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Properties

Label 18.18.5703857685...9216.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{14}\cdot 3^{21}\cdot 11^{12}\cdot 13^{9}$
Root discriminant $110.16$
Ramified primes $2, 3, 11, 13$
Class number $1$ (GRH)
Class group Trivial (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![-1319078, 10514490, 9855147, -32457542, -17858889, 28021338, 14757756, -9346950, -5362239, 1336398, 908568, -91218, -80911, 2910, 3936, -34, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 99*x^16 - 34*x^15 + 3936*x^14 + 2910*x^13 - 80911*x^12 - 91218*x^11 + 908568*x^10 + 1336398*x^9 - 5362239*x^8 - 9346950*x^7 + 14757756*x^6 + 28021338*x^5 - 17858889*x^4 - 32457542*x^3 + 9855147*x^2 + 10514490*x - 1319078)
 
gp: K = bnfinit(x^18 - 99*x^16 - 34*x^15 + 3936*x^14 + 2910*x^13 - 80911*x^12 - 91218*x^11 + 908568*x^10 + 1336398*x^9 - 5362239*x^8 - 9346950*x^7 + 14757756*x^6 + 28021338*x^5 - 17858889*x^4 - 32457542*x^3 + 9855147*x^2 + 10514490*x - 1319078, 1)
 

Normalized defining polynomial

\( x^{18} - 99 x^{16} - 34 x^{15} + 3936 x^{14} + 2910 x^{13} - 80911 x^{12} - 91218 x^{11} + 908568 x^{10} + 1336398 x^{9} - 5362239 x^{8} - 9346950 x^{7} + 14757756 x^{6} + 28021338 x^{5} - 17858889 x^{4} - 32457542 x^{3} + 9855147 x^{2} + 10514490 x - 1319078 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5703857685299429613102692113176969216=2^{14}\cdot 3^{21}\cdot 11^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $110.16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{5}{16} a^{2} + \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{3}{16} a^{9} + \frac{1}{8} a^{8} + \frac{5}{16} a^{7} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{5}{16} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{3}{8} a^{7} - \frac{5}{32} a^{6} - \frac{3}{8} a^{5} - \frac{9}{32} a^{4} + \frac{7}{16} a^{3} + \frac{11}{32} a^{2} - \frac{7}{16} a + \frac{1}{16}$, $\frac{1}{160} a^{15} - \frac{1}{32} a^{13} + \frac{1}{40} a^{12} + \frac{1}{160} a^{11} + \frac{3}{80} a^{10} - \frac{1}{4} a^{9} + \frac{7}{80} a^{8} - \frac{49}{160} a^{7} + \frac{17}{80} a^{6} - \frac{73}{160} a^{5} - \frac{3}{80} a^{4} - \frac{49}{160} a^{3} - \frac{1}{2} a^{2} - \frac{29}{80} a + \frac{19}{40}$, $\frac{1}{603520} a^{16} + \frac{5}{15088} a^{15} + \frac{743}{60352} a^{14} - \frac{7053}{301760} a^{13} + \frac{4663}{301760} a^{12} + \frac{5449}{150880} a^{11} + \frac{13579}{120704} a^{10} - \frac{55803}{301760} a^{9} + \frac{105151}{603520} a^{8} + \frac{63}{4715} a^{7} - \frac{29751}{75440} a^{6} - \frac{22803}{301760} a^{5} - \frac{60681}{150880} a^{4} + \frac{4353}{30176} a^{3} - \frac{225493}{603520} a^{2} + \frac{137723}{301760} a - \frac{4409}{60352}$, $\frac{1}{10604905034572566970900030214267098880} a^{17} - \frac{567464527483899102284848388555}{2120981006914513394180006042853419776} a^{16} - \frac{335738789890937414158655052585613}{757493216755183355064287872447649920} a^{15} + \frac{1947026393834952412558516919371653}{1325613129321570871362503776783387360} a^{14} + \frac{5377182466772102324824355692256901}{662806564660785435681251888391693680} a^{13} - \frac{87450174310061313118684982385898461}{5302452517286283485450015107133549440} a^{12} + \frac{371019049599534685474940446346433743}{10604905034572566970900030214267098880} a^{11} + \frac{421568323710937549314514116019219637}{10604905034572566970900030214267098880} a^{10} + \frac{784908369779249693842977212647382841}{10604905034572566970900030214267098880} a^{9} + \frac{2123438227382040768128425952520478951}{10604905034572566970900030214267098880} a^{8} - \frac{7939240806859214298051611253675179}{530245251728628348545001510713354944} a^{7} + \frac{66315393188852236993801585341769169}{757493216755183355064287872447649920} a^{6} + \frac{818455558899495984711048642216619901}{5302452517286283485450015107133549440} a^{5} + \frac{931049490758792711428769592606061403}{2651226258643141742725007553566774720} a^{4} - \frac{234821280751494520070312680503402185}{2120981006914513394180006042853419776} a^{3} - \frac{4081988479454891197347501981078913339}{10604905034572566970900030214267098880} a^{2} + \frac{441669225924841494144990668949205979}{2651226258643141742725007553566774720} a + \frac{524074448161377329116033143485967239}{5302452517286283485450015107133549440}$

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:  $17$
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:  \( 30071457300200 \) (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

3.3.1573.1, 6.6.3473957916.1, 9.9.4902949660999104.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
3Data not computed
$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$$[\ ]_{3}^{2}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.12.9.1$x^{12} - 104 x^{8} - 45968 x^{4} - 2847312$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$