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Properties

Label 18.0.21603338316...1875.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $62.57$
Ramified primes $5, 7, 13$
Class number $2808$ (GRH)
Class group $[6, 6, 78]$ (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![3521939, 2753785, 2217367, 597223, 378444, -52128, 74956, 14520, 44804, 14123, 4994, -355, -15, 20, 124, -1, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + x^16 - x^15 + 124*x^14 + 20*x^13 - 15*x^12 - 355*x^11 + 4994*x^10 + 14123*x^9 + 44804*x^8 + 14520*x^7 + 74956*x^6 - 52128*x^5 + 378444*x^4 + 597223*x^3 + 2217367*x^2 + 2753785*x + 3521939)
 
gp: K = bnfinit(x^18 - 3*x^17 + x^16 - x^15 + 124*x^14 + 20*x^13 - 15*x^12 - 355*x^11 + 4994*x^10 + 14123*x^9 + 44804*x^8 + 14520*x^7 + 74956*x^6 - 52128*x^5 + 378444*x^4 + 597223*x^3 + 2217367*x^2 + 2753785*x + 3521939, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + x^{16} - x^{15} + 124 x^{14} + 20 x^{13} - 15 x^{12} - 355 x^{11} + 4994 x^{10} + 14123 x^{9} + 44804 x^{8} + 14520 x^{7} + 74956 x^{6} - 52128 x^{5} + 378444 x^{4} + 597223 x^{3} + 2217367 x^{2} + 2753785 x + 3521939 \)

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:  \(-216033383169661016512360513671875=-\,5^{9}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.57$
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$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(261,·)$, $\chi_{455}(326,·)$, $\chi_{455}(139,·)$, $\chi_{455}(269,·)$, $\chi_{455}(334,·)$, $\chi_{455}(16,·)$, $\chi_{455}(209,·)$, $\chi_{455}(339,·)$, $\chi_{455}(404,·)$, $\chi_{455}(94,·)$, $\chi_{455}(159,·)$, $\chi_{455}(419,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{6}{13} a^{11} + \frac{2}{13} a^{10} + \frac{1}{13} a^{9} + \frac{4}{13} a^{8} - \frac{3}{13} a^{7} - \frac{5}{13} a^{6} + \frac{2}{13} a^{5} + \frac{5}{13} a^{3} + \frac{6}{13} a^{2} + \frac{2}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{13} + \frac{5}{13} a^{11} - \frac{3}{13} a^{9} - \frac{5}{13} a^{8} + \frac{3}{13} a^{7} - \frac{2}{13} a^{6} - \frac{1}{13} a^{5} + \frac{5}{13} a^{4} - \frac{3}{13} a^{3} - \frac{1}{13} a^{2} + \frac{4}{13} a + \frac{4}{13}$, $\frac{1}{169} a^{14} - \frac{5}{169} a^{13} + \frac{5}{169} a^{12} - \frac{25}{169} a^{11} - \frac{81}{169} a^{10} - \frac{16}{169} a^{9} + \frac{67}{169} a^{8} - \frac{82}{169} a^{7} - \frac{43}{169} a^{6} - \frac{81}{169} a^{5} + \frac{76}{169} a^{4} + \frac{1}{169} a^{3} - \frac{69}{169} a^{2} - \frac{16}{169} a - \frac{7}{169}$, $\frac{1}{169} a^{15} + \frac{6}{169} a^{13} - \frac{76}{169} a^{11} - \frac{83}{169} a^{10} + \frac{6}{13} a^{9} - \frac{46}{169} a^{8} - \frac{37}{169} a^{7} - \frac{10}{169} a^{6} - \frac{17}{169} a^{5} + \frac{4}{169} a^{4} + \frac{27}{169} a^{3} - \frac{49}{169} a^{2} + \frac{17}{169} a + \frac{69}{169}$, $\frac{1}{507} a^{16} - \frac{1}{507} a^{15} + \frac{1}{507} a^{14} - \frac{7}{507} a^{13} + \frac{16}{507} a^{12} + \frac{131}{507} a^{11} - \frac{15}{169} a^{10} - \frac{6}{169} a^{9} - \frac{235}{507} a^{8} + \frac{59}{169} a^{7} - \frac{4}{13} a^{6} - \frac{53}{169} a^{5} - \frac{149}{507} a^{4} - \frac{94}{507} a^{3} - \frac{71}{169} a^{2} - \frac{76}{507} a - \frac{229}{507}$, $\frac{1}{6748202930264749029249080514165770241025544757} a^{17} - \frac{2233316172174133583640192784016851574652371}{6748202930264749029249080514165770241025544757} a^{16} + \frac{6160649735237359419148018056822127690198391}{6748202930264749029249080514165770241025544757} a^{15} + \frac{7017023048136673434671170351014356334242605}{6748202930264749029249080514165770241025544757} a^{14} + \frac{8031878243196017998658275303927098974888033}{519092533097288386865313885705059249309657289} a^{13} - \frac{223986862347734409922893245150483142783618488}{6748202930264749029249080514165770241025544757} a^{12} - \frac{3013511902803694264103197806355966268152507553}{6748202930264749029249080514165770241025544757} a^{11} + \frac{628224936295707149641392416175711777066628250}{2249400976754916343083026838055256747008514919} a^{10} - \frac{2279186410004624885766630345425731323662284477}{6748202930264749029249080514165770241025544757} a^{9} - \frac{61670009393752141276422945724989615685597049}{519092533097288386865313885705059249309657289} a^{8} - \frac{262170073499459879343945488718151601549977297}{749800325584972114361008946018418915669504973} a^{7} - \frac{3134889009694333790842469268974765491752380}{2249400976754916343083026838055256747008514919} a^{6} - \frac{218018576459007501882876489396435692723370185}{519092533097288386865313885705059249309657289} a^{5} + \frac{558099425845067882129706653164060407765934573}{6748202930264749029249080514165770241025544757} a^{4} - \frac{1929984801974298100833754254729948571326320}{39930194853637568220408760438850711485358253} a^{3} + \frac{202634797415796277654601915604401448660146366}{519092533097288386865313885705059249309657289} a^{2} - \frac{771248610316899120356178657754830459551275948}{2249400976754916343083026838055256747008514919} a - \frac{3275334092729137421349306502445008585990482986}{6748202930264749029249080514165770241025544757}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{78}$, which has order $2808$ (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:  \( 205236.825908 \) (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

\(\Q(\sqrt{-35}) \), 3.3.8281.1, 3.3.8281.2, 3.3.169.1, \(\Q(\zeta_{7})^+\), 6.0.60003090875.1, 6.0.60003090875.2, 6.0.1224552875.2, 6.0.2100875.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$