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Properties

Label 18.6.20297970987...5697.1
Degree $18$
Signature $[6, 6]$
Discriminant $11^{8}\cdot 17^{9}\cdot 41^{8}$
Root discriminant $62.35$
Ramified primes $11, 17, 41$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25432, -90508, 47036, 187467, -51154, -182058, 41661, 106680, -22879, -41562, 8177, 11030, -1969, -1852, 285, 180, -25, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 25*x^16 + 180*x^15 + 285*x^14 - 1852*x^13 - 1969*x^12 + 11030*x^11 + 8177*x^10 - 41562*x^9 - 22879*x^8 + 106680*x^7 + 41661*x^6 - 182058*x^5 - 51154*x^4 + 187467*x^3 + 47036*x^2 - 90508*x - 25432)
 
gp: K = bnfinit(x^18 - 7*x^17 - 25*x^16 + 180*x^15 + 285*x^14 - 1852*x^13 - 1969*x^12 + 11030*x^11 + 8177*x^10 - 41562*x^9 - 22879*x^8 + 106680*x^7 + 41661*x^6 - 182058*x^5 - 51154*x^4 + 187467*x^3 + 47036*x^2 - 90508*x - 25432, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 25 x^{16} + 180 x^{15} + 285 x^{14} - 1852 x^{13} - 1969 x^{12} + 11030 x^{11} + 8177 x^{10} - 41562 x^{9} - 22879 x^{8} + 106680 x^{7} + 41661 x^{6} - 182058 x^{5} - 51154 x^{4} + 187467 x^{3} + 47036 x^{2} - 90508 x - 25432 \)

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:  \(202979709877915393317843809755697=11^{8}\cdot 17^{9}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.35$
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:  $11, 17, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{15194570553154259549143779274147768} a^{17} + \frac{424217396038291727217259125461969}{7597285276577129774571889637073884} a^{16} + \frac{880347693774104293369106370777877}{15194570553154259549143779274147768} a^{15} + \frac{86031677784303704502434936006929}{1899321319144282443642972409268471} a^{14} - \frac{135485885865968298404987808356699}{15194570553154259549143779274147768} a^{13} - \frac{1550464511351839127511778739878255}{7597285276577129774571889637073884} a^{12} + \frac{1523402754116783098170425701410699}{15194570553154259549143779274147768} a^{11} - \frac{1585656288741314117035266730165103}{7597285276577129774571889637073884} a^{10} - \frac{169340890081073234336961121022057}{893798267832603502890810545538104} a^{9} + \frac{168829280783757576520622881528689}{3798642638288564887285944818536942} a^{8} - \frac{2301372518161120817181768852070851}{15194570553154259549143779274147768} a^{7} + \frac{485792471631201578871735503598935}{3798642638288564887285944818536942} a^{6} + \frac{7550084827308240066049071142141195}{15194570553154259549143779274147768} a^{5} - \frac{850653392816845481130068543808239}{7597285276577129774571889637073884} a^{4} + \frac{1095381766973500572615291126695059}{7597285276577129774571889637073884} a^{3} + \frac{552833928626006847541106231449108}{1899321319144282443642972409268471} a^{2} - \frac{340946059860268117976943371550191}{3798642638288564887285944818536942} a - \frac{16889934092934800770848302295905}{111724783479075437861351318192263}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 456143236.434 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.697.1, 6.2.8258753.2, 9.9.203260472893313.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$