↑

Properties

Label 18.2.77358185988...1792.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{34}\cdot 3^{37}$
Root discriminant $35.43$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_9$ (as 18T50)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144, -216, -810, -252, 1782, 2592, 3231, 2052, 972, 336, 45, -108, -186, -72, -27, -12, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 - 27*x^14 - 72*x^13 - 186*x^12 - 108*x^11 + 45*x^10 + 336*x^9 + 972*x^8 + 2052*x^7 + 3231*x^6 + 2592*x^5 + 1782*x^4 - 252*x^3 - 810*x^2 - 216*x + 144)
 
gp: K = bnfinit(x^18 - 12*x^15 - 27*x^14 - 72*x^13 - 186*x^12 - 108*x^11 + 45*x^10 + 336*x^9 + 972*x^8 + 2052*x^7 + 3231*x^6 + 2592*x^5 + 1782*x^4 - 252*x^3 - 810*x^2 - 216*x + 144, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{15} - 27 x^{14} - 72 x^{13} - 186 x^{12} - 108 x^{11} + 45 x^{10} + 336 x^{9} + 972 x^{8} + 2052 x^{7} + 3231 x^{6} + 2592 x^{5} + 1782 x^{4} - 252 x^{3} - 810 x^{2} - 216 x + 144 \)

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:  \(7735818598867901659628961792=2^{34}\cdot 3^{37}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.43$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{18} a^{9} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{9} - \frac{1}{6} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{36} a^{11} - \frac{1}{36} a^{9} - \frac{1}{6} a^{8} + \frac{1}{4} a^{7} - \frac{1}{3} a^{6} - \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{72} a^{12} - \frac{1}{72} a^{10} - \frac{1}{36} a^{9} + \frac{1}{8} a^{8} + \frac{1}{3} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{11} - \frac{1}{72} a^{9} - \frac{1}{6} a^{8} - \frac{1}{24} a^{7} + \frac{1}{3} a^{6} + \frac{5}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{12} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{72} a^{14} + \frac{1}{12} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} + \frac{1}{3} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{72} a^{15} - \frac{1}{36} a^{9} - \frac{3}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{12} a^{3} + \frac{1}{3}$, $\frac{1}{22464} a^{16} - \frac{5}{1248} a^{15} - \frac{1}{2496} a^{14} - \frac{1}{234} a^{13} - \frac{25}{3744} a^{12} + \frac{1}{624} a^{11} - \frac{11}{1872} a^{10} - \frac{25}{936} a^{9} + \frac{553}{2496} a^{8} - \frac{1475}{3744} a^{7} - \frac{71}{832} a^{6} + \frac{9}{52} a^{5} - \frac{7}{312} a^{4} + \frac{37}{156} a^{3} - \frac{139}{416} a^{2} + \frac{137}{312} a + \frac{17}{156}$, $\frac{1}{41952502934784} a^{17} - \frac{7976273}{1553796404992} a^{16} + \frac{49001407051}{13984167644928} a^{15} - \frac{17268568559}{4661389214976} a^{14} - \frac{195407467}{118509895296} a^{13} - \frac{474623633}{776898202496} a^{12} + \frac{2453790271}{1748020955616} a^{11} + \frac{21365903053}{3496041911232} a^{10} - \frac{112187042093}{13984167644928} a^{9} - \frac{2050568129665}{13984167644928} a^{8} + \frac{1995167764301}{4661389214976} a^{7} - \frac{534054794217}{1553796404992} a^{6} + \frac{1689441439}{9875824608} a^{5} - \frac{73490258801}{194224550624} a^{4} - \frac{294887807129}{2330694607488} a^{3} + \frac{178411819095}{776898202496} a^{2} - \frac{30692998403}{582673651872} a - \frac{109325204341}{291336825936}$

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:  \( 178921364.387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.1.216.1, 6.2.2239488.3

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

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 $18$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{9}$

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$$[2]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed