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Properties

Label 18.6.23109419877...0608.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{16}\cdot 3^{37}\cdot 23^{8}$
Root discriminant $71.37$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10622, -186912, 334836, -288762, 549954, -212940, 448401, -169380, 229878, -75340, 49131, -5670, -1650, 1728, -585, -36, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^15 - 585*x^14 + 1728*x^13 - 1650*x^12 - 5670*x^11 + 49131*x^10 - 75340*x^9 + 229878*x^8 - 169380*x^7 + 448401*x^6 - 212940*x^5 + 549954*x^4 - 288762*x^3 + 334836*x^2 - 186912*x - 10622)
 
gp: K = bnfinit(x^18 - 36*x^15 - 585*x^14 + 1728*x^13 - 1650*x^12 - 5670*x^11 + 49131*x^10 - 75340*x^9 + 229878*x^8 - 169380*x^7 + 448401*x^6 - 212940*x^5 + 549954*x^4 - 288762*x^3 + 334836*x^2 - 186912*x - 10622, 1)
 

Normalized defining polynomial

\( x^{18} - 36 x^{15} - 585 x^{14} + 1728 x^{13} - 1650 x^{12} - 5670 x^{11} + 49131 x^{10} - 75340 x^{9} + 229878 x^{8} - 169380 x^{7} + 448401 x^{6} - 212940 x^{5} + 549954 x^{4} - 288762 x^{3} + 334836 x^{2} - 186912 x - 10622 \)

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:  \(2310941987733575020334469218500608=2^{16}\cdot 3^{37}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.37$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{92} a^{14} + \frac{5}{46} a^{13} - \frac{5}{46} a^{12} - \frac{3}{46} a^{11} + \frac{9}{46} a^{10} - \frac{11}{46} a^{9} + \frac{9}{23} a^{8} - \frac{2}{23} a^{7} - \frac{3}{92} a^{6} + \frac{10}{23} a^{5} - \frac{9}{23} a^{4} - \frac{17}{46} a^{3} + \frac{9}{23} a^{2} - \frac{3}{23} a - \frac{5}{46}$, $\frac{1}{184} a^{15} - \frac{1}{184} a^{14} + \frac{9}{92} a^{13} + \frac{3}{46} a^{12} + \frac{19}{92} a^{11} - \frac{9}{46} a^{10} - \frac{11}{46} a^{9} - \frac{9}{46} a^{8} + \frac{39}{184} a^{7} - \frac{19}{184} a^{6} + \frac{19}{46} a^{5} - \frac{3}{92} a^{4} - \frac{25}{92} a^{3} + \frac{13}{46} a^{2} - \frac{31}{92} a + \frac{9}{92}$, $\frac{1}{1288} a^{16} + \frac{1}{644} a^{15} - \frac{5}{1288} a^{14} - \frac{67}{644} a^{13} - \frac{47}{644} a^{12} + \frac{99}{644} a^{11} + \frac{5}{161} a^{10} + \frac{34}{161} a^{9} + \frac{45}{184} a^{8} + \frac{313}{644} a^{7} + \frac{263}{1288} a^{6} + \frac{263}{644} a^{5} - \frac{113}{322} a^{4} + \frac{107}{644} a^{3} - \frac{129}{644} a^{2} - \frac{60}{161} a + \frac{127}{644}$, $\frac{1}{6659852162446263770301665957338808153741974936} a^{17} - \frac{204834784033139347894413979652592901268173}{951407451778037681471666565334115450534567848} a^{16} - \frac{7382082815145228520465745195803273064187313}{3329926081223131885150832978669404076870987468} a^{15} - \frac{751628039754744533202021376054885370364549}{1664963040611565942575416489334702038435493734} a^{14} - \frac{403725666724818656470948100881771721194541819}{3329926081223131885150832978669404076870987468} a^{13} - \frac{114475215907850624394356610531114872366351541}{1664963040611565942575416489334702038435493734} a^{12} - \frac{382781767612614234559971476595663289939892161}{1664963040611565942575416489334702038435493734} a^{11} - \frac{371521525556223403107818256252259455598244253}{1664963040611565942575416489334702038435493734} a^{10} - \frac{1363511045755015651875572025931915917274567221}{6659852162446263770301665957338808153741974936} a^{9} + \frac{748190372012515028669820986515748934101874315}{6659852162446263770301665957338808153741974936} a^{8} - \frac{120689346739180271543912879302367806635979281}{1664963040611565942575416489334702038435493734} a^{7} + \frac{195152796879315225862257933652348546408616647}{475703725889018840735833282667057725267283924} a^{6} - \frac{1434436894723626936713969578612724287675481595}{3329926081223131885150832978669404076870987468} a^{5} + \frac{311083975410226770970625643491765591080606477}{1664963040611565942575416489334702038435493734} a^{4} - \frac{26472995089376916409871009414177153210668793}{475703725889018840735833282667057725267283924} a^{3} - \frac{718798545312492983040270779147212965492648561}{3329926081223131885150832978669404076870987468} a^{2} - \frac{141223455304263920961406110072385729030496669}{832481520305782971287708244667351019217746867} a - \frac{52149965496029725429059076934961509997810}{17712372772463467474206558397177681259952061}$

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:  \( 27167646192.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.22356.2, 3.3.22356.1, 3.3.22356.3, 3.3.621.1, 6.2.4627692.1, 9.9.6938632771983936.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
3Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$