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Properties

Label 18.0.64774404297...3168.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 11^{12}$
Root discriminant $18.51$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, -72, -12, -60, 316, -120, -39, -320, 267, -4, 114, -104, 7, -14, 0, 2, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 + 2*x^15 - 14*x^13 + 7*x^12 - 104*x^11 + 114*x^10 - 4*x^9 + 267*x^8 - 320*x^7 - 39*x^6 - 120*x^5 + 316*x^4 - 60*x^3 - 12*x^2 - 72*x + 36)
 
gp: K = bnfinit(x^18 - 2*x^17 + 5*x^16 + 2*x^15 - 14*x^13 + 7*x^12 - 104*x^11 + 114*x^10 - 4*x^9 + 267*x^8 - 320*x^7 - 39*x^6 - 120*x^5 + 316*x^4 - 60*x^3 - 12*x^2 - 72*x + 36, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 5 x^{16} + 2 x^{15} - 14 x^{13} + 7 x^{12} - 104 x^{11} + 114 x^{10} - 4 x^{9} + 267 x^{8} - 320 x^{7} - 39 x^{6} - 120 x^{5} + 316 x^{4} - 60 x^{3} - 12 x^{2} - 72 x + 36 \)

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:  \(-64774404297457079943168=-\,2^{20}\cdot 3^{9}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.51$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{36} a^{15} - \frac{1}{18} a^{14} + \frac{1}{36} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{3} a^{10} - \frac{13}{36} a^{9} - \frac{2}{9} a^{8} - \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{11}{36} a^{5} + \frac{1}{3} a^{4} - \frac{11}{36} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{5004} a^{16} - \frac{67}{5004} a^{15} - \frac{421}{5004} a^{14} + \frac{611}{5004} a^{13} + \frac{10}{417} a^{12} + \frac{383}{1251} a^{11} + \frac{23}{5004} a^{10} + \frac{221}{556} a^{9} - \frac{1139}{2502} a^{8} - \frac{97}{1251} a^{7} + \frac{775}{1668} a^{6} + \frac{833}{5004} a^{5} - \frac{2471}{5004} a^{4} + \frac{1117}{5004} a^{3} + \frac{85}{834} a^{2} + \frac{95}{834} a + \frac{43}{278}$, $\frac{1}{1573414095096} a^{17} + \frac{750397}{786707047548} a^{16} - \frac{406528769}{174823788344} a^{15} + \frac{785305487}{131117841258} a^{14} + \frac{41110537955}{786707047548} a^{13} + \frac{7784014715}{131117841258} a^{12} + \frac{44045231887}{1573414095096} a^{11} + \frac{17810014841}{43705947086} a^{10} - \frac{76293962566}{196676761887} a^{9} - \frac{118005979021}{262235682516} a^{8} - \frac{1162328699}{11319525864} a^{7} - \frac{830805251}{131117841258} a^{6} + \frac{49371988873}{524471365032} a^{5} + \frac{294756530789}{786707047548} a^{4} - \frac{368685278809}{786707047548} a^{3} + \frac{88988077775}{262235682516} a^{2} - \frac{20030686558}{65558920629} a - \frac{10789416013}{43705947086}$

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:  \( -\frac{1202581207}{3773175288} a^{17} + \frac{2274077873}{5659762932} a^{16} - \frac{14579681003}{11319525864} a^{15} - \frac{2257593601}{1414940733} a^{14} - \frac{6453489799}{5659762932} a^{13} + \frac{10420276097}{2829881466} a^{12} + \frac{2251537223}{3773175288} a^{11} + \frac{47572540717}{1414940733} a^{10} - \frac{16040002834}{1414940733} a^{9} - \frac{44462828365}{5659762932} a^{8} - \frac{1029125604179}{11319525864} a^{7} + \frac{48352453399}{1414940733} a^{6} + \frac{49449474177}{1257725096} a^{5} + \frac{376386064349}{5659762932} a^{4} - \frac{32323516635}{628862548} a^{3} - \frac{12751216365}{628862548} a^{2} - \frac{1479450890}{157215637} a + \frac{5067668697}{314431274} \) (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:  \( 117833.92441176013 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

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

\(\Q(\sqrt{-3}) \), 3.1.484.1, 3.1.1452.1 x3, 6.0.6324912.2, 6.0.6324912.1, 9.1.48980118528.1 x3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$