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Properties

Label 18.0.16403701976...4064.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 11^{9}\cdot 13^{12}$
Root discriminant $41.98$
Ramified primes $2, 3, 11, 13$
Class number $126$ (GRH)
Class group $[126]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![90787, -145088, 586289, -469025, 444163, -210997, 162275, -55927, 35125, -7238, 6780, -1431, 1177, -91, 115, -2, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 13*x^16 - 2*x^15 + 115*x^14 - 91*x^13 + 1177*x^12 - 1431*x^11 + 6780*x^10 - 7238*x^9 + 35125*x^8 - 55927*x^7 + 162275*x^6 - 210997*x^5 + 444163*x^4 - 469025*x^3 + 586289*x^2 - 145088*x + 90787)
 
gp: K = bnfinit(x^18 - x^17 + 13*x^16 - 2*x^15 + 115*x^14 - 91*x^13 + 1177*x^12 - 1431*x^11 + 6780*x^10 - 7238*x^9 + 35125*x^8 - 55927*x^7 + 162275*x^6 - 210997*x^5 + 444163*x^4 - 469025*x^3 + 586289*x^2 - 145088*x + 90787, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 13 x^{16} - 2 x^{15} + 115 x^{14} - 91 x^{13} + 1177 x^{12} - 1431 x^{11} + 6780 x^{10} - 7238 x^{9} + 35125 x^{8} - 55927 x^{7} + 162275 x^{6} - 210997 x^{5} + 444163 x^{4} - 469025 x^{3} + 586289 x^{2} - 145088 x + 90787 \)

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:  \(-164037019762900412649715544064=-\,2^{12}\cdot 3^{6}\cdot 11^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.98$
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:  $2, 3, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{7}{15} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{15} a^{8} - \frac{4}{15} a^{7} - \frac{4}{15} a^{6} + \frac{7}{15} a^{5} + \frac{1}{15} a^{4} + \frac{1}{15} a^{3} + \frac{1}{15} a^{2} + \frac{4}{15}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{11} - \frac{2}{5} a^{10} + \frac{1}{3} a^{9} + \frac{4}{15} a^{8} - \frac{2}{15} a^{7} - \frac{2}{5} a^{6} + \frac{1}{3} a^{5} - \frac{7}{15} a^{4} - \frac{7}{15} a^{3} + \frac{7}{15} a^{2} + \frac{4}{15} a - \frac{2}{15}$, $\frac{1}{15} a^{14} + \frac{2}{15} a^{11} - \frac{7}{15} a^{10} + \frac{7}{15} a^{9} - \frac{1}{5} a^{8} + \frac{1}{3} a^{7} + \frac{1}{15} a^{6} - \frac{2}{5} a^{4} - \frac{7}{15} a^{3} + \frac{1}{3} a^{2} - \frac{2}{15} a + \frac{4}{15}$, $\frac{1}{15} a^{15} + \frac{7}{15} a^{11} + \frac{1}{15} a^{10} + \frac{2}{5} a^{9} + \frac{7}{15} a^{8} - \frac{2}{5} a^{7} - \frac{7}{15} a^{6} - \frac{1}{3} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{4}{15} a^{2} + \frac{4}{15} a + \frac{7}{15}$, $\frac{1}{75} a^{16} - \frac{1}{75} a^{15} - \frac{1}{75} a^{14} + \frac{2}{75} a^{13} - \frac{1}{75} a^{12} + \frac{31}{75} a^{11} + \frac{7}{25} a^{10} - \frac{1}{15} a^{9} - \frac{8}{25} a^{8} - \frac{23}{75} a^{7} - \frac{8}{25} a^{6} + \frac{2}{15} a^{5} - \frac{19}{75} a^{4} + \frac{23}{75} a^{3} - \frac{7}{25} a^{2} + \frac{13}{75} a + \frac{28}{75}$, $\frac{1}{9102487367694912004193215043520257325} a^{17} + \frac{18759074355459183669817447176022286}{3034162455898304001397738347840085775} a^{16} + \frac{22219797757216711931100649200003059}{1820497473538982400838643008704051465} a^{15} - \frac{224729455168326737174959656739895512}{9102487367694912004193215043520257325} a^{14} + \frac{72534221511665876131318016098000502}{9102487367694912004193215043520257325} a^{13} - \frac{152357677241708704682632713608152258}{9102487367694912004193215043520257325} a^{12} + \frac{10052698934106251080270049596122524}{121366498235932160055909533913603431} a^{11} + \frac{97136157891329936920784812629643183}{3034162455898304001397738347840085775} a^{10} - \frac{149958264840056352366040975101884189}{9102487367694912004193215043520257325} a^{9} - \frac{1233907040193387312487193994450364448}{3034162455898304001397738347840085775} a^{8} + \frac{4489316443359947822103033710777935664}{9102487367694912004193215043520257325} a^{7} - \frac{90985536994857143092617919680957837}{3034162455898304001397738347840085775} a^{6} + \frac{3751150365517063298355803236326764011}{9102487367694912004193215043520257325} a^{5} - \frac{2375676097237630271710906267275958738}{9102487367694912004193215043520257325} a^{4} + \frac{1251059238715486513547151840610629536}{9102487367694912004193215043520257325} a^{3} + \frac{911090568098134025265899934529819698}{3034162455898304001397738347840085775} a^{2} - \frac{499321508994386273787947616982614892}{1820497473538982400838643008704051465} a - \frac{2679594218538398386338336477022061848}{9102487367694912004193215043520257325}$

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

Class group and class number

$C_{126}$, which has order $126$ (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:  \( 368294.41393950605 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.3.169.1, 3.1.2028.1, 6.0.5474115504.1, 6.0.38014691.1, 9.3.8340725952.1

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 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{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}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
13Data not computed