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Properties

Label 18.2.58498535041...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{33}\cdot 3^{20}\cdot 5^{9}$
Root discriminant $27.01$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (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![-729, 0, 1188, 114, -2700, 1140, 2259, -1308, -972, 620, 468, -228, -171, 60, 60, -6, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 6*x^15 + 60*x^14 + 60*x^13 - 171*x^12 - 228*x^11 + 468*x^10 + 620*x^9 - 972*x^8 - 1308*x^7 + 2259*x^6 + 1140*x^5 - 2700*x^4 + 114*x^3 + 1188*x^2 - 729)
 
gp: K = bnfinit(x^18 - 12*x^16 - 6*x^15 + 60*x^14 + 60*x^13 - 171*x^12 - 228*x^11 + 468*x^10 + 620*x^9 - 972*x^8 - 1308*x^7 + 2259*x^6 + 1140*x^5 - 2700*x^4 + 114*x^3 + 1188*x^2 - 729, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 6 x^{15} + 60 x^{14} + 60 x^{13} - 171 x^{12} - 228 x^{11} + 468 x^{10} + 620 x^{9} - 972 x^{8} - 1308 x^{7} + 2259 x^{6} + 1140 x^{5} - 2700 x^{4} + 114 x^{3} + 1188 x^{2} - 729 \)

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:  \(58498535041007616000000000=2^{33}\cdot 3^{20}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.01$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} + \frac{1}{6} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{60} a^{14} - \frac{1}{15} a^{13} - \frac{1}{30} a^{12} - \frac{2}{15} a^{11} - \frac{1}{60} a^{10} + \frac{1}{15} a^{9} + \frac{13}{60} a^{8} + \frac{2}{15} a^{7} + \frac{13}{60} a^{6} - \frac{1}{10} a^{5} + \frac{3}{20} a^{4} - \frac{3}{10} a^{3} + \frac{3}{10} a^{2} + \frac{1}{10} a + \frac{7}{20}$, $\frac{1}{25560} a^{15} - \frac{59}{25560} a^{14} + \frac{989}{12780} a^{13} + \frac{179}{3195} a^{12} + \frac{6049}{25560} a^{11} + \frac{1759}{25560} a^{10} + \frac{2713}{25560} a^{9} - \frac{4577}{25560} a^{8} - \frac{10727}{25560} a^{7} - \frac{569}{2840} a^{6} + \frac{581}{2840} a^{5} + \frac{4229}{8520} a^{4} - \frac{368}{1065} a^{3} + \frac{1121}{4260} a^{2} - \frac{13}{120} a - \frac{249}{568}$, $\frac{1}{70034400} a^{16} - \frac{91}{35017200} a^{15} + \frac{337681}{70034400} a^{14} - \frac{822553}{35017200} a^{13} + \frac{3164641}{70034400} a^{12} + \frac{64921}{17508600} a^{11} - \frac{1521353}{7003440} a^{10} + \frac{1161907}{17508600} a^{9} - \frac{8566999}{35017200} a^{8} + \frac{1206049}{5836200} a^{7} - \frac{35347}{466896} a^{6} - \frac{13793}{82200} a^{5} + \frac{9731987}{23344800} a^{4} - \frac{5012041}{11672400} a^{3} - \frac{1696973}{23344800} a^{2} - \frac{1254589}{3890800} a - \frac{376951}{7781600}$, $\frac{1}{29437349248800} a^{17} - \frac{479}{136284024300} a^{16} + \frac{10950607}{1962489949920} a^{15} + \frac{1357243103}{1226556218700} a^{14} - \frac{812090550049}{9812449749600} a^{13} + \frac{170422151347}{4906224874800} a^{12} + \frac{366086878729}{1635408291600} a^{11} - \frac{48196084861}{2453112437400} a^{10} - \frac{372314748673}{1635408291600} a^{9} + \frac{1391660044571}{7359337312200} a^{8} + \frac{7106469037}{181712032400} a^{7} - \frac{258242487373}{2453112437400} a^{6} - \frac{1094419157923}{3270816583200} a^{5} - \frac{485130754939}{2453112437400} a^{4} + \frac{258148152959}{1090272194400} a^{3} + \frac{854439081593}{2453112437400} a^{2} + \frac{1758607547}{43610887776} a + \frac{49951757009}{181712032400}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2138977.3943718825 \) (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

\(\Q(\sqrt{10}) \), 3.1.108.1, 3.1.1080.1, 6.2.20736000.2 x2, 6.2.186624000.3, 6.2.46656000.4, 9.1.60466176000.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$