↑

Properties

Label 18.0.72207389626...2659.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{9}\cdot 31^{15}$
Root discriminant $240.45$
Ramified primes $3, 7, 31$
Class number $13026944$ (GRH)
Class group $[2, 4, 28, 58156]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3320473161664, 1262975131200, 1630678178304, 213915299096, 285834467088, 23995814184, 23514110103, 1712290440, 1386184545, 40776861, 63962538, -473373, 1885484, -21558, 32481, -161, 294, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 294*x^16 - 161*x^15 + 32481*x^14 - 21558*x^13 + 1885484*x^12 - 473373*x^11 + 63962538*x^10 + 40776861*x^9 + 1386184545*x^8 + 1712290440*x^7 + 23514110103*x^6 + 23995814184*x^5 + 285834467088*x^4 + 213915299096*x^3 + 1630678178304*x^2 + 1262975131200*x + 3320473161664)
 
gp: K = bnfinit(x^18 + 294*x^16 - 161*x^15 + 32481*x^14 - 21558*x^13 + 1885484*x^12 - 473373*x^11 + 63962538*x^10 + 40776861*x^9 + 1386184545*x^8 + 1712290440*x^7 + 23514110103*x^6 + 23995814184*x^5 + 285834467088*x^4 + 213915299096*x^3 + 1630678178304*x^2 + 1262975131200*x + 3320473161664, 1)
 

Normalized defining polynomial

\( x^{18} + 294 x^{16} - 161 x^{15} + 32481 x^{14} - 21558 x^{13} + 1885484 x^{12} - 473373 x^{11} + 63962538 x^{10} + 40776861 x^{9} + 1386184545 x^{8} + 1712290440 x^{7} + 23514110103 x^{6} + 23995814184 x^{5} + 285834467088 x^{4} + 213915299096 x^{3} + 1630678178304 x^{2} + 1262975131200 x + 3320473161664 \)

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:  \(-7220738962643911588750494056035314570262659=-\,3^{27}\cdot 7^{9}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.45$
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:  $3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1953=3^{2}\cdot 7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1953}(1,·)$, $\chi_{1953}(1091,·)$, $\chi_{1953}(904,·)$, $\chi_{1953}(650,·)$, $\chi_{1953}(652,·)$, $\chi_{1953}(1742,·)$, $\chi_{1953}(211,·)$, $\chi_{1953}(398,·)$, $\chi_{1953}(1303,·)$, $\chi_{1953}(1049,·)$, $\chi_{1953}(862,·)$, $\chi_{1953}(1952,·)$, $\chi_{1953}(1700,·)$, $\chi_{1953}(1513,·)$, $\chi_{1953}(1555,·)$, $\chi_{1953}(440,·)$, $\chi_{1953}(253,·)$, $\chi_{1953}(1301,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{119944} a^{15} - \frac{1}{44} a^{14} - \frac{1513}{59972} a^{13} + \frac{26667}{119944} a^{12} + \frac{6231}{119944} a^{11} + \frac{6031}{29986} a^{10} - \frac{37}{319} a^{9} + \frac{17003}{119944} a^{8} - \frac{6371}{14993} a^{7} + \frac{46897}{119944} a^{6} + \frac{20307}{119944} a^{5} - \frac{9375}{59972} a^{4} + \frac{26967}{119944} a^{3} + \frac{4899}{59972} a^{2} - \frac{4162}{14993} a - \frac{179}{14993}$, $\frac{1}{7086051632} a^{16} - \frac{3727}{3543025816} a^{15} - \frac{382175809}{3543025816} a^{14} - \frac{1759997997}{7086051632} a^{13} - \frac{49570875}{644186512} a^{12} + \frac{549579}{5553332} a^{11} + \frac{91891417}{885756454} a^{10} + \frac{482358451}{7086051632} a^{9} - \frac{14904852}{40261657} a^{8} - \frac{1587162847}{7086051632} a^{7} - \frac{3480707133}{7086051632} a^{6} - \frac{285324327}{3543025816} a^{5} + \frac{1109040575}{7086051632} a^{4} + \frac{1546922915}{3543025816} a^{3} + \frac{2876847}{30543326} a^{2} + \frac{3810582}{40261657} a + \frac{23227509}{442878227}$, $\frac{1}{23969657738026152751594460936329839312759980875663988901871509651208475383585648140497888} a^{17} - \frac{21507874177192434818362794096343650684083828076971867051101812648249906700157}{11984828869013076375797230468164919656379990437831994450935754825604237691792824070248944} a^{16} + \frac{26458663825332070776628915635402611743292410667822786373211188805723205221470108483}{11984828869013076375797230468164919656379990437831994450935754825604237691792824070248944} a^{15} + \frac{659966465434892312940138803545384236403189069398536165791164779670199234273506121624483}{23969657738026152751594460936329839312759980875663988901871509651208475383585648140497888} a^{14} + \frac{4594214388648101737267987617767111639702884701730967801686397957030646076707594931064155}{23969657738026152751594460936329839312759980875663988901871509651208475383585648140497888} a^{13} + \frac{1812075034121497143940841725204329443820824933542744354663641377047327809141838442371}{13744069803914078412611502830464357404105493621366966113458434433032382674074339530102} a^{12} - \frac{698021490402451644239849171562594641784945735408015082944238655950527891331531297602397}{2996207217253269093949307617041229914094997609457998612733938706401059422948206017562236} a^{11} - \frac{3115372261533916788923838559504431280704317604237611841414227186537784589583306315105813}{23969657738026152751594460936329839312759980875663988901871509651208475383585648140497888} a^{10} + \frac{12014443596763594594524632410982340853321437287661661377428592980381426586961638500501}{127498179457585918891459898597499145280638196147148877137614413038342954168008766704776} a^{9} + \frac{739665072473009060657179913987672988854945771478393032221823967862583882080136953020235}{2179059794366013886508587357848167210250907352333089900170137241018952307598695285499808} a^{8} + \frac{8099702722017777166717801065155489808209849771028257296391678958298267333063680771634703}{23969657738026152751594460936329839312759980875663988901871509651208475383585648140497888} a^{7} + \frac{135888898046339034991433718652815334199635082002606231208156048129679994663387544903951}{11984828869013076375797230468164919656379990437831994450935754825604237691792824070248944} a^{6} - \frac{1759925455709032777255295996450923506356911099960826701387797914298251209625903836749337}{23969657738026152751594460936329839312759980875663988901871509651208475383585648140497888} a^{5} + \frac{4289800341489977574300549318828360791324513616512596247248432929381739621379925625057305}{11984828869013076375797230468164919656379990437831994450935754825604237691792824070248944} a^{4} + \frac{124378209632554984353080098181401638319255910929129310722644476259860214151826021760803}{1498103608626634546974653808520614957047498804728999306366969353200529711474103008781118} a^{3} - \frac{89470898169055461093481306187029931924377745550411454170989869078024380108291011646733}{272382474295751735813573419731020901281363419041636237521267155127369038449836910687476} a^{2} - \frac{278057843675508438560728143293965580438732337676991158263452735590984291562545379721071}{1498103608626634546974653808520614957047498804728999306366969353200529711474103008781118} a - \frac{220730753564739235324763783338830482459451224886660121108284167786458631263772063505570}{749051804313317273487326904260307478523749402364499653183484676600264855737051504390559}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{28}\times C_{58156}$, which has order $13026944$ (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:  \( 4617140.625511786 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-651}) \), \(\Q(\zeta_{9})^+\), 3.3.961.1, 3.3.77841.1, 3.3.77841.2, 6.0.201127054779.4, 6.0.265134567411.1, 6.0.193283099642619.3, 6.0.193283099642619.2, 9.9.471655843734321.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31Data not computed