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Properties

Label 18.0.11020871256...1727.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 47^{9}$
Root discriminant $100.54$
Ramified primes $3, 47$
Class number $1254285$ (GRH)
Class group $[9, 139365]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18936946911, -9291550797, 11461225869, -4653356364, 3146390811, -1074264939, 515548185, -148974714, 55596501, -13554338, 4088970, -828522, 204504, -33264, 6660, -804, 126, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 126*x^16 - 804*x^15 + 6660*x^14 - 33264*x^13 + 204504*x^12 - 828522*x^11 + 4088970*x^10 - 13554338*x^9 + 55596501*x^8 - 148974714*x^7 + 515548185*x^6 - 1074264939*x^5 + 3146390811*x^4 - 4653356364*x^3 + 11461225869*x^2 - 9291550797*x + 18936946911)
 
gp: K = bnfinit(x^18 - 9*x^17 + 126*x^16 - 804*x^15 + 6660*x^14 - 33264*x^13 + 204504*x^12 - 828522*x^11 + 4088970*x^10 - 13554338*x^9 + 55596501*x^8 - 148974714*x^7 + 515548185*x^6 - 1074264939*x^5 + 3146390811*x^4 - 4653356364*x^3 + 11461225869*x^2 - 9291550797*x + 18936946911, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 126 x^{16} - 804 x^{15} + 6660 x^{14} - 33264 x^{13} + 204504 x^{12} - 828522 x^{11} + 4088970 x^{10} - 13554338 x^{9} + 55596501 x^{8} - 148974714 x^{7} + 515548185 x^{6} - 1074264939 x^{5} + 3146390811 x^{4} - 4653356364 x^{3} + 11461225869 x^{2} - 9291550797 x + 18936946911 \)

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:  \(-1102087125658583523206929858282481727=-\,3^{44}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.54$
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:  $3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1269=3^{3}\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{1269}(1,·)$, $\chi_{1269}(706,·)$, $\chi_{1269}(328,·)$, $\chi_{1269}(1033,·)$, $\chi_{1269}(142,·)$, $\chi_{1269}(847,·)$, $\chi_{1269}(469,·)$, $\chi_{1269}(1174,·)$, $\chi_{1269}(283,·)$, $\chi_{1269}(988,·)$, $\chi_{1269}(610,·)$, $\chi_{1269}(424,·)$, $\chi_{1269}(1129,·)$, $\chi_{1269}(46,·)$, $\chi_{1269}(751,·)$, $\chi_{1269}(565,·)$, $\chi_{1269}(187,·)$, $\chi_{1269}(892,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{53} a^{16} - \frac{15}{53} a^{15} + \frac{23}{53} a^{14} - \frac{8}{53} a^{13} - \frac{10}{53} a^{12} - \frac{19}{53} a^{11} + \frac{7}{53} a^{10} - \frac{5}{53} a^{9} + \frac{24}{53} a^{8} + \frac{14}{53} a^{7} - \frac{21}{53} a^{6} - \frac{14}{53} a^{5} + \frac{16}{53} a^{4} + \frac{20}{53} a^{3} - \frac{9}{53} a^{2} - \frac{26}{53} a + \frac{22}{53}$, $\frac{1}{697934848279328471749668646932198230318986545300020191906013} a^{17} - \frac{5457332218129822488539181516716049617705267835434182481719}{697934848279328471749668646932198230318986545300020191906013} a^{16} - \frac{190311352464839950599245715746210617791006058194306776834832}{697934848279328471749668646932198230318986545300020191906013} a^{15} + \frac{316879169156388937235295520948517789011765613457926940566910}{697934848279328471749668646932198230318986545300020191906013} a^{14} + \frac{19351536209519041971217392652022524910620199179366816783973}{697934848279328471749668646932198230318986545300020191906013} a^{13} + \frac{110326343249149136804361858469474002996701799885569857713194}{697934848279328471749668646932198230318986545300020191906013} a^{12} - \frac{110323279821876293216110070542177977037535703680869359103689}{697934848279328471749668646932198230318986545300020191906013} a^{11} + \frac{200931379479515504024443733443597544107900572111849359207940}{697934848279328471749668646932198230318986545300020191906013} a^{10} - \frac{330950881159654508308787366057583313945878087097707948464295}{697934848279328471749668646932198230318986545300020191906013} a^{9} + \frac{229285070811000180626437475515781441965040474922662064380360}{697934848279328471749668646932198230318986545300020191906013} a^{8} - \frac{169826462651469322537110510309688517425938916647212309118807}{697934848279328471749668646932198230318986545300020191906013} a^{7} + \frac{323431629310582765747651084651716981568993740394472938710920}{697934848279328471749668646932198230318986545300020191906013} a^{6} + \frac{268369835631758121546011363222891749046596736782720357592934}{697934848279328471749668646932198230318986545300020191906013} a^{5} - \frac{67644589571804460138155382906871615275352761927324030440390}{697934848279328471749668646932198230318986545300020191906013} a^{4} - \frac{78868688601510990022073821836683284287178688123645825594777}{697934848279328471749668646932198230318986545300020191906013} a^{3} + \frac{190140355951701254052341813322199312440847305826517668603147}{697934848279328471749668646932198230318986545300020191906013} a^{2} + \frac{297820919521503363105891167703977093248530240145948848508123}{697934848279328471749668646932198230318986545300020191906013} a - \frac{214697222850104216178074211059325576799270492342066097997899}{697934848279328471749668646932198230318986545300020191906013}$

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

Class group and class number

$C_{9}\times C_{139365}$, which has order $1254285$ (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:  \( 40934.03294431194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\zeta_{9})^+\), 6.0.681182703.2, \(\Q(\zeta_{27})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ $18$ R ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
47Data not computed