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Properties

Label 18.0.21515434065...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{9}\cdot 11^{6}\cdot 19^{15}$
Root discriminant $91.82$
Ramified primes $2, 5, 11, 19$
Class number $68992$ (GRH)
Class group $[2, 2, 4, 4312]$ (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![13067109, -31834602, 52690863, -55016363, 41420961, -20620332, 5997076, 29214, -580776, 42506, 148668, -75980, 22374, -5332, 1517, -172, 47, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 47*x^16 - 172*x^15 + 1517*x^14 - 5332*x^13 + 22374*x^12 - 75980*x^11 + 148668*x^10 + 42506*x^9 - 580776*x^8 + 29214*x^7 + 5997076*x^6 - 20620332*x^5 + 41420961*x^4 - 55016363*x^3 + 52690863*x^2 - 31834602*x + 13067109)
 
gp: K = bnfinit(x^18 - 3*x^17 + 47*x^16 - 172*x^15 + 1517*x^14 - 5332*x^13 + 22374*x^12 - 75980*x^11 + 148668*x^10 + 42506*x^9 - 580776*x^8 + 29214*x^7 + 5997076*x^6 - 20620332*x^5 + 41420961*x^4 - 55016363*x^3 + 52690863*x^2 - 31834602*x + 13067109, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 47 x^{16} - 172 x^{15} + 1517 x^{14} - 5332 x^{13} + 22374 x^{12} - 75980 x^{11} + 148668 x^{10} + 42506 x^{9} - 580776 x^{8} + 29214 x^{7} + 5997076 x^{6} - 20620332 x^{5} + 41420961 x^{4} - 55016363 x^{3} + 52690863 x^{2} - 31834602 x + 13067109 \)

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:  \(-215154340657376220394997912000000000=-\,2^{12}\cdot 5^{9}\cdot 11^{6}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.82$
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:  $2, 5, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{14} + \frac{5}{11} a^{11} - \frac{3}{11} a^{10} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} + \frac{2}{11} a^{7} - \frac{3}{11} a^{6} + \frac{3}{11} a^{5} + \frac{1}{11} a^{3} - \frac{2}{11} a^{2} - \frac{2}{11} a - \frac{1}{2}$, $\frac{1}{22} a^{15} + \frac{5}{11} a^{12} - \frac{3}{11} a^{11} + \frac{5}{11} a^{10} - \frac{1}{11} a^{9} + \frac{2}{11} a^{8} - \frac{3}{11} a^{7} + \frac{3}{11} a^{6} + \frac{1}{11} a^{4} - \frac{2}{11} a^{3} - \frac{2}{11} a^{2} - \frac{1}{2} a$, $\frac{1}{66} a^{16} - \frac{1}{66} a^{14} - \frac{1}{66} a^{13} + \frac{5}{66} a^{12} + \frac{1}{6} a^{11} + \frac{5}{22} a^{10} - \frac{17}{66} a^{9} - \frac{5}{22} a^{8} - \frac{31}{66} a^{7} - \frac{9}{22} a^{6} - \frac{5}{22} a^{5} + \frac{7}{66} a^{4} + \frac{9}{22} a^{3} - \frac{3}{11} a^{2} - \frac{29}{66} a$, $\frac{1}{12422153753427658754065837740747858504806011010888355695957508} a^{17} + \frac{1581908561701918634508053343555154035214039855001254945823}{1035179479452304896172153145062321542067167584240696307996459} a^{16} + \frac{265384277489897729548798341856173389600843371100653753200533}{12422153753427658754065837740747858504806011010888355695957508} a^{15} - \frac{71474871773996767890569786632917045726975166364924019875149}{12422153753427658754065837740747858504806011010888355695957508} a^{14} - \frac{210371761752792955271165517385384940052373060914498346938424}{3105538438356914688516459435186964626201502752722088923989377} a^{13} + \frac{1094745492256137938823299747340004669219877019656152618959746}{3105538438356914688516459435186964626201502752722088923989377} a^{12} - \frac{15550564546569482901270693119107104982997077108679542995101}{690119652968203264114768763374881028044778389493797538664306} a^{11} + \frac{2614487459710570096684280376032782919170267215411533142757403}{6211076876713829377032918870373929252403005505444177847978754} a^{10} - \frac{395799719672368190387408048140620199542266950335491308930583}{2070358958904609792344306290124643084134335168481392615992918} a^{9} - \frac{1328330222166371818248131689357253914290886096856811293353540}{3105538438356914688516459435186964626201502752722088923989377} a^{8} + \frac{290094877433515483949337791859199538422114964243879150299531}{1035179479452304896172153145062321542067167584240696307996459} a^{7} - \frac{60394470300949717397789767779120092238128428243386573318419}{690119652968203264114768763374881028044778389493797538664306} a^{6} + \frac{2874089693681120731872060968942445921317028208855333609618919}{6211076876713829377032918870373929252403005505444177847978754} a^{5} - \frac{256103728146200835004139394305812317042797541485477976437119}{690119652968203264114768763374881028044778389493797538664306} a^{4} - \frac{45720741051783181132796622024653319885290513043497442733045}{1380239305936406528229537526749762056089556778987595077328612} a^{3} - \frac{2585316533328142244305394786160432224709997991208544824824209}{6211076876713829377032918870373929252403005505444177847978754} a^{2} + \frac{1651638057006165579033433487388318403918162312179806879297185}{4140717917809219584688612580249286168268670336962785231985836} a + \frac{42038916681930711378972003594828920808925384823387154682625}{125476300539673320748139775159069277826323343544326825211692}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4312}$, which has order $68992$ (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:  \( 654963.5034721284 \) (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

\(\Q(\sqrt{-95}) \), 3.3.15884.1, 3.3.361.1, 6.0.599215958000.1, 6.0.309512375.1, 9.9.4007556327104.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
19Data not computed