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Properties

Label 18.0.41709759864...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 5^{9}\cdot 19^{16}$
Root discriminant $265.06$
Ramified primes $2, 3, 5, 19$
Class number $95020254$ (GRH)
Class group $[57, 1667022]$ (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![13658762789, 11377676520, 8852161824, 3360208862, 1018042977, 63660336, -10081135, -2498994, 7371690, 2377242, 240744, -162564, -26645, 5700, 2439, -76, -69, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 69*x^16 - 76*x^15 + 2439*x^14 + 5700*x^13 - 26645*x^12 - 162564*x^11 + 240744*x^10 + 2377242*x^9 + 7371690*x^8 - 2498994*x^7 - 10081135*x^6 + 63660336*x^5 + 1018042977*x^4 + 3360208862*x^3 + 8852161824*x^2 + 11377676520*x + 13658762789)
 
gp: K = bnfinit(x^18 - 69*x^16 - 76*x^15 + 2439*x^14 + 5700*x^13 - 26645*x^12 - 162564*x^11 + 240744*x^10 + 2377242*x^9 + 7371690*x^8 - 2498994*x^7 - 10081135*x^6 + 63660336*x^5 + 1018042977*x^4 + 3360208862*x^3 + 8852161824*x^2 + 11377676520*x + 13658762789, 1)
 

Normalized defining polynomial

\( x^{18} - 69 x^{16} - 76 x^{15} + 2439 x^{14} + 5700 x^{13} - 26645 x^{12} - 162564 x^{11} + 240744 x^{10} + 2377242 x^{9} + 7371690 x^{8} - 2498994 x^{7} - 10081135 x^{6} + 63660336 x^{5} + 1018042977 x^{4} + 3360208862 x^{3} + 8852161824 x^{2} + 11377676520 x + 13658762789 \)

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:  \(-41709759864743767520229789527327232000000000=-\,2^{18}\cdot 3^{24}\cdot 5^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $265.06$
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:  $2, 3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3420=2^{2}\cdot 3^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3420}(1,·)$, $\chi_{3420}(2821,·)$, $\chi_{3420}(3079,·)$, $\chi_{3420}(139,·)$, $\chi_{3420}(2221,·)$, $\chi_{3420}(2899,·)$, $\chi_{3420}(919,·)$, $\chi_{3420}(2581,·)$, $\chi_{3420}(1879,·)$, $\chi_{3420}(2779,·)$, $\chi_{3420}(481,·)$, $\chi_{3420}(859,·)$, $\chi_{3420}(3121,·)$, $\chi_{3420}(3241,·)$, $\chi_{3420}(1261,·)$, $\chi_{3420}(2479,·)$, $\chi_{3420}(1201,·)$, $\chi_{3420}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{49} a^{10} - \frac{2}{49} a^{9} + \frac{3}{49} a^{8} - \frac{1}{49} a^{7} + \frac{3}{49} a^{6} - \frac{9}{49} a^{5} + \frac{12}{49} a^{4} - \frac{16}{49} a^{3} - \frac{19}{49} a^{2} - \frac{3}{7} a$, $\frac{1}{49} a^{11} - \frac{1}{49} a^{9} - \frac{2}{49} a^{8} + \frac{1}{49} a^{7} - \frac{3}{49} a^{6} - \frac{6}{49} a^{5} + \frac{8}{49} a^{4} - \frac{2}{49} a^{3} - \frac{3}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{49} a^{12} + \frac{3}{49} a^{9} - \frac{3}{49} a^{8} + \frac{3}{49} a^{7} - \frac{3}{49} a^{6} - \frac{1}{49} a^{5} + \frac{10}{49} a^{4} + \frac{23}{49} a^{3} - \frac{5}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{343} a^{13} + \frac{3}{343} a^{12} - \frac{1}{343} a^{11} - \frac{3}{343} a^{10} - \frac{16}{343} a^{9} - \frac{22}{343} a^{8} + \frac{18}{343} a^{7} + \frac{10}{343} a^{6} - \frac{136}{343} a^{5} - \frac{104}{343} a^{4} + \frac{85}{343} a^{3} - \frac{129}{343} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{343} a^{14} - \frac{3}{343} a^{12} - \frac{16}{343} a^{9} - \frac{2}{49} a^{8} + \frac{19}{343} a^{7} - \frac{19}{343} a^{6} + \frac{136}{343} a^{5} + \frac{159}{343} a^{4} - \frac{139}{343} a^{3} - \frac{75}{343} a^{2} - \frac{1}{7}$, $\frac{1}{343} a^{15} + \frac{2}{343} a^{12} - \frac{3}{343} a^{11} + \frac{3}{343} a^{10} + \frac{8}{343} a^{9} + \frac{9}{343} a^{8} - \frac{2}{49} a^{7} - \frac{23}{343} a^{6} + \frac{45}{343} a^{5} - \frac{87}{343} a^{4} + \frac{159}{343} a^{3} + \frac{47}{343} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{373737259} a^{16} + \frac{208660}{373737259} a^{15} - \frac{498793}{373737259} a^{14} - \frac{277702}{373737259} a^{13} - \frac{1982728}{373737259} a^{12} + \frac{1002965}{373737259} a^{11} + \frac{3609689}{373737259} a^{10} + \frac{1101864}{373737259} a^{9} - \frac{24979240}{373737259} a^{8} - \frac{2386789}{53391037} a^{7} - \frac{2645242}{53391037} a^{6} + \frac{56684753}{373737259} a^{5} - \frac{129650526}{373737259} a^{4} - \frac{138785959}{373737259} a^{3} + \frac{131294342}{373737259} a^{2} - \frac{3296805}{7627291} a - \frac{792465}{7627291}$, $\frac{1}{669375521285522837450741166095415230850606119791243373} a^{17} + \frac{607474335542510050416372269571894404651922824}{669375521285522837450741166095415230850606119791243373} a^{16} - \frac{111252968328424149141911030618478175619739506309941}{669375521285522837450741166095415230850606119791243373} a^{15} - \frac{102300214180253970856918561746453188280371735897916}{669375521285522837450741166095415230850606119791243373} a^{14} - \frac{947621362235044237200636375887739260969983494965816}{669375521285522837450741166095415230850606119791243373} a^{13} + \frac{5036575947683159979481503820456550240841973074101680}{669375521285522837450741166095415230850606119791243373} a^{12} - \frac{1693207028530393404398567155514510096618530733237752}{669375521285522837450741166095415230850606119791243373} a^{11} + \frac{1898703836821273837553550117712874841913815353709363}{669375521285522837450741166095415230850606119791243373} a^{10} + \frac{36431998536524536011412551338804356937698198316080424}{669375521285522837450741166095415230850606119791243373} a^{9} - \frac{17587303498375025780685373238734392174638388280295721}{669375521285522837450741166095415230850606119791243373} a^{8} + \frac{157670232013915251643711346863969626695630365619842}{13660724924194343621443697267253372058175635097780477} a^{7} - \frac{28132826978523284936590460594824118154916984078673739}{669375521285522837450741166095415230850606119791243373} a^{6} - \frac{217159927113992048203827905786614503009152667307006864}{669375521285522837450741166095415230850606119791243373} a^{5} + \frac{250054822647760904945112534634045219167225933670890718}{669375521285522837450741166095415230850606119791243373} a^{4} + \frac{188221859300307725944770351564975673001356195637431458}{669375521285522837450741166095415230850606119791243373} a^{3} + \frac{49851886445191991785462762752869992782169703775839883}{669375521285522837450741166095415230850606119791243373} a^{2} - \frac{549920797435725093528725948944180805082499001526288}{1951532132027763374491956752464767436882233585397211} a + \frac{2289420187939694533761282882270507668932842179061227}{13660724924194343621443697267253372058175635097780477}$

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

Class group and class number

$C_{57}\times C_{1667022}$, which has order $95020254$ (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:  \( 22027035.20428972 \) (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

\(\Q(\sqrt{-5}) \), 3.3.361.1, 6.0.1042568000.1, 9.9.9025761726072081.2

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 ${\href{/LocalNumberField/7.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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$3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
5Data not computed
19Data not computed