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Properties

Label 18.18.6298349363...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $45.24$
Ramified primes $5, 7, 13$
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![-1, 855, 5802, 3653, -33166, -27278, 61756, 47600, -51161, -32862, 22609, 10595, -5490, -1665, 709, 119, -44, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 44*x^16 + 119*x^15 + 709*x^14 - 1665*x^13 - 5490*x^12 + 10595*x^11 + 22609*x^10 - 32862*x^9 - 51161*x^8 + 47600*x^7 + 61756*x^6 - 27278*x^5 - 33166*x^4 + 3653*x^3 + 5802*x^2 + 855*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 44*x^16 + 119*x^15 + 709*x^14 - 1665*x^13 - 5490*x^12 + 10595*x^11 + 22609*x^10 - 32862*x^9 - 51161*x^8 + 47600*x^7 + 61756*x^6 - 27278*x^5 - 33166*x^4 + 3653*x^3 + 5802*x^2 + 855*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 44 x^{16} + 119 x^{15} + 709 x^{14} - 1665 x^{13} - 5490 x^{12} + 10595 x^{11} + 22609 x^{10} - 32862 x^{9} - 51161 x^{8} + 47600 x^{7} + 61756 x^{6} - 27278 x^{5} - 33166 x^{4} + 3653 x^{3} + 5802 x^{2} + 855 x - 1 \)

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:  \(629834936354696841143908203125=5^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.24$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(261,·)$, $\chi_{455}(326,·)$, $\chi_{455}(9,·)$, $\chi_{455}(74,·)$, $\chi_{455}(204,·)$, $\chi_{455}(79,·)$, $\chi_{455}(16,·)$, $\chi_{455}(81,·)$, $\chi_{455}(274,·)$, $\chi_{455}(211,·)$, $\chi_{455}(29,·)$, $\chi_{455}(289,·)$, $\chi_{455}(354,·)$, $\chi_{455}(144,·)$, $\chi_{455}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7221800709783303356449094441} a^{17} + \frac{1493932516236388646398169725}{7221800709783303356449094441} a^{16} - \frac{1001630244059772906127245623}{7221800709783303356449094441} a^{15} - \frac{1809033464762245459950923790}{7221800709783303356449094441} a^{14} + \frac{1514924199254007379548817043}{7221800709783303356449094441} a^{13} - \frac{2304724186199734489774064102}{7221800709783303356449094441} a^{12} + \frac{152093471486647529537120380}{7221800709783303356449094441} a^{11} - \frac{3439271246226345247363439286}{7221800709783303356449094441} a^{10} - \frac{3213311786542134685606189121}{7221800709783303356449094441} a^{9} - \frac{122939315278969138355133952}{7221800709783303356449094441} a^{8} + \frac{2250025051915986381978577529}{7221800709783303356449094441} a^{7} - \frac{2644726428249349914946644720}{7221800709783303356449094441} a^{6} - \frac{2687523150722624377785371137}{7221800709783303356449094441} a^{5} - \frac{598658815763909289161176908}{7221800709783303356449094441} a^{4} + \frac{2546277371750837404276083792}{7221800709783303356449094441} a^{3} + \frac{1713506642614156722627031902}{7221800709783303356449094441} a^{2} - \frac{2718685002378467476777260737}{7221800709783303356449094441} a - \frac{3181589512211668260174480904}{7221800709783303356449094441}$

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:  \( 774354039.422 \) (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

\(\Q(\sqrt{5}) \), 3.3.8281.1, 3.3.8281.2, 3.3.169.1, \(\Q(\zeta_{7})^+\), 6.6.8571870125.2, 6.6.8571870125.1, 6.6.3570125.1, 6.6.300125.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$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}$