↑

Properties

Label 18.0.20296288612...9728.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{45}\cdot 13^{15}$
Root discriminant $373.80$
Ramified primes $2, 3, 13$
Class number $424777908$ (GRH)
Class group $[2, 212388954]$ (GRH)
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3422001643008, 0, 16914992230656, 0, 4337177495040, 0, 428157265536, 0, 21172612032, 0, 588128112, 0, 9596496, 0, 91260, 0, 468, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 468*x^16 + 91260*x^14 + 9596496*x^12 + 588128112*x^10 + 21172612032*x^8 + 428157265536*x^6 + 4337177495040*x^4 + 16914992230656*x^2 + 3422001643008)
 
gp: K = bnfinit(x^18 + 468*x^16 + 91260*x^14 + 9596496*x^12 + 588128112*x^10 + 21172612032*x^8 + 428157265536*x^6 + 4337177495040*x^4 + 16914992230656*x^2 + 3422001643008, 1)
 

Normalized defining polynomial

\( x^{18} + 468 x^{16} + 91260 x^{14} + 9596496 x^{12} + 588128112 x^{10} + 21172612032 x^{8} + 428157265536 x^{6} + 4337177495040 x^{4} + 16914992230656 x^{2} + 3422001643008 \)

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:  \(-20296288612744454761949989145129188927343689728=-\,2^{27}\cdot 3^{45}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $373.80$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2808=2^{3}\cdot 3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2808}(1,·)$, $\chi_{2808}(2785,·)$, $\chi_{2808}(2573,·)$, $\chi_{2808}(913,·)$, $\chi_{2808}(725,·)$, $\chi_{2808}(1849,·)$, $\chi_{2808}(1661,·)$, $\chi_{2808}(2597,·)$, $\chi_{2808}(2401,·)$, $\chi_{2808}(1637,·)$, $\chi_{2808}(529,·)$, $\chi_{2808}(937,·)$, $\chi_{2808}(173,·)$, $\chi_{2808}(701,·)$, $\chi_{2808}(1465,·)$, $\chi_{2808}(2045,·)$, $\chi_{2808}(1873,·)$, $\chi_{2808}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{104} a^{6}$, $\frac{1}{104} a^{7}$, $\frac{1}{208} a^{8}$, $\frac{1}{209456} a^{9} + \frac{9}{8056} a^{7} + \frac{351}{4028} a^{5} + \frac{35}{2014} a^{3} - \frac{367}{1007} a$, $\frac{1}{1689472096} a^{10} - \frac{1645321}{844736048} a^{8} - \frac{549287}{422368024} a^{6} - \frac{1379555}{16244924} a^{4} + \frac{1864597}{8122462} a^{2} - \frac{1056}{4033}$, $\frac{1}{1689472096} a^{11} + \frac{11}{64979696} a^{9} + \frac{136518}{52796003} a^{7} - \frac{516493}{16244924} a^{5} - \frac{731314}{4061231} a^{3} + \frac{174739}{4061231} a$, $\frac{1}{43926274496} a^{12} - \frac{561601}{844736048} a^{8} + \frac{1464433}{422368024} a^{6} + \frac{1941695}{16244924} a^{4} + \frac{1058639}{8122462} a^{2} - \frac{483}{4033}$, $\frac{1}{43926274496} a^{13} - \frac{39}{32489848} a^{9} - \frac{1987815}{422368024} a^{7} - \frac{292587}{16244924} a^{5} + \frac{373029}{8122462} a^{3} + \frac{900971}{4061231} a$, $\frac{1}{87852548992} a^{14} - \frac{147171}{105592006} a^{8} - \frac{17409}{22229896} a^{6} - \frac{75383}{854996} a^{4} - \frac{931317}{8122462} a^{2} + \frac{1994}{4033}$, $\frac{1}{87852548992} a^{15} + \frac{67}{211184012} a^{9} - \frac{48401}{32489848} a^{7} + \frac{433407}{4061231} a^{5} - \frac{326367}{8122462} a^{3} + \frac{306032}{4061231} a$, $\frac{1}{175705097984} a^{16} + \frac{1703867}{844736048} a^{8} - \frac{412811}{211184012} a^{6} - \frac{44412}{4061231} a^{4} + \frac{125449}{8122462} a^{2} + \frac{698}{4033}$, $\frac{1}{175705097984} a^{17} + \frac{1941}{844736048} a^{9} - \frac{950645}{422368024} a^{7} - \frac{552717}{16244924} a^{5} + \frac{738252}{4061231} a^{3} - \frac{119846}{4061231} a$

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

Class group and class number

$C_{2}\times C_{212388954}$, which has order $424777908$ (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:  \( 54961806.57802202 \) (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

\(\Q(\sqrt{-78}) \), \(\Q(\zeta_{9})^+\), 6.0.22140698112.9, 9.9.151470380950257681.2

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 $18$ $18$ $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
3Data not computed
13Data not computed