 + 2268*x^5 + 990*x^4 - 36*x^3 + 108*x + 36)
 
gp: K = bnfinit(x^18 - 3*x^16 - 6*x^15 - 125*x^12 - 270*x^11 - 180*x^10 - 60*x^9 - 333*x^8 - 90*x^7 + 1539*x^6 + 2268*x^5 + 990*x^4 - 36*x^3 + 108*x + 36, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 6 x^{15} - 125 x^{12} - 270 x^{11} - 180 x^{10} - 60 x^{9} - 333 x^{8} - 90 x^{7} + 1539 x^{6} + 2268 x^{5} + 990 x^{4} - 36 x^{3} + 108 x + 36 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(317733228541125000000000000=2^{12}\cdot 3^{26}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{12} a^{9} - \frac{1}{3} a^{8} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{132} a^{16} + \frac{1}{66} a^{14} + \frac{5}{132} a^{13} - \frac{1}{132} a^{12} - \frac{5}{22} a^{11} - \frac{5}{33} a^{10} + \frac{3}{44} a^{9} + \frac{17}{132} a^{8} + \frac{31}{66} a^{7} + \frac{19}{66} a^{6} + \frac{1}{4} a^{5} + \frac{2}{11} a^{4} - \frac{7}{22} a^{3} + \frac{9}{22} a^{2} - \frac{4}{11} a - \frac{5}{11}$, $\frac{1}{4378359121902324372} a^{17} + \frac{400500780062260}{364863260158527031} a^{16} + \frac{40773872675153807}{4378359121902324372} a^{15} + \frac{102050788374208295}{2189179560951162186} a^{14} + \frac{7689807809599939}{199016323722832926} a^{13} - \frac{84355161461797123}{1094589780475581093} a^{12} + \frac{108126459996697699}{4378359121902324372} a^{11} + \frac{31116770070085364}{364863260158527031} a^{10} + \frac{75647568462368075}{1094589780475581093} a^{9} - \frac{283702804850790025}{2189179560951162186} a^{8} + \frac{1049186848997031835}{4378359121902324372} a^{7} - \frac{535988083022434817}{2189179560951162186} a^{6} + \frac{62336099839751119}{1459453040634108124} a^{5} - \frac{330917665131954113}{729726520317054062} a^{4} + \frac{358469383626477751}{729726520317054062} a^{3} - \frac{275718736112240623}{729726520317054062} a^{2} + \frac{168790437289141803}{364863260158527031} a - \frac{7868301034851247}{364863260158527031}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 6275708.277649186 \) (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

\(\Q(\sqrt{5}) \), 3.1.300.1, 6.2.450000.1, 9.3.1594323000000.4

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.4.1$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$$[\ ]_{3}^{2}$
$3$3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
3.12.18.36$x^{12} + 27 x^{11} + 36 x^{10} - 12 x^{9} + 9 x^{8} + 9 x^{6} + 36 x^{5} + 18 x^{4} - 36 x^{3} + 27 x - 18$$6$$2$$18$$C_6\times S_3$$[3/2, 2]_{2}^{2}$
5Data not computed