↑

Properties

Label 18.0.33875468057...4256.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 11^{9}\cdot 19^{17}$
Root discriminant $107.01$
Ramified primes $2, 11, 19$
Class number $1209920$ (GRH)
Class group $[2, 2, 4, 75620]$ (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![44801006129, 0, 61092281085, 0, 24436912434, 0, 4443074988, 0, 437575567, 0, 25314289, 0, 885115, 0, 18392, 0, 209, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 209*x^16 + 18392*x^14 + 885115*x^12 + 25314289*x^10 + 437575567*x^8 + 4443074988*x^6 + 24436912434*x^4 + 61092281085*x^2 + 44801006129)
 
gp: K = bnfinit(x^18 + 209*x^16 + 18392*x^14 + 885115*x^12 + 25314289*x^10 + 437575567*x^8 + 4443074988*x^6 + 24436912434*x^4 + 61092281085*x^2 + 44801006129, 1)
 

Normalized defining polynomial

\( x^{18} + 209 x^{16} + 18392 x^{14} + 885115 x^{12} + 25314289 x^{10} + 437575567 x^{8} + 4443074988 x^{6} + 24436912434 x^{4} + 61092281085 x^{2} + 44801006129 \)

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:  \(-3387546805757652419012622474765664256=-\,2^{18}\cdot 11^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.01$
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:  $2, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(836=2^{2}\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{836}(1,·)$, $\chi_{836}(835,·)$, $\chi_{836}(395,·)$, $\chi_{836}(397,·)$, $\chi_{836}(527,·)$, $\chi_{836}(529,·)$, $\chi_{836}(659,·)$, $\chi_{836}(791,·)$, $\chi_{836}(219,·)$, $\chi_{836}(353,·)$, $\chi_{836}(483,·)$, $\chi_{836}(617,·)$, $\chi_{836}(45,·)$, $\chi_{836}(177,·)$, $\chi_{836}(307,·)$, $\chi_{836}(309,·)$, $\chi_{836}(439,·)$, $\chi_{836}(441,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{121} a^{4}$, $\frac{1}{121} a^{5}$, $\frac{1}{1331} a^{6}$, $\frac{1}{1331} a^{7}$, $\frac{1}{14641} a^{8}$, $\frac{1}{14641} a^{9}$, $\frac{1}{161051} a^{10}$, $\frac{1}{161051} a^{11}$, $\frac{1}{1771561} a^{12}$, $\frac{1}{1771561} a^{13}$, $\frac{1}{19487171} a^{14}$, $\frac{1}{19487171} a^{15}$, $\frac{1}{214358881} a^{16}$, $\frac{1}{214358881} a^{17}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{75620}$, which has order $1209920$ (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:  \( 22305.895079162343 \) (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

\(\Q(\sqrt{-209}) \), 3.3.361.1, 6.0.210924017216.2, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed