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Properties

Label 18.2.43584805012...0000.2
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{20}\cdot 5^{15}$
Root discriminant $20.57$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 0, -3, 0, 0, 30, 0, 0, 145, 0, 0, -30, 0, 0, -3, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^15 - 30*x^12 + 145*x^9 + 30*x^6 - 3*x^3 - 1)
 
gp: K = bnfinit(x^18 - 3*x^15 - 30*x^12 + 145*x^9 + 30*x^6 - 3*x^3 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{15} - 30 x^{12} + 145 x^{9} + 30 x^{6} - 3 x^{3} - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(435848050125000000000000=2^{12}\cdot 3^{20}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.57$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{4}{9} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{4}{9} a^{4} - \frac{2}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{4}{9} a^{5} - \frac{2}{9} a^{2}$, $\frac{1}{657} a^{15} - \frac{10}{219} a^{12} - \frac{23}{657} a^{9} + \frac{4}{73} a^{6} - \frac{139}{657} a^{3} + \frac{319}{657}$, $\frac{1}{1971} a^{16} - \frac{1}{1971} a^{15} - \frac{1}{27} a^{14} - \frac{10}{657} a^{13} - \frac{43}{1971} a^{12} + \frac{1}{27} a^{11} + \frac{196}{1971} a^{10} - \frac{41}{657} a^{9} + \frac{4}{27} a^{8} + \frac{4}{219} a^{7} + \frac{256}{1971} a^{6} - \frac{1}{27} a^{5} + \frac{737}{1971} a^{4} - \frac{30}{73} a^{3} - \frac{1}{27} a^{2} - \frac{776}{1971} a + \frac{703}{1971}$, $\frac{1}{1971} a^{17} + \frac{1}{1971} a^{15} - \frac{103}{1971} a^{14} - \frac{1}{27} a^{13} - \frac{10}{657} a^{12} + \frac{269}{1971} a^{11} + \frac{1}{27} a^{10} + \frac{196}{1971} a^{9} + \frac{328}{1971} a^{8} + \frac{4}{27} a^{7} + \frac{4}{219} a^{6} + \frac{664}{1971} a^{5} - \frac{1}{27} a^{4} + \frac{737}{1971} a^{3} - \frac{283}{657} a^{2} - \frac{1}{27} a - \frac{776}{1971}$

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:  $9$
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:  \( 44339.29194183518 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.135.1, 3.1.300.1, 6.2.36450000.5 x2, 6.2.91125.1, 6.2.450000.1, 9.1.295245000000.2

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$3.6.6.4$x^{6} + 3 x^{4} + 6 x^{3} + 9 x^{2} + 63 x + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$