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Properties

Label 18.6.39226324511...0000.3
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{22}\cdot 5^{15}$
Root discriminant $23.24$
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![9, -171, 900, 2277, -1035, -3771, 39, 2490, 507, -795, -420, 60, 160, 65, -20, -18, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 18*x^15 - 20*x^14 + 65*x^13 + 160*x^12 + 60*x^11 - 420*x^10 - 795*x^9 + 507*x^8 + 2490*x^7 + 39*x^6 - 3771*x^5 - 1035*x^4 + 2277*x^3 + 900*x^2 - 171*x + 9)
 
gp: K = bnfinit(x^18 - 3*x^16 - 18*x^15 - 20*x^14 + 65*x^13 + 160*x^12 + 60*x^11 - 420*x^10 - 795*x^9 + 507*x^8 + 2490*x^7 + 39*x^6 - 3771*x^5 - 1035*x^4 + 2277*x^3 + 900*x^2 - 171*x + 9, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 18 x^{15} - 20 x^{14} + 65 x^{13} + 160 x^{12} + 60 x^{11} - 420 x^{10} - 795 x^{9} + 507 x^{8} + 2490 x^{7} + 39 x^{6} - 3771 x^{5} - 1035 x^{4} + 2277 x^{3} + 900 x^{2} - 171 x + 9 \)

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:  \(3922632451125000000000000=2^{12}\cdot 3^{22}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{3} a^{8} - \frac{4}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{13} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{3} a^{9} - \frac{4}{15} a^{8} - \frac{1}{3} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{15} a^{14} + \frac{4}{15} a^{10} - \frac{4}{15} a^{9} - \frac{1}{3} a^{8} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5}$, $\frac{1}{45} a^{15} - \frac{11}{45} a^{11} + \frac{11}{45} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{7}{15} a^{6} - \frac{4}{15} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{2}{5} a$, $\frac{1}{45} a^{16} + \frac{1}{45} a^{12} - \frac{7}{45} a^{11} - \frac{14}{45} a^{10} - \frac{1}{3} a^{9} + \frac{7}{15} a^{7} + \frac{7}{15} a^{5} + \frac{1}{3} a^{4} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{56285537113738845} a^{17} - \frac{203894663040364}{18761845704579615} a^{16} - \frac{195104233890388}{18761845704579615} a^{15} + \frac{11310143597168}{1705622336779965} a^{14} - \frac{1594942790084774}{56285537113738845} a^{13} - \frac{32992655144723}{11257107422747769} a^{12} + \frac{6020643348431293}{56285537113738845} a^{11} + \frac{3199547600428814}{18761845704579615} a^{10} - \frac{8132876108543357}{18761845704579615} a^{9} + \frac{236143973272177}{18761845704579615} a^{8} + \frac{2630485620846679}{18761845704579615} a^{7} - \frac{124083122938219}{568540778926655} a^{6} + \frac{2665278416538773}{18761845704579615} a^{5} - \frac{2885493193437557}{6253948568193205} a^{4} + \frac{542740889756162}{6253948568193205} a^{3} + \frac{1514291601922863}{6253948568193205} a^{2} - \frac{130171372122439}{1250789713638641} a - \frac{336257653624248}{6253948568193205}$

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:  \( 485986.166592576 \) (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, 9.3.177147000000.3

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.22.65$x^{12} + 27 x^{11} + 18 x^{10} + 33 x^{9} - 9 x^{8} + 36 x^{7} + 18 x^{6} + 18 x^{3} - 27 x - 18$$6$$2$$22$$C_6\times S_3$$[5/2]_{2}^{6}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$