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Properties

Label 18.0.33819058577...3283.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 107^{8}$
Root discriminant $13.82$
Ramified primes $3, 107$
Class number $1$
Class group Trivial
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![1, 1, -3, 24, 20, -48, 54, -98, 116, -140, 117, -96, 77, -49, 31, -12, 7, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 7*x^16 - 12*x^15 + 31*x^14 - 49*x^13 + 77*x^12 - 96*x^11 + 117*x^10 - 140*x^9 + 116*x^8 - 98*x^7 + 54*x^6 - 48*x^5 + 20*x^4 + 24*x^3 - 3*x^2 + x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 7*x^16 - 12*x^15 + 31*x^14 - 49*x^13 + 77*x^12 - 96*x^11 + 117*x^10 - 140*x^9 + 116*x^8 - 98*x^7 + 54*x^6 - 48*x^5 + 20*x^4 + 24*x^3 - 3*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 7 x^{16} - 12 x^{15} + 31 x^{14} - 49 x^{13} + 77 x^{12} - 96 x^{11} + 117 x^{10} - 140 x^{9} + 116 x^{8} - 98 x^{7} + 54 x^{6} - 48 x^{5} + 20 x^{4} + 24 x^{3} - 3 x^{2} + x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-338190585776316833283=-\,3^{9}\cdot 107^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.82$
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:  $3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{16} + \frac{2}{21} a^{15} + \frac{1}{7} a^{14} - \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{10}{21} a^{11} - \frac{1}{21} a^{10} - \frac{1}{7} a^{9} + \frac{5}{21} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{2}{21} a^{4} + \frac{1}{3} a^{3} - \frac{5}{21} a^{2} + \frac{4}{21} a - \frac{4}{21}$, $\frac{1}{1579137} a^{17} + \frac{22501}{1579137} a^{16} - \frac{111785}{1579137} a^{15} - \frac{11860}{225591} a^{14} + \frac{73610}{526379} a^{13} + \frac{2699}{225591} a^{12} - \frac{465764}{1579137} a^{11} - \frac{54331}{1579137} a^{10} - \frac{283141}{1579137} a^{9} - \frac{177145}{526379} a^{8} + \frac{3486}{75197} a^{7} - \frac{35149}{75197} a^{6} + \frac{298891}{1579137} a^{5} - \frac{653116}{1579137} a^{4} + \frac{635504}{1579137} a^{3} + \frac{608243}{1579137} a^{2} + \frac{3751}{7779} a + \frac{171081}{526379}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2289647}{1579137} a^{17} + \frac{859034}{225591} a^{16} - \frac{18939043}{1579137} a^{15} + \frac{38032763}{1579137} a^{14} - \frac{89032183}{1579137} a^{13} + \frac{159920650}{1579137} a^{12} - \frac{35888431}{225591} a^{11} + \frac{114738247}{526379} a^{10} - \frac{140826312}{526379} a^{9} + \frac{173313178}{526379} a^{8} - \frac{71133584}{225591} a^{7} + \frac{62684864}{225591} a^{6} - \frac{311158744}{1579137} a^{5} + \frac{76608927}{526379} a^{4} - \frac{150900581}{1579137} a^{3} - \frac{1107949}{225591} a^{2} + \frac{170679}{18151} a - \frac{7185601}{1579137} \) (order $6$)
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:  \( 1816.53595065 \)
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

\(\Q(\sqrt{-3}) \), 3.1.107.1, 3.3.321.1, 6.0.309123.2 x2, 6.0.309123.3, 6.0.309123.1, 9.3.3539149227.1

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: 6.0.309123.2
Degree 9 sibling: 9.3.3539149227.1
Degree 12 sibling: 12.0.1094032426497921.1
Degree 18 siblings: 18.6.36186392678065901161281.1, Deg 18

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$