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Properties

Label 18.0.12538688094...1552.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 13^{15}$
Root discriminant $148.64$
Ramified primes $2, 3, 7, 13$
Class number $6196736$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 14, 3458]$ (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![14832537993, 0, 19776717324, 0, 9888358662, 0, 2456997543, 0, 336051261, 0, 26159679, 0, 1142505, 0, 26208, 0, 273, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 273*x^16 + 26208*x^14 + 1142505*x^12 + 26159679*x^10 + 336051261*x^8 + 2456997543*x^6 + 9888358662*x^4 + 19776717324*x^2 + 14832537993)
 
gp: K = bnfinit(x^18 + 273*x^16 + 26208*x^14 + 1142505*x^12 + 26159679*x^10 + 336051261*x^8 + 2456997543*x^6 + 9888358662*x^4 + 19776717324*x^2 + 14832537993, 1)
 

Normalized defining polynomial

\( x^{18} + 273 x^{16} + 26208 x^{14} + 1142505 x^{12} + 26159679 x^{10} + 336051261 x^{8} + 2456997543 x^{6} + 9888358662 x^{4} + 19776717324 x^{2} + 14832537993 \)

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:  \(-1253868809453171177940640399286136471552=-\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.64$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1092=2^{2}\cdot 3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1092}(1,·)$, $\chi_{1092}(1091,·)$, $\chi_{1092}(647,·)$, $\chi_{1092}(841,·)$, $\chi_{1092}(781,·)$, $\chi_{1092}(335,·)$, $\chi_{1092}(529,·)$, $\chi_{1092}(467,·)$, $\chi_{1092}(719,·)$, $\chi_{1092}(289,·)$, $\chi_{1092}(803,·)$, $\chi_{1092}(625,·)$, $\chi_{1092}(563,·)$, $\chi_{1092}(373,·)$, $\chi_{1092}(311,·)$, $\chi_{1092}(251,·)$, $\chi_{1092}(445,·)$, $\chi_{1092}(757,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{2457} a^{6}$, $\frac{1}{2457} a^{7}$, $\frac{1}{7371} a^{8}$, $\frac{1}{7371} a^{9}$, $\frac{1}{22113} a^{10}$, $\frac{1}{22113} a^{11}$, $\frac{1}{362210940} a^{12} - \frac{5}{265356} a^{10} - \frac{1}{49140} a^{8} + \frac{1}{9828} a^{6} + \frac{13}{540} a^{4} - \frac{1}{12} a^{2} - \frac{11}{60}$, $\frac{1}{362210940} a^{13} - \frac{5}{265356} a^{11} - \frac{1}{49140} a^{9} + \frac{1}{9828} a^{7} + \frac{13}{540} a^{5} - \frac{1}{12} a^{3} - \frac{11}{60} a$, $\frac{1}{1086632820} a^{14} - \frac{1}{331695} a^{10} + \frac{1}{5265} a^{6} - \frac{1}{27} a^{4} + \frac{1}{45} a^{2} - \frac{1}{12}$, $\frac{1}{1086632820} a^{15} - \frac{1}{331695} a^{11} + \frac{1}{5265} a^{7} - \frac{1}{27} a^{5} + \frac{1}{45} a^{3} - \frac{1}{12} a$, $\frac{1}{42378679980} a^{16} + \frac{1}{66339} a^{10} + \frac{1}{110565} a^{8} - \frac{1}{7371} a^{6} + \frac{14}{1755} a^{4} - \frac{1}{36} a^{2} - \frac{2}{15}$, $\frac{1}{42378679980} a^{17} + \frac{1}{66339} a^{11} + \frac{1}{110565} a^{9} - \frac{1}{7371} a^{7} + \frac{14}{1755} a^{5} - \frac{1}{36} a^{3} - \frac{2}{15} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{3458}$, which has order $6196736$ (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:  \( 205236.82590785183 \) (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

\(\Q(\sqrt{-273}) \), 3.3.8281.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 3.3.169.1, 6.0.10783275467328.3, 6.0.63806363712.2, 6.0.10783275467328.4, 6.0.220066846272.2, 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[2]^{3}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$