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Properties

Label 18.0.50198942259...6347.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 37^{12}$
Root discriminant $57.70$
Ramified primes $3, 37$
Class number $189$ (GRH)
Class group $[3, 3, 21]$ (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![1771561, 0, 0, 1289739, 0, 0, 845791, 0, 0, 70492, 0, 0, 5869, 0, 0, -70, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 70*x^15 + 5869*x^12 + 70492*x^9 + 845791*x^6 + 1289739*x^3 + 1771561)
 
gp: K = bnfinit(x^18 - 70*x^15 + 5869*x^12 + 70492*x^9 + 845791*x^6 + 1289739*x^3 + 1771561, 1)
 

Normalized defining polynomial

\( x^{18} - 70 x^{15} + 5869 x^{12} + 70492 x^{9} + 845791 x^{6} + 1289739 x^{3} + 1771561 \)

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:  \(-50198942259523899975028947826347=-\,3^{27}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.70$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(333=3^{2}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{333}(1,·)$, $\chi_{333}(322,·)$, $\chi_{333}(260,·)$, $\chi_{333}(137,·)$, $\chi_{333}(10,·)$, $\chi_{333}(269,·)$, $\chi_{333}(211,·)$, $\chi_{333}(149,·)$, $\chi_{333}(26,·)$, $\chi_{333}(158,·)$, $\chi_{333}(223,·)$, $\chi_{333}(100,·)$, $\chi_{333}(38,·)$, $\chi_{333}(232,·)$, $\chi_{333}(47,·)$, $\chi_{333}(112,·)$, $\chi_{333}(248,·)$, $\chi_{333}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1861} a^{12} - \frac{203}{1861} a^{9} + \frac{291}{1861} a^{6} - \frac{711}{1861} a^{3} + \frac{576}{1861}$, $\frac{1}{20471} a^{13} - \frac{9508}{20471} a^{10} - \frac{9014}{20471} a^{7} - \frac{711}{20471} a^{4} + \frac{8020}{20471} a$, $\frac{1}{225181} a^{14} - \frac{9508}{225181} a^{11} - \frac{9014}{225181} a^{8} - \frac{62124}{225181} a^{5} + \frac{89904}{225181} a^{2}$, $\frac{1}{12519280014931019} a^{15} - \frac{969617753009}{12519280014931019} a^{12} - \frac{5362818191581735}{12519280014931019} a^{9} + \frac{3586093171955111}{12519280014931019} a^{6} + \frac{2761656850520723}{12519280014931019} a^{3} + \frac{643816988343}{9405920371849}$, $\frac{1}{137712080164241209} a^{16} - \frac{969617753009}{137712080164241209} a^{13} - \frac{30401378221443773}{137712080164241209} a^{10} - \frac{8933186842975908}{137712080164241209} a^{7} + \frac{52838776910244799}{137712080164241209} a^{4} - \frac{18168023755355}{103465124090339} a$, $\frac{1}{1514832881806653299} a^{17} - \frac{969617753009}{1514832881806653299} a^{14} + \frac{658159022599762272}{1514832881806653299} a^{11} - \frac{8933186842975908}{1514832881806653299} a^{8} + \frac{465975017402968426}{1514832881806653299} a^{5} + \frac{499157596696340}{1138116364993729} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{21}$, which has order $189$ (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:  \( \frac{102536530042}{137712080164241209} a^{16} - \frac{7230725308800}{137712080164241209} a^{13} + \frac{606244669104960}{137712080164241209} a^{10} + \frac{6866938080315449}{137712080164241209} a^{7} + \frac{87366891280789440}{137712080164241209} a^{4} + \frac{100093897488960}{103465124090339} a \) (order $18$)
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:  \( 8790607.76401 \) (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

\(\Q(\sqrt{-3}) \), 3.3.110889.2, 3.3.1369.1, 3.3.110889.1, \(\Q(\zeta_{9})^+\), 6.0.36889110963.1, 6.0.50602347.1, 6.0.36889110963.2, \(\Q(\zeta_{9})\), 9.9.1363532208525369.2

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$