↑

Properties

Label 18.18.2106844814...8125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{9}\cdot 5^{9}\cdot 19^{17}$
Root discriminant $62.48$
Ramified primes $3, 5, 19$
Class number $2$ (GRH)
Class group $[2]$ (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![1044469, -6025205, 6025205, 12652555, -12652555, -7892981, 7892981, 2379787, -2379787, -402421, 402421, 40203, -40203, -2357, 2357, 75, -75, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 75*x^16 + 75*x^15 + 2357*x^14 - 2357*x^13 - 40203*x^12 + 40203*x^11 + 402421*x^10 - 402421*x^9 - 2379787*x^8 + 2379787*x^7 + 7892981*x^6 - 7892981*x^5 - 12652555*x^4 + 12652555*x^3 + 6025205*x^2 - 6025205*x + 1044469)
 
gp: K = bnfinit(x^18 - x^17 - 75*x^16 + 75*x^15 + 2357*x^14 - 2357*x^13 - 40203*x^12 + 40203*x^11 + 402421*x^10 - 402421*x^9 - 2379787*x^8 + 2379787*x^7 + 7892981*x^6 - 7892981*x^5 - 12652555*x^4 + 12652555*x^3 + 6025205*x^2 - 6025205*x + 1044469, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 75 x^{16} + 75 x^{15} + 2357 x^{14} - 2357 x^{13} - 40203 x^{12} + 40203 x^{11} + 402421 x^{10} - 402421 x^{9} - 2379787 x^{8} + 2379787 x^{7} + 7892981 x^{6} - 7892981 x^{5} - 12652555 x^{4} + 12652555 x^{3} + 6025205 x^{2} - 6025205 x + 1044469 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(210684481487848166847338548828125=3^{9}\cdot 5^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.48$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(285=3\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{285}(256,·)$, $\chi_{285}(1,·)$, $\chi_{285}(196,·)$, $\chi_{285}(269,·)$, $\chi_{285}(14,·)$, $\chi_{285}(271,·)$, $\chi_{285}(16,·)$, $\chi_{285}(89,·)$, $\chi_{285}(284,·)$, $\chi_{285}(29,·)$, $\chi_{285}(224,·)$, $\chi_{285}(226,·)$, $\chi_{285}(164,·)$, $\chi_{285}(106,·)$, $\chi_{285}(179,·)$, $\chi_{285}(121,·)$, $\chi_{285}(59,·)$, $\chi_{285}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23939} a^{10} - \frac{5178}{23939} a^{9} - \frac{40}{23939} a^{8} - \frac{5104}{23939} a^{7} + \frac{560}{23939} a^{6} - \frac{10569}{23939} a^{5} - \frac{3200}{23939} a^{4} + \frac{7075}{23939} a^{3} + \frac{6400}{23939} a^{2} - \frac{8490}{23939} a - \frac{2048}{23939}$, $\frac{1}{23939} a^{11} - \frac{44}{23939} a^{9} + \frac{3227}{23939} a^{8} + \frac{704}{23939} a^{7} - \frac{7508}{23939} a^{6} - \frac{4928}{23939} a^{5} + \frac{3263}{23939} a^{4} - \frac{9859}{23939} a^{3} - \frac{866}{23939} a^{2} - \frac{11264}{23939} a + \frac{433}{23939}$, $\frac{1}{23939} a^{12} - \frac{9154}{23939} a^{9} - \frac{1056}{23939} a^{8} + \frac{7306}{23939} a^{7} - \frac{4227}{23939} a^{6} - \frac{6932}{23939} a^{5} - \frac{7025}{23939} a^{4} - \frac{773}{23939} a^{3} + \frac{7007}{23939} a^{2} + \frac{9897}{23939} a + \frac{5644}{23939}$, $\frac{1}{23939} a^{13} - \frac{1248}{23939} a^{9} + \frac{231}{23939} a^{8} + \frac{2685}{23939} a^{7} - \frac{3638}{23939} a^{6} + \frac{5787}{23939} a^{5} + \frac{7763}{23939} a^{4} - \frac{7377}{23939} a^{3} - \frac{7175}{23939} a^{2} - \frac{5822}{23939} a - \frac{3155}{23939}$, $\frac{1}{23939} a^{14} + \frac{1617}{23939} a^{9} + \frac{643}{23939} a^{8} - \frac{5656}{23939} a^{7} + \frac{10436}{23939} a^{6} + \frac{8040}{23939} a^{5} - \frac{3164}{23939} a^{4} - \frac{11066}{23939} a^{3} + \frac{9691}{23939} a^{2} + \frac{6302}{23939} a + \frac{5569}{23939}$, $\frac{1}{23939} a^{15} - \frac{5181}{23939} a^{9} + \frac{11146}{23939} a^{8} + \frac{4649}{23939} a^{7} - \frac{11737}{23939} a^{6} - \frac{5537}{23939} a^{5} - \frac{7490}{23939} a^{4} - \frac{11681}{23939} a^{3} - \frac{850}{23939} a^{2} - \frac{7087}{23939} a + \frac{8034}{23939}$, $\frac{1}{23939} a^{16} - \frac{4392}{23939} a^{9} - \frac{11079}{23939} a^{8} - \frac{2966}{23939} a^{7} - \frac{796}{23939} a^{6} + \frac{6953}{23939} a^{5} - \frac{1154}{23939} a^{4} + \frac{4116}{23939} a^{3} - \frac{4202}{23939} a^{2} - \frac{2713}{23939} a - \frac{5711}{23939}$, $\frac{1}{23939} a^{17} - \frac{10805}{23939} a^{9} - \frac{11073}{23939} a^{8} - \frac{10660}{23939} a^{7} + \frac{756}{23939} a^{6} - \frac{2481}{23939} a^{5} + \frac{1909}{23939} a^{4} - \frac{3624}{23939} a^{3} + \frac{1701}{23939} a^{2} + \frac{3171}{23939} a + \frac{6248}{23939}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $17$
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:  \( 9738921681.01 \) (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

\(\Q(\sqrt{285}) \), 3.3.361.1, 6.6.8356834125.1, \(\Q(\zeta_{19})^+\)

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 $18$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
5Data not computed
19Data not computed