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Properties

Label 18.0.12303951687...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 19^{12}$
Root discriminant $247.68$
Ramified primes $2, 3, 5, 19$
Class number $6561$ (GRH)
Class group $[3, 3, 3, 3, 9, 9]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1328391828, 35801892, 33435936, -85483590, 20343942, -236412, -4869773, 1417041, -379995, -168870, 190365, -55755, 2680, 3141, -615, -18, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 615*x^14 + 3141*x^13 + 2680*x^12 - 55755*x^11 + 190365*x^10 - 168870*x^9 - 379995*x^8 + 1417041*x^7 - 4869773*x^6 - 236412*x^5 + 20343942*x^4 - 85483590*x^3 + 33435936*x^2 + 35801892*x + 1328391828)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 615*x^14 + 3141*x^13 + 2680*x^12 - 55755*x^11 + 190365*x^10 - 168870*x^9 - 379995*x^8 + 1417041*x^7 - 4869773*x^6 - 236412*x^5 + 20343942*x^4 - 85483590*x^3 + 33435936*x^2 + 35801892*x + 1328391828, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 18 x^{15} - 615 x^{14} + 3141 x^{13} + 2680 x^{12} - 55755 x^{11} + 190365 x^{10} - 168870 x^{9} - 379995 x^{8} + 1417041 x^{7} - 4869773 x^{6} - 236412 x^{5} + 20343942 x^{4} - 85483590 x^{3} + 33435936 x^{2} + 35801892 x + 1328391828 \)

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:  \(-12303951687949724503891516957203000000000000=-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $247.68$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{3}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{9} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{7}{18} a^{7} - \frac{1}{18} a^{6} + \frac{4}{9} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{11} - \frac{1}{18} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{5}{18} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6679602} a^{15} - \frac{46769}{3339801} a^{14} - \frac{3205}{742178} a^{13} + \frac{521207}{6679602} a^{12} - \frac{344221}{3339801} a^{11} - \frac{74069}{742178} a^{10} - \frac{47935}{351558} a^{9} + \frac{1569293}{3339801} a^{8} - \frac{212131}{742178} a^{7} + \frac{2188177}{6679602} a^{6} - \frac{115922}{3339801} a^{5} + \frac{130865}{742178} a^{4} - \frac{46294}{1113267} a^{3} - \frac{531916}{1113267} a^{2} - \frac{160800}{371089} a + \frac{42960}{371089}$, $\frac{1}{78991171989679441458558} a^{16} - \frac{16733507735005}{26330390663226480486186} a^{15} - \frac{222849591735197528735}{26330390663226480486186} a^{14} + \frac{1060492300472168991377}{78991171989679441458558} a^{13} + \frac{257169578171286410261}{4388398443871080081031} a^{12} + \frac{532227764845362593089}{26330390663226480486186} a^{11} - \frac{12229399142607205984693}{78991171989679441458558} a^{10} + \frac{248867502066841509519}{4388398443871080081031} a^{9} - \frac{6794770279230914452667}{26330390663226480486186} a^{8} + \frac{349123070757372102805}{4157430104719970603082} a^{7} - \frac{2439180786050350466005}{13165195331613240243093} a^{6} - \frac{3039510063756278731653}{8776796887742160162062} a^{5} - \frac{113149916644934756071}{4388398443871080081031} a^{4} + \frac{1143956333631693438961}{8776796887742160162062} a^{3} - \frac{768028117021643343572}{4388398443871080081031} a^{2} - \frac{1486069818961264904122}{4388398443871080081031} a - \frac{1665894553330064003485}{4388398443871080081031}$, $\frac{1}{481498983045618524603966866984837808814} a^{17} + \frac{105165869067493}{28323469590918736741409815704990459342} a^{16} + \frac{4569891021257754793758213152894}{80249830507603087433994477830806301469} a^{15} - \frac{1705798619290547342157921598594141303}{240749491522809262301983433492418904407} a^{14} - \frac{7673358547227854948333555376865278427}{481498983045618524603966866984837808814} a^{13} + \frac{28652112184723743863706740319179762875}{481498983045618524603966866984837808814} a^{12} + \frac{10388976790395723565240628901396753026}{240749491522809262301983433492418904407} a^{11} + \frac{51224667892539477264048931642895875327}{481498983045618524603966866984837808814} a^{10} - \frac{5999012941119845938646660025066777097}{53499887005068724955996318553870867646} a^{9} + \frac{3732981958379530157765598308867404231}{7766112629768040719418820435239319497} a^{8} + \frac{96263331676367385895109414470421904277}{481498983045618524603966866984837808814} a^{7} - \frac{7118349297333668423474738645952252727}{25342051739243080242314045630780937306} a^{6} - \frac{172236978364691560256251992367801481669}{481498983045618524603966866984837808814} a^{5} + \frac{9174457734414853737696330474117494933}{80249830507603087433994477830806301469} a^{4} - \frac{52835900888925694916832164342726710423}{160499661015206174867988955661612602938} a^{3} + \frac{13707098098153919626666670908029496759}{80249830507603087433994477830806301469} a^{2} + \frac{12557943529489816449836327510603010594}{26749943502534362477998159276935433823} a - \frac{7602563754370867993431065052570683879}{26749943502534362477998159276935433823}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $6561$ (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:  \( -\frac{22440187401181765}{13888992248105591326932837} a^{17} + \frac{332049770834039095}{27777984496211182653865674} a^{16} - \frac{217102279354920751}{4629664082701863775644279} a^{15} - \frac{664477177689968455}{27777984496211182653865674} a^{14} + \frac{24307710972222590849}{27777984496211182653865674} a^{13} - \frac{37175763196288622657}{9259328165403727551288558} a^{12} - \frac{201951426086533250131}{27777984496211182653865674} a^{11} + \frac{1828888255331672463833}{27777984496211182653865674} a^{10} - \frac{2114318862660214460551}{9259328165403727551288558} a^{9} + \frac{4689417337473041671807}{27777984496211182653865674} a^{8} + \frac{2484426049565908716247}{27777984496211182653865674} a^{7} - \frac{36581557324596667378541}{9259328165403727551288558} a^{6} + \frac{36170251815412125670929}{3086442721801242517096186} a^{5} - \frac{15994904008961996355136}{4629664082701863775644279} a^{4} - \frac{290940460756262398318607}{9259328165403727551288558} a^{3} + \frac{192141637203033070982713}{1543221360900621258548093} a^{2} + \frac{75209860526457702971276}{1543221360900621258548093} a + \frac{104202528176610034828618}{90777727111801250502829} \) (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:  \( 315019930628.8986 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 6.0.1771470000.6, Deg 6, Deg 6, 6.0.3518667.2

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$