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Properties

Label 18.18.5305655318...1509.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 19^{12}\cdot 229^{9}$
Root discriminant $394.29$
Ramified primes $7, 19, 229$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3\times D_9$ (as 18T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1164062910163, -1197940138849, 2453632857591, 1050009163838, -1350644250998, -231330970625, 274397510025, 10719756413, -21732949914, -33481138, 811072595, -4948990, -15388414, 31278, 147470, 637, -635, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 635*x^16 + 637*x^15 + 147470*x^14 + 31278*x^13 - 15388414*x^12 - 4948990*x^11 + 811072595*x^10 - 33481138*x^9 - 21732949914*x^8 + 10719756413*x^7 + 274397510025*x^6 - 231330970625*x^5 - 1350644250998*x^4 + 1050009163838*x^3 + 2453632857591*x^2 - 1197940138849*x - 1164062910163)
 
gp: K = bnfinit(x^18 - 3*x^17 - 635*x^16 + 637*x^15 + 147470*x^14 + 31278*x^13 - 15388414*x^12 - 4948990*x^11 + 811072595*x^10 - 33481138*x^9 - 21732949914*x^8 + 10719756413*x^7 + 274397510025*x^6 - 231330970625*x^5 - 1350644250998*x^4 + 1050009163838*x^3 + 2453632857591*x^2 - 1197940138849*x - 1164062910163, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 635 x^{16} + 637 x^{15} + 147470 x^{14} + 31278 x^{13} - 15388414 x^{12} - 4948990 x^{11} + 811072595 x^{10} - 33481138 x^{9} - 21732949914 x^{8} + 10719756413 x^{7} + 274397510025 x^{6} - 231330970625 x^{5} - 1350644250998 x^{4} + 1050009163838 x^{3} + 2453632857591 x^{2} - 1197940138849 x - 1164062910163 \)

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:  \(53056553186191280403749586075084940824853751509=7^{12}\cdot 19^{12}\cdot 229^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $394.29$
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:  $7, 19, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{37} a^{15} + \frac{15}{37} a^{14} + \frac{18}{37} a^{13} + \frac{8}{37} a^{12} + \frac{18}{37} a^{11} + \frac{16}{37} a^{10} - \frac{7}{37} a^{9} + \frac{9}{37} a^{8} + \frac{16}{37} a^{7} - \frac{18}{37} a^{6} - \frac{3}{37} a^{5} + \frac{3}{37} a^{4} + \frac{18}{37} a^{3} - \frac{14}{37} a^{2} - \frac{7}{37} a - \frac{5}{37}$, $\frac{1}{481} a^{16} - \frac{4}{481} a^{15} - \frac{119}{481} a^{14} + \frac{73}{481} a^{13} + \frac{162}{481} a^{12} + \frac{229}{481} a^{11} + \frac{59}{481} a^{10} - \frac{9}{37} a^{9} + \frac{30}{481} a^{8} + \frac{85}{481} a^{7} + \frac{9}{37} a^{6} - \frac{236}{481} a^{5} - \frac{113}{481} a^{4} - \frac{171}{481} a^{3} - \frac{3}{13} a^{2} - \frac{57}{481} a + \frac{132}{481}$, $\frac{1}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{17} + \frac{97313469853227573716582614833457808256305067192379773802501241428014480171383451200130205226179268}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{16} - \frac{1322530298845726823300579262634120537094782785247907795379006767575319390866318444959014421717749973}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{15} - \frac{49117150999870578677193481099406166326514948556762036688504682108906435736988277862181941834055275131}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{14} + \frac{7501974021005723657097893881632825649344095302376249569053260460018466678358625786274760038830489618}{23952688427139403239521013085107652022473490161043939362317460810889988775232461904701354594812452681} a^{13} + \frac{41020668712759290410373802466903662946187063323830603386557717117978356451733402038024310090603623270}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{12} + \frac{72411137518943043548419170597551739191051750657917695279856392127099222493798990427506088706787760901}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{11} + \frac{24784447363799233240940522315020454272067686700380839405664701330864840349496174914804830138696980444}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{10} - \frac{97591107177057897022691959352965752468608360192802286514576115413104924651726815509808208725290811444}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{9} - \frac{33950954906340428314909634877328963485406168101024966885616787252869160429931474583114759503113069320}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{8} - \frac{144056387570188466817126111828018036625701231931277505177562228248350890061514773232307785410088950962}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{7} - \frac{148949898886773336829265809674362123297415096429552106191335033823506508758558570418483592023993435889}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{6} - \frac{2232221984532165455499184931834130876450830038548295490485326992736761900326008513569969831610747463}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{5} - \frac{111327686542266728098417949993377094886140679217032152873192160909708465762842571128738794139369036598}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{4} - \frac{34015936101049073418175412649089094171394177198300428084266182278859546815223714842984940344023546100}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{3} + \frac{122078750539740528223097727059121523063459020636756758569246992289128480566464915721662268112660810909}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a^{2} - \frac{16156883622378745116381053601290306833886401257219289089478153668050107091985273963169486688824790317}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853} a + \frac{70663985701919140595152190599952663835690798545346218197062646862278589700711616512864970257865571934}{311384949552812242113773170106399476292155372093571211710126990541569854078022004761117609732561884853}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 39968959368600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_9$ (as 18T19):

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 $C_3\times D_9$
Character table for $C_3\times D_9$

Intermediate fields

\(\Q(\sqrt{229}) \), 3.3.229.1 x3, 6.6.12008989.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$19$19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
229Data not computed