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Properties

Label 18.0.16748395404...8288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{16}\cdot 19^{12}$
Root discriminant $477.65$
Ramified primes $2, 3, 7, 19$
Class number $177147$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 3, 9, 9]$ (GRH)
Galois group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5898348532, -2807652156, 3278636388, -975956418, 573490734, -104067936, 36908697, -1489695, -162243, 385808, -106503, 15903, -2436, -273, 207, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 207*x^14 - 273*x^13 - 2436*x^12 + 15903*x^11 - 106503*x^10 + 385808*x^9 - 162243*x^8 - 1489695*x^7 + 36908697*x^6 - 104067936*x^5 + 573490734*x^4 - 975956418*x^3 + 3278636388*x^2 - 2807652156*x + 5898348532, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 207 x^{14} - 273 x^{13} - 2436 x^{12} + 15903 x^{11} - 106503 x^{10} + 385808 x^{9} - 162243 x^{8} - 1489695 x^{7} + 36908697 x^{6} - 104067936 x^{5} + 573490734 x^{4} - 975956418 x^{3} + 3278636388 x^{2} - 2807652156 x + 5898348532 \)

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:  \(-1674839540438459282902886554720879596051603468288=-\,2^{12}\cdot 3^{33}\cdot 7^{16}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $477.65$
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:  $2, 3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{38} a^{15} + \frac{1}{19} a^{14} + \frac{4}{19} a^{13} + \frac{8}{19} a^{11} + \frac{9}{19} a^{10} - \frac{4}{19} a^{9} - \frac{7}{19} a^{8} - \frac{8}{19} a^{7} + \frac{4}{19} a^{6} + \frac{3}{19} a^{5} - \frac{9}{19} a^{4} + \frac{17}{38} a^{3} - \frac{9}{19} a^{2} + \frac{6}{19} a + \frac{6}{19}$, $\frac{1}{30344147649224529638611951977190} a^{16} - \frac{4}{15172073824612264819305975988595} a^{15} - \frac{199414300836952233611609617301}{15172073824612264819305975988595} a^{14} + \frac{1395900105858665635281267321177}{15172073824612264819305975988595} a^{13} + \frac{3734111046333453448846408440736}{15172073824612264819305975988595} a^{12} + \frac{4964853819673420506183002145886}{15172073824612264819305975988595} a^{11} + \frac{89043772492980023652522446719}{3034414764922452963861195197719} a^{10} - \frac{6882264890726525180800721689104}{15172073824612264819305975988595} a^{9} - \frac{219434574252964473244549393869}{892474930859544989370939764035} a^{8} - \frac{405426642814649250393973057866}{892474930859544989370939764035} a^{7} - \frac{3009498777658118929407577039407}{15172073824612264819305975988595} a^{6} - \frac{7172689642749779688382132145786}{15172073824612264819305975988595} a^{5} - \frac{11487965056858096214562857484693}{30344147649224529638611951977190} a^{4} + \frac{5998777775519940184491807316486}{15172073824612264819305975988595} a^{3} - \frac{1002807427327015676164137272429}{3034414764922452963861195197719} a^{2} + \frac{6933592532722291109787963829113}{15172073824612264819305975988595} a - \frac{4480576135865031778301377171448}{15172073824612264819305975988595}$, $\frac{1}{34126211868076225788238907059674984410} a^{17} + \frac{562311}{34126211868076225788238907059674984410} a^{16} - \frac{32505968318431966636417209747010612}{17063105934038112894119453529837492205} a^{15} + \frac{263320424897510376489531117779640841}{34126211868076225788238907059674984410} a^{14} - \frac{4267665448819960142384486506346373977}{34126211868076225788238907059674984410} a^{13} + \frac{124455342868149315962719149872360967}{6825242373615245157647781411934996882} a^{12} - \frac{2064944779861836054882006879227583837}{34126211868076225788238907059674984410} a^{11} + \frac{4339988769302951203295284408795558647}{34126211868076225788238907059674984410} a^{10} - \frac{8283124509598379584813742438956448773}{34126211868076225788238907059674984410} a^{9} + \frac{208227242239816362868236782856638291}{2007424227533895634602288650569116730} a^{8} - \frac{2173345362164623890076976282723758847}{6825242373615245157647781411934996882} a^{7} - \frac{2169336702167928893269994733271698293}{34126211868076225788238907059674984410} a^{6} + \frac{3412192907039972623801422288662660437}{17063105934038112894119453529837492205} a^{5} + \frac{969445600144367659822343992256900890}{3412621186807622578823890705967498441} a^{4} - \frac{833247786212867138817622921491807361}{2007424227533895634602288650569116730} a^{3} + \frac{151967408568229028934388409589438442}{898058207054637520743129133149341695} a^{2} + \frac{1407317994461287180935986513267811134}{17063105934038112894119453529837492205} a - \frac{7088106619371175222226310481142464}{233741177178604286220814431915582085}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $177147$ (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:  \( -\frac{8539081304383941}{3249680685137461488687535} a^{17} + \frac{145164382174526997}{6499361370274922977375070} a^{16} - \frac{241574864271440927}{3249680685137461488687535} a^{15} + \frac{7582477058748147}{68414330213420241867106} a^{14} - \frac{100394457125936127}{1299872274054984595475014} a^{13} + \frac{1144373346367940143}{6499361370274922977375070} a^{12} + \frac{2231411869837008663}{342071651067101209335530} a^{11} - \frac{249413467545902726319}{6499361370274922977375070} a^{10} + \frac{1421200407834866675749}{6499361370274922977375070} a^{9} - \frac{4495147123916233287903}{6499361370274922977375070} a^{8} - \frac{8046716405628002399301}{6499361370274922977375070} a^{7} + \frac{9470765870387327153221}{1299872274054984595475014} a^{6} - \frac{541460084975402384041131}{6499361370274922977375070} a^{5} + \frac{613027579448857705827009}{3249680685137461488687535} a^{4} - \frac{5815192154771732169935003}{6499361370274922977375070} a^{3} + \frac{3761511589663562119912014}{3249680685137461488687535} a^{2} - \frac{9573899203763918987852106}{3249680685137461488687535} a + \frac{5804676630515993811543427}{3249680685137461488687535} \) (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:  \( 3161751653565.267 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1323.1 x3, 6.0.5250987.1, 9.1.747181267707388232384064.1 x3

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

Sibling fields

Degree 9 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$7$7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$