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Properties

Label 18.0.20015044241...000.16
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $70.81$
Ramified primes $2, 3, 5, 7$
Class number $23328$ (GRH)
Class group $[6, 6, 6, 6, 18]$ (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![82566296, 9405912, 65929308, 6369416, 23403354, 2018526, 4862043, 378498, 674097, 43582, 69687, 2670, 5655, 24, 333, -8, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 8*x^15 + 333*x^14 + 24*x^13 + 5655*x^12 + 2670*x^11 + 69687*x^10 + 43582*x^9 + 674097*x^8 + 378498*x^7 + 4862043*x^6 + 2018526*x^5 + 23403354*x^4 + 6369416*x^3 + 65929308*x^2 + 9405912*x + 82566296)
 
gp: K = bnfinit(x^18 + 15*x^16 - 8*x^15 + 333*x^14 + 24*x^13 + 5655*x^12 + 2670*x^11 + 69687*x^10 + 43582*x^9 + 674097*x^8 + 378498*x^7 + 4862043*x^6 + 2018526*x^5 + 23403354*x^4 + 6369416*x^3 + 65929308*x^2 + 9405912*x + 82566296, 1)
 

Normalized defining polynomial

\( x^{18} + 15 x^{16} - 8 x^{15} + 333 x^{14} + 24 x^{13} + 5655 x^{12} + 2670 x^{11} + 69687 x^{10} + 43582 x^{9} + 674097 x^{8} + 378498 x^{7} + 4862043 x^{6} + 2018526 x^{5} + 23403354 x^{4} + 6369416 x^{3} + 65929308 x^{2} + 9405912 x + 82566296 \)

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:  \(-2001504424181159874396672000000000=-\,2^{18}\cdot 3^{24}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.81$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1260=2^{2}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1260}(1,·)$, $\chi_{1260}(1219,·)$, $\chi_{1260}(961,·)$, $\chi_{1260}(841,·)$, $\chi_{1260}(781,·)$, $\chi_{1260}(79,·)$, $\chi_{1260}(919,·)$, $\chi_{1260}(541,·)$, $\chi_{1260}(799,·)$, $\chi_{1260}(739,·)$, $\chi_{1260}(421,·)$, $\chi_{1260}(361,·)$, $\chi_{1260}(1159,·)$, $\chi_{1260}(1201,·)$, $\chi_{1260}(499,·)$, $\chi_{1260}(121,·)$, $\chi_{1260}(379,·)$, $\chi_{1260}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2032} a^{16} - \frac{69}{1016} a^{15} + \frac{69}{2032} a^{14} - \frac{15}{1016} a^{13} + \frac{739}{2032} a^{12} + \frac{191}{1016} a^{11} - \frac{943}{2032} a^{10} + \frac{49}{127} a^{9} - \frac{343}{2032} a^{8} - \frac{55}{508} a^{7} + \frac{363}{2032} a^{6} - \frac{193}{508} a^{5} - \frac{487}{2032} a^{4} + \frac{75}{508} a^{3} + \frac{137}{508} a^{2} - \frac{77}{254} a + \frac{21}{508}$, $\frac{1}{48873621264658762942584345382404200239296310461234992} a^{17} - \frac{2724805036758102687479907034859828032049093476463}{48873621264658762942584345382404200239296310461234992} a^{16} - \frac{3709880930530592591731021627962069995670407601504873}{48873621264658762942584345382404200239296310461234992} a^{15} + \frac{3808299837207453726721407884945105217477181867408133}{48873621264658762942584345382404200239296310461234992} a^{14} + \frac{12259445274044431752878223244307724962419258226619753}{48873621264658762942584345382404200239296310461234992} a^{13} + \frac{13989072384562200387972253898467959040972007167455619}{48873621264658762942584345382404200239296310461234992} a^{12} + \frac{4965605181416861581359509564557554758026117180080875}{48873621264658762942584345382404200239296310461234992} a^{11} - \frac{14173632703954256740319171286975813122206314243846153}{48873621264658762942584345382404200239296310461234992} a^{10} + \frac{21405224406850688901344201660345383994451104270595721}{48873621264658762942584345382404200239296310461234992} a^{9} + \frac{23218267939543850375457562218322462807734798625853323}{48873621264658762942584345382404200239296310461234992} a^{8} + \frac{18375140189784050568991602491729225380357977629155823}{48873621264658762942584345382404200239296310461234992} a^{7} - \frac{17897314207376941626473486329801202280764518425970999}{48873621264658762942584345382404200239296310461234992} a^{6} + \frac{6478104014305480319567508358156012108260638091293173}{48873621264658762942584345382404200239296310461234992} a^{5} + \frac{17547665269571092912425760905917936010258324225126571}{48873621264658762942584345382404200239296310461234992} a^{4} + \frac{1127802486924894656835527160708472370385291434827783}{3054601329041172683911521586400262514956019403827187} a^{3} - \frac{727754284425404234392878204765815137977837023120681}{6109202658082345367823043172800525029912038807654374} a^{2} - \frac{2333224704625489938064226866218988915118062162204631}{12218405316164690735646086345601050059824077615308748} a + \frac{2219912712624290237935652519852828167449713586133061}{12218405316164690735646086345601050059824077615308748}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{6}\times C_{6}\times C_{18}$, which has order $23328$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.52488000.1, 6.0.126023688000.1, 6.0.19208000.1, 6.0.126023688000.14, 9.9.62523502209.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$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}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$