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Properties

Label 18.0.16386765794...1504.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 19^{12}$
Root discriminant $61.62$
Ramified primes $2, 3, 19$
Class number $592$ (GRH)
Class group $[4, 148]$ (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![1369, 0, 22743, 0, 107217, 0, 211985, 0, 189069, 0, 72789, 0, 13428, 0, 1254, 0, 57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 57*x^16 + 1254*x^14 + 13428*x^12 + 72789*x^10 + 189069*x^8 + 211985*x^6 + 107217*x^4 + 22743*x^2 + 1369)
 
gp: K = bnfinit(x^18 + 57*x^16 + 1254*x^14 + 13428*x^12 + 72789*x^10 + 189069*x^8 + 211985*x^6 + 107217*x^4 + 22743*x^2 + 1369, 1)
 

Normalized defining polynomial

\( x^{18} + 57 x^{16} + 1254 x^{14} + 13428 x^{12} + 72789 x^{10} + 189069 x^{8} + 211985 x^{6} + 107217 x^{4} + 22743 x^{2} + 1369 \)

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:  \(-163867657942686205372561998741504=-\,2^{18}\cdot 3^{24}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.62$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(684=2^{2}\cdot 3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{684}(1,·)$, $\chi_{684}(619,·)$, $\chi_{684}(391,·)$, $\chi_{684}(457,·)$, $\chi_{684}(463,·)$, $\chi_{684}(277,·)$, $\chi_{684}(343,·)$, $\chi_{684}(7,·)$, $\chi_{684}(349,·)$, $\chi_{684}(163,·)$, $\chi_{684}(229,·)$, $\chi_{684}(235,·)$, $\chi_{684}(49,·)$, $\chi_{684}(115,·)$, $\chi_{684}(577,·)$, $\chi_{684}(121,·)$, $\chi_{684}(505,·)$, $\chi_{684}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{20} a^{10} - \frac{1}{10} a^{8} - \frac{2}{5} a^{6} + \frac{7}{20} a^{4} - \frac{2}{5} a^{2} + \frac{9}{20}$, $\frac{1}{20} a^{13} - \frac{1}{20} a^{11} - \frac{1}{10} a^{9} - \frac{2}{5} a^{7} + \frac{7}{20} a^{5} - \frac{2}{5} a^{3} + \frac{9}{20} a$, $\frac{1}{30620} a^{14} + \frac{727}{30620} a^{12} + \frac{165}{3062} a^{10} - \frac{977}{15310} a^{8} + \frac{4863}{30620} a^{6} + \frac{5619}{15310} a^{4} - \frac{1413}{6124} a^{2} - \frac{3799}{15310}$, $\frac{1}{1132940} a^{15} + \frac{16037}{1132940} a^{13} + \frac{3227}{113294} a^{11} - \frac{19626}{283235} a^{9} + \frac{234513}{1132940} a^{7} + \frac{29602}{283235} a^{5} + \frac{35331}{226588} a^{3} + \frac{57441}{566470} a$, $\frac{1}{9303703280} a^{16} + \frac{18541}{2325925820} a^{14} - \frac{114326001}{4651851640} a^{12} + \frac{361400021}{4651851640} a^{10} + \frac{1051090087}{9303703280} a^{8} - \frac{1391501139}{4651851640} a^{6} - \frac{1368096961}{9303703280} a^{4} - \frac{106035011}{930370328} a^{2} + \frac{16705197}{50290288}$, $\frac{1}{9303703280} a^{17} + \frac{16}{581481455} a^{15} - \frac{9179553}{4651851640} a^{13} - \frac{73184079}{930370328} a^{11} - \frac{117959781}{1860740656} a^{9} + \frac{1804527141}{4651851640} a^{7} - \frac{2676432801}{9303703280} a^{5} - \frac{2173839703}{4651851640} a^{3} - \frac{2143941263}{9303703280} a$

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

Class group and class number

$C_{4}\times C_{148}$, which has order $592$ (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:  \( -\frac{45507}{25145144} a^{17} - \frac{23778505}{232592582} a^{15} - \frac{1031396553}{465185164} a^{13} - \frac{10776392397}{465185164} a^{11} - \frac{111480678133}{930370328} a^{9} - \frac{130313920641}{465185164} a^{7} - \frac{219528264969}{930370328} a^{5} - \frac{29047893071}{465185164} a^{3} - \frac{781451079}{930370328} a \) (order $4$)
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:  \( 1472619.0824 \) (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

\(\Q(\sqrt{-1}) \), 3.3.29241.1, 3.3.29241.2, \(\Q(\zeta_{9})^+\), 3.3.361.1, 6.0.54722309184.5, 6.0.54722309184.1, 6.0.419904.1, 6.0.8340544.1, 9.9.25002110044521.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$19$19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$