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Properties

Label 18.0.85310662079...9259.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 59^{9}$
Root discriminant $112.65$
Ramified primes $3, 59$
Class number $4140423$ (GRH)
Class group $[4140423]$ (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![111738707823, -47273111433, 56678803083, -19961087904, 12957858630, -3847455189, 1756678785, -441398538, 155739987, -32934506, 9357696, -1636470, 380112, -52920, 10008, -1020, 153, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 153*x^16 - 1020*x^15 + 10008*x^14 - 52920*x^13 + 380112*x^12 - 1636470*x^11 + 9357696*x^10 - 32934506*x^9 + 155739987*x^8 - 441398538*x^7 + 1756678785*x^6 - 3847455189*x^5 + 12957858630*x^4 - 19961087904*x^3 + 56678803083*x^2 - 47273111433*x + 111738707823)
 
gp: K = bnfinit(x^18 - 9*x^17 + 153*x^16 - 1020*x^15 + 10008*x^14 - 52920*x^13 + 380112*x^12 - 1636470*x^11 + 9357696*x^10 - 32934506*x^9 + 155739987*x^8 - 441398538*x^7 + 1756678785*x^6 - 3847455189*x^5 + 12957858630*x^4 - 19961087904*x^3 + 56678803083*x^2 - 47273111433*x + 111738707823, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 153 x^{16} - 1020 x^{15} + 10008 x^{14} - 52920 x^{13} + 380112 x^{12} - 1636470 x^{11} + 9357696 x^{10} - 32934506 x^{9} + 155739987 x^{8} - 441398538 x^{7} + 1756678785 x^{6} - 3847455189 x^{5} + 12957858630 x^{4} - 19961087904 x^{3} + 56678803083 x^{2} - 47273111433 x + 111738707823 \)

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:  \(-8531066207949676048491159248909849259=-\,3^{44}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.65$
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:  $3, 59$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1593=3^{3}\cdot 59\)
Dirichlet character group:    $\lbrace$$\chi_{1593}(1,·)$, $\chi_{1593}(1474,·)$, $\chi_{1593}(709,·)$, $\chi_{1593}(1417,·)$, $\chi_{1593}(589,·)$, $\chi_{1593}(1297,·)$, $\chi_{1593}(532,·)$, $\chi_{1593}(1240,·)$, $\chi_{1593}(412,·)$, $\chi_{1593}(1120,·)$, $\chi_{1593}(355,·)$, $\chi_{1593}(1063,·)$, $\chi_{1593}(235,·)$, $\chi_{1593}(943,·)$, $\chi_{1593}(178,·)$, $\chi_{1593}(886,·)$, $\chi_{1593}(58,·)$, $\chi_{1593}(766,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{72793838858411957827636788005895881920980254945155555755331008097} a^{17} - \frac{33153666407187712793946527426894516515073682562230361600361870552}{72793838858411957827636788005895881920980254945155555755331008097} a^{16} - \frac{12986995166326504504301305504471219127494268729279444361231264766}{72793838858411957827636788005895881920980254945155555755331008097} a^{15} - \frac{20551975574648852127787061207064214754548753357017604748526843994}{72793838858411957827636788005895881920980254945155555755331008097} a^{14} + \frac{18361543181276343700979771944048282911399831949966762178033253634}{72793838858411957827636788005895881920980254945155555755331008097} a^{13} + \frac{31018267645696258349707667115229261128914711509953263365588355779}{72793838858411957827636788005895881920980254945155555755331008097} a^{12} - \frac{160163775336223142298386382690992102896064136122499711871636507}{72793838858411957827636788005895881920980254945155555755331008097} a^{11} + \frac{5605423820588706499199556571131411387279406871013074851641324221}{72793838858411957827636788005895881920980254945155555755331008097} a^{10} + \frac{22158990491105760063827765409220715728156186731855439664319052692}{72793838858411957827636788005895881920980254945155555755331008097} a^{9} + \frac{20990663755681641327581335526160681008396632563775633380193967248}{72793838858411957827636788005895881920980254945155555755331008097} a^{8} + \frac{26900858387646247640178321829260025053639820325879593517969892412}{72793838858411957827636788005895881920980254945155555755331008097} a^{7} - \frac{15975729878741841856167932626587784771471852356516448092192654076}{72793838858411957827636788005895881920980254945155555755331008097} a^{6} + \frac{35498053809403255248111790932469113174351509944789325548884629359}{72793838858411957827636788005895881920980254945155555755331008097} a^{5} + \frac{7420366810390473432687995296846341724425778630049347969225241614}{72793838858411957827636788005895881920980254945155555755331008097} a^{4} + \frac{16322079467558656332712644261917571722284731422094948177454115984}{72793838858411957827636788005895881920980254945155555755331008097} a^{3} + \frac{29872140898889054412664422064327004385508571538167151186331167094}{72793838858411957827636788005895881920980254945155555755331008097} a^{2} - \frac{3934414074570091889458760671841035493187612098441735426343424696}{72793838858411957827636788005895881920980254945155555755331008097} a + \frac{11014467074120477734827776481760159739849549153095327411145136}{680316251013195867548007364541083008607292102291173418274121571}$

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

Class group and class number

$C_{4140423}$, which has order $4140423$ (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

\(\Q(\sqrt{-59}) \), \(\Q(\zeta_{9})^+\), 6.0.1347491619.1, \(\Q(\zeta_{27})^+\)

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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ R

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]$
59Data not computed