↑

Properties

Label 18.0.34087453954...5847.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 11^{9}\cdot 13^{15}$
Root discriminant $121.66$
Ramified primes $3, 11, 13$
Class number $1179360$ (GRH)
Class group $[2, 36, 16380]$ (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![707555512, -268220532, 606840420, -215794195, 239214930, -51988584, 40584179, -5879268, 3866553, -112362, 427959, -105270, 53789, -8670, 3117, -256, 87, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 87*x^16 - 256*x^15 + 3117*x^14 - 8670*x^13 + 53789*x^12 - 105270*x^11 + 427959*x^10 - 112362*x^9 + 3866553*x^8 - 5879268*x^7 + 40584179*x^6 - 51988584*x^5 + 239214930*x^4 - 215794195*x^3 + 606840420*x^2 - 268220532*x + 707555512)
 
gp: K = bnfinit(x^18 - 3*x^17 + 87*x^16 - 256*x^15 + 3117*x^14 - 8670*x^13 + 53789*x^12 - 105270*x^11 + 427959*x^10 - 112362*x^9 + 3866553*x^8 - 5879268*x^7 + 40584179*x^6 - 51988584*x^5 + 239214930*x^4 - 215794195*x^3 + 606840420*x^2 - 268220532*x + 707555512, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 87 x^{16} - 256 x^{15} + 3117 x^{14} - 8670 x^{13} + 53789 x^{12} - 105270 x^{11} + 427959 x^{10} - 112362 x^{9} + 3866553 x^{8} - 5879268 x^{7} + 40584179 x^{6} - 51988584 x^{5} + 239214930 x^{4} - 215794195 x^{3} + 606840420 x^{2} - 268220532 x + 707555512 \)

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:  \(-34087453954147523343197977587258425847=-\,3^{24}\cdot 11^{9}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.66$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1287=3^{2}\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1287}(1,·)$, $\chi_{1287}(133,·)$, $\chi_{1287}(10,·)$, $\chi_{1287}(142,·)$, $\chi_{1287}(529,·)$, $\chi_{1287}(472,·)$, $\chi_{1287}(868,·)$, $\chi_{1287}(859,·)$, $\chi_{1287}(991,·)$, $\chi_{1287}(100,·)$, $\chi_{1287}(1000,·)$, $\chi_{1287}(43,·)$, $\chi_{1287}(430,·)$, $\chi_{1287}(562,·)$, $\chi_{1287}(439,·)$, $\chi_{1287}(901,·)$, $\chi_{1287}(571,·)$, $\chi_{1287}(958,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{1228} a^{14} + \frac{58}{307} a^{13} + \frac{263}{1228} a^{12} - \frac{71}{307} a^{11} + \frac{133}{1228} a^{10} - \frac{41}{614} a^{9} - \frac{17}{1228} a^{8} + \frac{26}{307} a^{7} + \frac{201}{1228} a^{6} + \frac{261}{614} a^{5} - \frac{111}{1228} a^{4} - \frac{122}{307} a^{3} - \frac{97}{1228} a^{2} + \frac{165}{614} a - \frac{20}{307}$, $\frac{1}{126484} a^{15} + \frac{11}{63242} a^{14} - \frac{16529}{126484} a^{13} - \frac{6664}{31621} a^{12} - \frac{15135}{126484} a^{11} - \frac{3626}{31621} a^{10} - \frac{15953}{126484} a^{9} + \frac{7363}{63242} a^{8} + \frac{3535}{126484} a^{7} - \frac{21765}{63242} a^{6} + \frac{47453}{126484} a^{5} + \frac{7087}{31621} a^{4} + \frac{28089}{126484} a^{3} - \frac{4079}{63242} a^{2} + \frac{30087}{63242} a + \frac{1744}{31621}$, $\frac{1}{126484} a^{16} - \frac{9}{63242} a^{14} - \frac{10317}{63242} a^{13} - \frac{9189}{63242} a^{12} - \frac{4483}{31621} a^{11} - \frac{7363}{31621} a^{10} - \frac{8013}{63242} a^{9} + \frac{5767}{31621} a^{8} + \frac{472}{31621} a^{7} - \frac{1458}{31621} a^{6} + \frac{3441}{31621} a^{5} - \frac{7787}{63242} a^{4} - \frac{1277}{63242} a^{3} - \frac{17541}{126484} a^{2} - \frac{4479}{63242} a + \frac{1184}{31621}$, $\frac{1}{596875643430861044286050885344159221301254180385886887353603884} a^{17} + \frac{666669602098984318176811938067350293679181458784750036139}{596875643430861044286050885344159221301254180385886887353603884} a^{16} - \frac{340617189827468930740133581792909507805381204428950800275}{596875643430861044286050885344159221301254180385886887353603884} a^{15} + \frac{160697040281150927015676555521286354055318402501513432465401}{596875643430861044286050885344159221301254180385886887353603884} a^{14} - \frac{97591590363854379665880079876908817299361408857604937373699865}{596875643430861044286050885344159221301254180385886887353603884} a^{13} + \frac{136741827792781853245154387528276190375064926578327258656894685}{596875643430861044286050885344159221301254180385886887353603884} a^{12} - \frac{119255100592800339440967262155095799032295076850227418615624275}{596875643430861044286050885344159221301254180385886887353603884} a^{11} - \frac{132399560023191206417428222205311640855740212304197797102224583}{596875643430861044286050885344159221301254180385886887353603884} a^{10} + \frac{56402684393667774664842576328057778760329707413744176672861277}{596875643430861044286050885344159221301254180385886887353603884} a^{9} + \frac{134575901404030396000375355586394242720434198219995875205076003}{596875643430861044286050885344159221301254180385886887353603884} a^{8} - \frac{133347471800280948982480588457674577952403474971311863717796171}{596875643430861044286050885344159221301254180385886887353603884} a^{7} + \frac{25975709021542414558668778348035979760863565078493071490441975}{596875643430861044286050885344159221301254180385886887353603884} a^{6} + \frac{277239986030750125999987368793627326883494706542442798649399717}{596875643430861044286050885344159221301254180385886887353603884} a^{5} - \frac{289481642559450717994170206143282709546171808486836450316046323}{596875643430861044286050885344159221301254180385886887353603884} a^{4} + \frac{66306945538578131790530509032963981205095770892960477231750979}{149218910857715261071512721336039805325313545096471721838400971} a^{3} + \frac{134699213455775201061336903055724585305534610878990985892508307}{298437821715430522143025442672079610650627090192943443676801942} a^{2} + \frac{81946407115916519484906154958543811598555852049899874183798935}{298437821715430522143025442672079610650627090192943443676801942} a + \frac{801393923135409718837929427141150738249392267026693361150187}{2815451148258778510783258893132826515571953681065504185630207}$

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

Class group and class number

$C_{2}\times C_{36}\times C_{16380}$, which has order $1179360$ (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:  \( 400417.1364448253 \) (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

\(\Q(\sqrt{-143}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.169.1, 3.3.13689.1, 6.0.19185722127.10, 6.0.3242387039463.4, 6.0.494190983.1, 6.0.3242387039463.3, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed