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Properties

Label 18.0.41429799128...4784.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{18}\cdot 11^{15}$
Root discriminant $30.11$
Ramified primes $2, 3, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92, 132, -156, -314, 918, 228, -1597, 1359, 771, -1803, 1170, -216, -13, -138, 192, -119, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 192*x^14 - 138*x^13 - 13*x^12 - 216*x^11 + 1170*x^10 - 1803*x^9 + 771*x^8 + 1359*x^7 - 1597*x^6 + 228*x^5 + 918*x^4 - 314*x^3 - 156*x^2 + 132*x + 92)
 
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 119*x^15 + 192*x^14 - 138*x^13 - 13*x^12 - 216*x^11 + 1170*x^10 - 1803*x^9 + 771*x^8 + 1359*x^7 - 1597*x^6 + 228*x^5 + 918*x^4 - 314*x^3 - 156*x^2 + 132*x + 92, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 42 x^{16} - 119 x^{15} + 192 x^{14} - 138 x^{13} - 13 x^{12} - 216 x^{11} + 1170 x^{10} - 1803 x^{9} + 771 x^{8} + 1359 x^{7} - 1597 x^{6} + 228 x^{5} + 918 x^{4} - 314 x^{3} - 156 x^{2} + 132 x + 92 \)

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:  \(-414297991288157786796374784=-\,2^{8}\cdot 3^{18}\cdot 11^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.11$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{6} - \frac{1}{6} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{3} a^{7} - \frac{1}{6} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{18} a^{15} - \frac{1}{18} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{2}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{18} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9}$, $\frac{1}{90} a^{16} + \frac{1}{45} a^{15} + \frac{1}{15} a^{14} - \frac{2}{45} a^{13} - \frac{1}{18} a^{12} - \frac{2}{5} a^{11} + \frac{1}{90} a^{10} + \frac{19}{45} a^{9} - \frac{17}{90} a^{7} + \frac{7}{45} a^{6} + \frac{13}{30} a^{5} - \frac{1}{2} a^{4} - \frac{4}{15} a^{3} - \frac{2}{5} a^{2} - \frac{4}{45} a - \frac{8}{45}$, $\frac{1}{316034138859914024910} a^{17} + \frac{194209618065950551}{316034138859914024910} a^{16} - \frac{2907083273367227711}{316034138859914024910} a^{15} + \frac{258230091779686567}{63206827771982804982} a^{14} - \frac{5809992272848222448}{158017069429957012455} a^{13} + \frac{7270523951697530699}{316034138859914024910} a^{12} + \frac{17300717379255828821}{158017069429957012455} a^{11} + \frac{145756606801412145757}{316034138859914024910} a^{10} + \frac{29260987233484781951}{158017069429957012455} a^{9} - \frac{111175254540557291387}{316034138859914024910} a^{8} + \frac{154094574038258499661}{316034138859914024910} a^{7} + \frac{3871931336887293700}{31603413885991402491} a^{6} + \frac{10288256382632790779}{35114904317768224990} a^{5} - \frac{11179653455309467511}{35114904317768224990} a^{4} - \frac{3320845461728405147}{52672356476652337485} a^{3} - \frac{67320105393343061986}{158017069429957012455} a^{2} + \frac{47112050935277845286}{158017069429957012455} a - \frac{33865218744927759892}{158017069429957012455}$

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:  $8$
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:  \( 2125238.111612494 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.3267.1, 6.0.117406179.1, 9.3.557914162608.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
11Data not computed