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Properties

Label 18.0.50078803235...8288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 43^{8}$
Root discriminant $30.43$
Ramified primes $2, 3, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11881, 8175, 23406, -13723, 38337, -41841, 42489, -27198, 17385, -7743, 2886, -1152, 788, -333, 54, -2, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 - 2*x^15 + 54*x^14 - 333*x^13 + 788*x^12 - 1152*x^11 + 2886*x^10 - 7743*x^9 + 17385*x^8 - 27198*x^7 + 42489*x^6 - 41841*x^5 + 38337*x^4 - 13723*x^3 + 23406*x^2 + 8175*x + 11881)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 - 2*x^15 + 54*x^14 - 333*x^13 + 788*x^12 - 1152*x^11 + 2886*x^10 - 7743*x^9 + 17385*x^8 - 27198*x^7 + 42489*x^6 - 41841*x^5 + 38337*x^4 - 13723*x^3 + 23406*x^2 + 8175*x + 11881, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} - 2 x^{15} + 54 x^{14} - 333 x^{13} + 788 x^{12} - 1152 x^{11} + 2886 x^{10} - 7743 x^{9} + 17385 x^{8} - 27198 x^{7} + 42489 x^{6} - 41841 x^{5} + 38337 x^{4} - 13723 x^{3} + 23406 x^{2} + 8175 x + 11881 \)

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:  \(-500788032352702912536588288=-\,2^{12}\cdot 3^{21}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.43$
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:  $2, 3, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{12} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{185} a^{15} + \frac{3}{185} a^{14} + \frac{9}{185} a^{13} + \frac{57}{185} a^{12} + \frac{4}{37} a^{11} - \frac{88}{185} a^{10} + \frac{44}{185} a^{9} + \frac{62}{185} a^{8} - \frac{13}{185} a^{7} + \frac{18}{185} a^{6} - \frac{3}{37} a^{5} - \frac{59}{185} a^{4} - \frac{87}{185} a^{3} + \frac{79}{185} a^{2} + \frac{13}{37} a - \frac{54}{185}$, $\frac{1}{3145} a^{16} - \frac{1}{3145} a^{15} + \frac{29}{629} a^{14} + \frac{21}{3145} a^{13} + \frac{8}{37} a^{12} - \frac{279}{3145} a^{11} - \frac{1491}{3145} a^{10} - \frac{669}{3145} a^{9} - \frac{927}{3145} a^{8} + \frac{1328}{3145} a^{7} + \frac{61}{3145} a^{6} + \frac{334}{3145} a^{5} - \frac{1368}{3145} a^{4} - \frac{1349}{3145} a^{3} + \frac{81}{185} a^{2} + \frac{12}{185} a - \frac{1227}{3145}$, $\frac{1}{25659151896853340138611182223469855} a^{17} + \frac{3559663336808503690151125253691}{25659151896853340138611182223469855} a^{16} - \frac{38233318982374016827121026228208}{25659151896853340138611182223469855} a^{15} + \frac{597807414019897649517875230800392}{25659151896853340138611182223469855} a^{14} + \frac{2227792248671653499645676721048044}{25659151896853340138611182223469855} a^{13} - \frac{711552937011767619517652946130573}{5131830379370668027722236444693971} a^{12} - \frac{6789226159360305568086666679552852}{25659151896853340138611182223469855} a^{11} + \frac{7359654234405658549552228277767954}{25659151896853340138611182223469855} a^{10} - \frac{8137250543981171972903069766169587}{25659151896853340138611182223469855} a^{9} + \frac{9626607699862567683592622059779793}{25659151896853340138611182223469855} a^{8} - \frac{53910486849350124163852910296559}{301872375257098119277778614393763} a^{7} - \frac{1357674008770002120554666432242933}{5131830379370668027722236444693971} a^{6} - \frac{1505477397336132395307907597220549}{25659151896853340138611182223469855} a^{5} - \frac{515324587987712559026876708799196}{1509361876285490596388893071968815} a^{4} - \frac{5681085796077008013853497249884756}{25659151896853340138611182223469855} a^{3} - \frac{21611910245661069783324087656206}{1509361876285490596388893071968815} a^{2} + \frac{10752266864880458916718147113640666}{25659151896853340138611182223469855} a + \frac{85020019629061030382321160045186}{235405063273883854482671396545595}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1155337288623553688}{37822461257072030750605} a^{17} - \frac{5821504657861398699}{37822461257072030750605} a^{16} + \frac{976166747886386233}{7564492251414406150121} a^{15} + \frac{3837399075746344724}{37822461257072030750605} a^{14} + \frac{59469249585451717309}{37822461257072030750605} a^{13} - \frac{341864005454211197487}{37822461257072030750605} a^{12} + \frac{634160166731328424843}{37822461257072030750605} a^{11} - \frac{594128799352617252647}{37822461257072030750605} a^{10} + \frac{2356241219596915640037}{37822461257072030750605} a^{9} - \frac{7106655516415713444043}{37822461257072030750605} a^{8} + \frac{15337306880892868347261}{37822461257072030750605} a^{7} - \frac{15292934424128179155176}{37822461257072030750605} a^{6} + \frac{29712608343482674597024}{37822461257072030750605} a^{5} - \frac{27538052802163569256176}{37822461257072030750605} a^{4} + \frac{44994820263377383222728}{37822461257072030750605} a^{3} - \frac{1301743237218358073681}{7564492251414406150121} a^{2} + \frac{62328490252530464019103}{37822461257072030750605} a + \frac{286290108343499453791}{346995057404330557345} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2431126.772055431 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.4306704645312.2

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$43$43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.2$x^{3} + 387$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.3$x^{3} - 3483$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$