↑

Properties

Label 18.2.54226471004...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 3^{25}\cdot 5^{12}$
Root discriminant $26.90$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times C_3:S_3.C_2$ (as 18T27)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-47, 264, 162, -1116, 471, 1992, -3384, 2658, -1515, 730, 105, -648, 468, -48, -93, 36, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 36*x^15 - 93*x^14 - 48*x^13 + 468*x^12 - 648*x^11 + 105*x^10 + 730*x^9 - 1515*x^8 + 2658*x^7 - 3384*x^6 + 1992*x^5 + 471*x^4 - 1116*x^3 + 162*x^2 + 264*x - 47)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 36*x^15 - 93*x^14 - 48*x^13 + 468*x^12 - 648*x^11 + 105*x^10 + 730*x^9 - 1515*x^8 + 2658*x^7 - 3384*x^6 + 1992*x^5 + 471*x^4 - 1116*x^3 + 162*x^2 + 264*x - 47, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 36 x^{15} - 93 x^{14} - 48 x^{13} + 468 x^{12} - 648 x^{11} + 105 x^{10} + 730 x^{9} - 1515 x^{8} + 2658 x^{7} - 3384 x^{6} + 1992 x^{5} + 471 x^{4} - 1116 x^{3} + 162 x^{2} + 264 x - 47 \)

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:  \(54226471004352000000000000=2^{18}\cdot 3^{25}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.90$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{7}{18} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{7}{18} a^{2} + \frac{1}{18} a - \frac{4}{9}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{12} - \frac{1}{6} a^{11} + \frac{1}{18} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{5}{18} a^{6} - \frac{1}{2} a^{4} + \frac{5}{18} a^{3} - \frac{1}{3} a^{2} + \frac{5}{18}$, $\frac{1}{738} a^{16} - \frac{1}{246} a^{15} + \frac{7}{369} a^{14} + \frac{3}{82} a^{13} + \frac{31}{369} a^{12} - \frac{59}{738} a^{11} - \frac{49}{738} a^{10} + \frac{44}{369} a^{9} + \frac{32}{123} a^{8} + \frac{130}{369} a^{7} + \frac{23}{246} a^{6} - \frac{341}{738} a^{5} + \frac{35}{123} a^{4} + \frac{187}{738} a^{3} + \frac{169}{369} a^{2} + \frac{289}{738} a - \frac{277}{738}$, $\frac{1}{51207689880342} a^{17} - \frac{14636651662}{25603844940171} a^{16} - \frac{161713422595}{17069229960114} a^{15} - \frac{85676312845}{51207689880342} a^{14} + \frac{1153654435301}{51207689880342} a^{13} - \frac{3447793400723}{51207689880342} a^{12} - \frac{23258395880}{2844871660019} a^{11} + \frac{250738107189}{5689743320038} a^{10} + \frac{2013029749774}{25603844940171} a^{9} + \frac{7051936718737}{25603844940171} a^{8} + \frac{8723643877985}{51207689880342} a^{7} - \frac{172021045151}{2844871660019} a^{6} - \frac{10378909380311}{51207689880342} a^{5} + \frac{14598586370299}{51207689880342} a^{4} - \frac{20394070154341}{51207689880342} a^{3} + \frac{3478613325761}{17069229960114} a^{2} - \frac{2846303012081}{8534614980057} a - \frac{3933815920049}{51207689880342}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1102355.96643 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 18T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), 9.1.2125764000000.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$