↑

Properties

Label 18.6.39226324511...0000.2
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)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, -72, -180, 504, 180, -1062, 189, 720, -63, 0, 0, 0, 55, 10, -20, -6, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 6*x^15 - 20*x^14 + 10*x^13 + 55*x^12 - 63*x^8 + 720*x^7 + 189*x^6 - 1062*x^5 + 180*x^4 + 504*x^3 - 180*x^2 - 72*x + 36)
 
gp: K = bnfinit(x^18 - 3*x^16 - 6*x^15 - 20*x^14 + 10*x^13 + 55*x^12 - 63*x^8 + 720*x^7 + 189*x^6 - 1062*x^5 + 180*x^4 + 504*x^3 - 180*x^2 - 72*x + 36, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 6 x^{15} - 20 x^{14} + 10 x^{13} + 55 x^{12} - 63 x^{8} + 720 x^{7} + 189 x^{6} - 1062 x^{5} + 180 x^{4} + 504 x^{3} - 180 x^{2} - 72 x + 36 \)

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:  \(3922632451125000000000000=2^{12}\cdot 3^{22}\cdot 5^{15}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} + \frac{5}{12} a^{8} + \frac{1}{3} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12036} a^{16} - \frac{41}{6018} a^{15} - \frac{47}{12036} a^{14} - \frac{25}{354} a^{13} + \frac{103}{2006} a^{12} - \frac{77}{1003} a^{11} - \frac{2525}{12036} a^{10} - \frac{1325}{6018} a^{9} + \frac{686}{3009} a^{8} + \frac{77}{177} a^{7} + \frac{4037}{12036} a^{6} + \frac{453}{1003} a^{5} + \frac{1697}{4012} a^{4} + \frac{495}{1003} a^{3} + \frac{623}{2006} a^{2} - \frac{129}{2006} a + \frac{333}{1003}$, $\frac{1}{571754333727372} a^{17} - \frac{792454279}{571754333727372} a^{16} + \frac{2106922809583}{95292388954562} a^{15} - \frac{9880824869381}{285877166863686} a^{14} - \frac{4289339468815}{190584777909124} a^{13} + \frac{2500839122879}{142938583431843} a^{12} + \frac{13439355849091}{95292388954562} a^{11} - \frac{2137365850091}{16816303933158} a^{10} - \frac{32584386986201}{571754333727372} a^{9} - \frac{44964947158425}{95292388954562} a^{8} - \frac{77483440655725}{285877166863686} a^{7} + \frac{40051749518869}{285877166863686} a^{6} - \frac{20232873878912}{47646194477281} a^{5} - \frac{72575039347323}{190584777909124} a^{4} + \frac{35210942884929}{95292388954562} a^{3} - \frac{1357785943895}{5605434644386} a^{2} + \frac{27718008228349}{95292388954562} a - \frac{3645149714661}{47646194477281}$

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:  \( 385463.69192704296 \) (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, 9.3.177147000000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$$[\ ]_{3}^{2}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.22.83$x^{12} + 27 x^{11} - 27 x^{10} + 3 x^{9} - 9 x^{8} - 36 x^{7} - 9 x^{6} + 27 x^{5} + 36 x^{3} + 27 x + 36$$6$$2$$22$$C_6\times S_3$$[5/2]_{2}^{6}$
5Data not computed