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Properties

Label 18.0.45150269416...8311.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 71^{9}$
Root discriminant $123.57$
Ramified primes $3, 71$
Class number $6813324$ (GRH)
Class group $[2, 3406662]$ (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![490347077061, -182157522921, 213718242393, -66458281248, 41823565245, -10993834143, 4833747609, -1075648194, 363852909, -68032562, 18488502, -2848122, 632832, -77112, 14004, -1236, 180, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 180*x^16 - 1236*x^15 + 14004*x^14 - 77112*x^13 + 632832*x^12 - 2848122*x^11 + 18488502*x^10 - 68032562*x^9 + 363852909*x^8 - 1075648194*x^7 + 4833747609*x^6 - 10993834143*x^5 + 41823565245*x^4 - 66458281248*x^3 + 213718242393*x^2 - 182157522921*x + 490347077061)
 
gp: K = bnfinit(x^18 - 9*x^17 + 180*x^16 - 1236*x^15 + 14004*x^14 - 77112*x^13 + 632832*x^12 - 2848122*x^11 + 18488502*x^10 - 68032562*x^9 + 363852909*x^8 - 1075648194*x^7 + 4833747609*x^6 - 10993834143*x^5 + 41823565245*x^4 - 66458281248*x^3 + 213718242393*x^2 - 182157522921*x + 490347077061, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 180 x^{16} - 1236 x^{15} + 14004 x^{14} - 77112 x^{13} + 632832 x^{12} - 2848122 x^{11} + 18488502 x^{10} - 68032562 x^{9} + 363852909 x^{8} - 1075648194 x^{7} + 4833747609 x^{6} - 10993834143 x^{5} + 41823565245 x^{4} - 66458281248 x^{3} + 213718242393 x^{2} - 182157522921 x + 490347077061 \)

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:  \(-45150269416273000041555802844069788311=-\,3^{44}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.57$
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:  $3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1917=3^{3}\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{1917}(640,·)$, $\chi_{1917}(1,·)$, $\chi_{1917}(1348,·)$, $\chi_{1917}(709,·)$, $\chi_{1917}(70,·)$, $\chi_{1917}(1492,·)$, $\chi_{1917}(853,·)$, $\chi_{1917}(214,·)$, $\chi_{1917}(1561,·)$, $\chi_{1917}(922,·)$, $\chi_{1917}(283,·)$, $\chi_{1917}(1705,·)$, $\chi_{1917}(1066,·)$, $\chi_{1917}(427,·)$, $\chi_{1917}(1774,·)$, $\chi_{1917}(1135,·)$, $\chi_{1917}(496,·)$, $\chi_{1917}(1279,·)$$\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}{37951806805409703891284842732184076574331971339612482132245148465889} a^{17} + \frac{12580888554000941482543658822326412636863586684433088315579204812954}{37951806805409703891284842732184076574331971339612482132245148465889} a^{16} + \frac{2692364109997352954765394806772817909932545518623982026197480843557}{37951806805409703891284842732184076574331971339612482132245148465889} a^{15} + \frac{6277932255954264492195147943588548075827788119083710512796696989459}{37951806805409703891284842732184076574331971339612482132245148465889} a^{14} - \frac{8745031453165538587216383839036350169013316277902171155735887880804}{37951806805409703891284842732184076574331971339612482132245148465889} a^{13} + \frac{11796088110054283017314836323144496482378995588613370514595730291026}{37951806805409703891284842732184076574331971339612482132245148465889} a^{12} + \frac{3892046123180999165916122645068836083007703618895696523158419986749}{37951806805409703891284842732184076574331971339612482132245148465889} a^{11} + \frac{15233102250962661734031490118828110029948085274153811188900444271539}{37951806805409703891284842732184076574331971339612482132245148465889} a^{10} - \frac{6642724537830176587479724253711906626535770595677794677732539173628}{37951806805409703891284842732184076574331971339612482132245148465889} a^{9} - \frac{2637217514023243076340085038642613427266732155718646140483713059617}{37951806805409703891284842732184076574331971339612482132245148465889} a^{8} - \frac{5196702757921046349198238620539241516377942167358649478280652065132}{37951806805409703891284842732184076574331971339612482132245148465889} a^{7} + \frac{5041128069481872343061058690441154441931629948808694185544596287917}{37951806805409703891284842732184076574331971339612482132245148465889} a^{6} + \frac{10260553315485176239268893183312557832773169255872233636467537491879}{37951806805409703891284842732184076574331971339612482132245148465889} a^{5} + \frac{13206821616168408935785588979235295824755042291179480936413012219929}{37951806805409703891284842732184076574331971339612482132245148465889} a^{4} + \frac{13234570176854743688512844548466501583183382121558713338127761995009}{37951806805409703891284842732184076574331971339612482132245148465889} a^{3} - \frac{47698727588621611991343315603682183901818212600743977238695518884}{37951806805409703891284842732184076574331971339612482132245148465889} a^{2} - \frac{16817666218393938219820320081315666235185150070282842914742583252137}{37951806805409703891284842732184076574331971339612482132245148465889} a + \frac{8207626157262482888129225652563461569807878800402865069886413936300}{37951806805409703891284842732184076574331971339612482132245148465889}$

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

Class group and class number

$C_{2}\times C_{3406662}$, which has order $6813324$ (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

\(\Q(\sqrt{-71}) \), \(\Q(\zeta_{9})^+\), 6.0.2348254071.3, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ $18$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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
$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]$
$71$71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$