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Properties

Label 18.18.2177118761...9189.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{9}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $48.47$
Ramified primes $3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![421, -5041, 369, 37573, -28872, -71560, 64848, 60622, -55814, -28087, 23838, 7639, -5461, -1198, 670, 97, -41, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 41*x^16 + 97*x^15 + 670*x^14 - 1198*x^13 - 5461*x^12 + 7639*x^11 + 23838*x^10 - 28087*x^9 - 55814*x^8 + 60622*x^7 + 64848*x^6 - 71560*x^5 - 28872*x^4 + 37573*x^3 + 369*x^2 - 5041*x + 421)
 
gp: K = bnfinit(x^18 - 3*x^17 - 41*x^16 + 97*x^15 + 670*x^14 - 1198*x^13 - 5461*x^12 + 7639*x^11 + 23838*x^10 - 28087*x^9 - 55814*x^8 + 60622*x^7 + 64848*x^6 - 71560*x^5 - 28872*x^4 + 37573*x^3 + 369*x^2 - 5041*x + 421, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 41 x^{16} + 97 x^{15} + 670 x^{14} - 1198 x^{13} - 5461 x^{12} + 7639 x^{11} + 23838 x^{10} - 28087 x^{9} - 55814 x^{8} + 60622 x^{7} + 64848 x^{6} - 71560 x^{5} - 28872 x^{4} + 37573 x^{3} + 369 x^{2} - 5041 x + 421 \)

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:  \(2177118761435360147462549499189=3^{9}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.47$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(131,·)$, $\chi_{273}(68,·)$, $\chi_{273}(269,·)$, $\chi_{273}(79,·)$, $\chi_{273}(16,·)$, $\chi_{273}(209,·)$, $\chi_{273}(146,·)$, $\chi_{273}(211,·)$, $\chi_{273}(22,·)$, $\chi_{273}(152,·)$, $\chi_{273}(100,·)$, $\chi_{273}(230,·)$, $\chi_{273}(235,·)$, $\chi_{273}(172,·)$, $\chi_{273}(248,·)$, $\chi_{273}(185,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{9} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{10} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{11} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{2627044618000583750837837127} a^{17} - \frac{4186536628419738999321043}{875681539333527916945945709} a^{16} + \frac{282553482859244117594641700}{2627044618000583750837837127} a^{15} - \frac{268746185037808480211419370}{2627044618000583750837837127} a^{14} + \frac{123814522609707925017519772}{2627044618000583750837837127} a^{13} + \frac{365586515557025719753698836}{2627044618000583750837837127} a^{12} - \frac{1169942412871495406230003601}{2627044618000583750837837127} a^{11} + \frac{335101617541977018341743898}{2627044618000583750837837127} a^{10} + \frac{707181082484897837325403780}{2627044618000583750837837127} a^{9} - \frac{87103556490211211178843814}{2627044618000583750837837127} a^{8} - \frac{842141032622771895082083118}{2627044618000583750837837127} a^{7} + \frac{218289390502062204071133452}{875681539333527916945945709} a^{6} - \frac{234614177699908424574072219}{875681539333527916945945709} a^{5} + \frac{346038046515295866717189848}{875681539333527916945945709} a^{4} - \frac{225000508123513104722073019}{2627044618000583750837837127} a^{3} + \frac{222459503623266132694692263}{875681539333527916945945709} a^{2} + \frac{187867497441971227735867838}{2627044618000583750837837127} a - \frac{445649156592071573902995872}{2627044618000583750837837127}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 1812653375.94 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{21}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 6.6.264503421.1, 6.6.12960667629.1, 6.6.12960667629.2, \(\Q(\zeta_{21})^+\), 9.9.567869252041.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$