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Properties

Label 18.18.1285169410...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{24}\cdot 5^{9}\cdot 13^{12}$
Root discriminant $53.49$
Ramified primes $3, 5, 13$
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![-181, -2403, 13995, 48055, -50556, -150630, 53034, 164055, -30768, -84569, 12312, 22752, -3224, -3231, 483, 226, -36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 36*x^16 + 226*x^15 + 483*x^14 - 3231*x^13 - 3224*x^12 + 22752*x^11 + 12312*x^10 - 84569*x^9 - 30768*x^8 + 164055*x^7 + 53034*x^6 - 150630*x^5 - 50556*x^4 + 48055*x^3 + 13995*x^2 - 2403*x - 181)
 
gp: K = bnfinit(x^18 - 6*x^17 - 36*x^16 + 226*x^15 + 483*x^14 - 3231*x^13 - 3224*x^12 + 22752*x^11 + 12312*x^10 - 84569*x^9 - 30768*x^8 + 164055*x^7 + 53034*x^6 - 150630*x^5 - 50556*x^4 + 48055*x^3 + 13995*x^2 - 2403*x - 181, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 36 x^{16} + 226 x^{15} + 483 x^{14} - 3231 x^{13} - 3224 x^{12} + 22752 x^{11} + 12312 x^{10} - 84569 x^{9} - 30768 x^{8} + 164055 x^{7} + 53034 x^{6} - 150630 x^{5} - 50556 x^{4} + 48055 x^{3} + 13995 x^{2} - 2403 x - 181 \)

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:  \(12851694105541388560018283203125=3^{24}\cdot 5^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.49$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(391,·)$, $\chi_{585}(139,·)$, $\chi_{585}(334,·)$, $\chi_{585}(79,·)$, $\chi_{585}(16,·)$, $\chi_{585}(529,·)$, $\chi_{585}(274,·)$, $\chi_{585}(211,·)$, $\chi_{585}(469,·)$, $\chi_{585}(406,·)$, $\chi_{585}(94,·)$, $\chi_{585}(289,·)$, $\chi_{585}(484,·)$, $\chi_{585}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{38} a^{15} - \frac{7}{38} a^{14} - \frac{3}{19} a^{13} - \frac{2}{19} a^{12} + \frac{15}{38} a^{10} + \frac{3}{19} a^{9} - \frac{13}{38} a^{8} - \frac{5}{38} a^{7} + \frac{8}{19} a^{6} - \frac{9}{19} a^{5} + \frac{5}{19} a^{4} - \frac{2}{19} a^{3} + \frac{9}{38} a^{2} + \frac{7}{38} a + \frac{5}{19}$, $\frac{1}{38} a^{16} + \frac{1}{19} a^{14} - \frac{4}{19} a^{13} - \frac{9}{38} a^{12} - \frac{2}{19} a^{11} - \frac{3}{38} a^{10} - \frac{9}{38} a^{9} + \frac{9}{19} a^{8} + \frac{9}{19} a^{6} + \frac{17}{38} a^{5} + \frac{9}{38} a^{4} - \frac{1}{2} a^{3} + \frac{13}{38} a^{2} - \frac{17}{38} a - \frac{3}{19}$, $\frac{1}{54349782523295134068331411043698} a^{17} - \frac{175981400420957258119009303963}{54349782523295134068331411043698} a^{16} - \frac{225554826796799699882803980887}{54349782523295134068331411043698} a^{15} - \frac{12188630033024458931355774080015}{54349782523295134068331411043698} a^{14} - \frac{890439303937850391631729766710}{27174891261647567034165705521849} a^{13} + \frac{13003594661388920836117626189497}{54349782523295134068331411043698} a^{12} - \frac{11460364425269473091374145106448}{27174891261647567034165705521849} a^{11} + \frac{7746770318311822627575619017332}{27174891261647567034165705521849} a^{10} + \frac{14136162213773646752397247677165}{54349782523295134068331411043698} a^{9} + \frac{6301209156376652043540350987305}{54349782523295134068331411043698} a^{8} - \frac{25193219672576383545696062908333}{54349782523295134068331411043698} a^{7} - \frac{6894507587265000788612061150491}{27174891261647567034165705521849} a^{6} + \frac{3106686200036035827100519350006}{27174891261647567034165705521849} a^{5} - \frac{5309735403730495039448192244761}{54349782523295134068331411043698} a^{4} - \frac{1241978605503711589514831236719}{54349782523295134068331411043698} a^{3} + \frac{2729730743193676201003644728849}{54349782523295134068331411043698} a^{2} + \frac{26797491965928311036736512853845}{54349782523295134068331411043698} a + \frac{4155576346844099109168295749013}{27174891261647567034165705521849}$

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:  \( 3649457632.69 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.13689.2, 3.3.169.1, 6.6.820125.1, 6.6.23423590125.1, 6.6.23423590125.2, 6.6.3570125.1, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$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}$
13Data not computed