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Properties

Label 18.0.69788144047...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $240.00$
Ramified primes $2, 3, 5, 7$
Class number $16723422$ (GRH)
Class group $[3, 3, 1858158]$ (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![65865831261, -10205696088, 7379910756, 3605893830, 167351805, -281435616, 129779769, 6167826, -15545754, -25382, 1676052, 0, -95193, 0, 3735, 0, -81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 81*x^16 + 3735*x^14 - 95193*x^12 + 1676052*x^10 - 25382*x^9 - 15545754*x^8 + 6167826*x^7 + 129779769*x^6 - 281435616*x^5 + 167351805*x^4 + 3605893830*x^3 + 7379910756*x^2 - 10205696088*x + 65865831261)
 
gp: K = bnfinit(x^18 - 81*x^16 + 3735*x^14 - 95193*x^12 + 1676052*x^10 - 25382*x^9 - 15545754*x^8 + 6167826*x^7 + 129779769*x^6 - 281435616*x^5 + 167351805*x^4 + 3605893830*x^3 + 7379910756*x^2 - 10205696088*x + 65865831261, 1)
 

Normalized defining polynomial

\( x^{18} - 81 x^{16} + 3735 x^{14} - 95193 x^{12} + 1676052 x^{10} - 25382 x^{9} - 15545754 x^{8} + 6167826 x^{7} + 129779769 x^{6} - 281435616 x^{5} + 167351805 x^{4} + 3605893830 x^{3} + 7379910756 x^{2} - 10205696088 x + 65865831261 \)

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:  \(-6978814404767355448133435215913472000000000=-\,2^{18}\cdot 3^{44}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $240.00$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3780=2^{2}\cdot 3^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{3780}(1,·)$, $\chi_{3780}(961,·)$, $\chi_{3780}(3721,·)$, $\chi_{3780}(1639,·)$, $\chi_{3780}(2221,·)$, $\chi_{3780}(2899,·)$, $\chi_{3780}(2839,·)$, $\chi_{3780}(2521,·)$, $\chi_{3780}(79,·)$, $\chi_{3780}(2461,·)$, $\chi_{3780}(3481,·)$, $\chi_{3780}(1339,·)$, $\chi_{3780}(2599,·)$, $\chi_{3780}(1579,·)$, $\chi_{3780}(1261,·)$, $\chi_{3780}(1201,·)$, $\chi_{3780}(379,·)$, $\chi_{3780}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{2}{7} a^{2} - \frac{1}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{2}{7} a^{3} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{3}{7} a^{4} + \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{7} a^{10} - \frac{3}{7} a^{4} + \frac{2}{7} a^{2} - \frac{3}{7}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{5} + \frac{2}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{49} a^{12} + \frac{2}{49} a^{10} - \frac{3}{49} a^{8} + \frac{1}{49} a^{6} + \frac{9}{49} a^{4} - \frac{10}{49} a^{2} - \frac{6}{49}$, $\frac{1}{49} a^{13} + \frac{2}{49} a^{11} - \frac{3}{49} a^{9} + \frac{1}{49} a^{7} + \frac{9}{49} a^{5} - \frac{10}{49} a^{3} - \frac{6}{49} a$, $\frac{1}{49} a^{14} + \frac{2}{7} a^{4} - \frac{2}{7} a^{2} - \frac{9}{49}$, $\frac{1}{49} a^{15} + \frac{2}{7} a^{5} - \frac{2}{7} a^{3} - \frac{9}{49} a$, $\frac{1}{49} a^{16} + \frac{3}{7} a^{4} + \frac{19}{49} a^{2} + \frac{2}{7}$, $\frac{1}{3826258349043559089884512402928375234027501725362009530994241} a^{17} - \frac{21639734654688427317921059405598810232117807240147214821510}{3826258349043559089884512402928375234027501725362009530994241} a^{16} - \frac{1893215141131343961438185463825283732339824999767854325033}{546608335577651298554930343275482176289643103623144218713463} a^{15} - \frac{21818736008796745615954728001552911737663391054621332948823}{3826258349043559089884512402928375234027501725362009530994241} a^{14} - \frac{16332406375166316576259955922316556183384444621296344719363}{3826258349043559089884512402928375234027501725362009530994241} a^{13} - \frac{2584402833647955899931876794960779708861270858441080252967}{3826258349043559089884512402928375234027501725362009530994241} a^{12} + \frac{171502361015111935055941125382936378198206066370523573189037}{3826258349043559089884512402928375234027501725362009530994241} a^{11} - \frac{21401979460514194558442276002827550642561690673484416118922}{3826258349043559089884512402928375234027501725362009530994241} a^{10} - \frac{257633957726817204408228178047780597532332407083000795400004}{3826258349043559089884512402928375234027501725362009530994241} a^{9} - \frac{113377784675601638974863527422383585768508083863312138945698}{3826258349043559089884512402928375234027501725362009530994241} a^{8} + \frac{186119678331719643704433363601731084448268964178548092949521}{3826258349043559089884512402928375234027501725362009530994241} a^{7} + \frac{260860763936397366889632704534704210223956990041334549279640}{3826258349043559089884512402928375234027501725362009530994241} a^{6} - \frac{1738804484848360123054516676995217963253329167694052935217817}{3826258349043559089884512402928375234027501725362009530994241} a^{5} + \frac{375893841952659562300484483342839772873138623222554870396937}{3826258349043559089884512402928375234027501725362009530994241} a^{4} + \frac{258930770691894833919067505598636700137282695115376523758554}{546608335577651298554930343275482176289643103623144218713463} a^{3} - \frac{168774702159914303005312084755897408481710058387967948330177}{546608335577651298554930343275482176289643103623144218713463} a^{2} - \frac{551283946709850283786566656074208319321041997934596981306294}{3826258349043559089884512402928375234027501725362009530994241} a - \frac{9812978546406888167781601662718097132789454257644071885358}{78086905082521614079275763325068882327091871946163459816209}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{1858158}$, which has order $16723422$ (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:  \( 4392158.291236831 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{9})^+\), 6.0.52488000.1, 9.9.3691950281939241.1

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 R R R R $18$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
2Data not computed
$3$3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
5Data not computed
$7$7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.2$x^{9} - 49 x^{3} + 686$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$