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Properties

Label 18.0.33400217127...2667.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{4}\cdot 643^{6}$
Root discriminant $23.04$
Ramified primes $3, 7, 643$
Class number $2$
Class group $[2]$
Galois group $C_2\times C_3\wr S_3$ (as 18T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![283, -1519, 3328, -3443, 1381, -54, 827, -1449, 707, 82, -90, -51, 23, 46, -32, 7, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 7*x^15 - 32*x^14 + 46*x^13 + 23*x^12 - 51*x^11 - 90*x^10 + 82*x^9 + 707*x^8 - 1449*x^7 + 827*x^6 - 54*x^5 + 1381*x^4 - 3443*x^3 + 3328*x^2 - 1519*x + 283)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 7*x^15 - 32*x^14 + 46*x^13 + 23*x^12 - 51*x^11 - 90*x^10 + 82*x^9 + 707*x^8 - 1449*x^7 + 827*x^6 - 54*x^5 + 1381*x^4 - 3443*x^3 + 3328*x^2 - 1519*x + 283, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + 7 x^{15} - 32 x^{14} + 46 x^{13} + 23 x^{12} - 51 x^{11} - 90 x^{10} + 82 x^{9} + 707 x^{8} - 1449 x^{7} + 827 x^{6} - 54 x^{5} + 1381 x^{4} - 3443 x^{3} + 3328 x^{2} - 1519 x + 283 \)

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:  \(-3340021712790174678422667=-\,3^{9}\cdot 7^{4}\cdot 643^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.04$
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:  $3, 7, 643$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{13} - \frac{3}{7} a^{11} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{4} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{238328215556667695632621} a^{17} - \frac{13344977105231835738886}{238328215556667695632621} a^{16} + \frac{10957901356323190478968}{238328215556667695632621} a^{15} - \frac{4639988538767742652766}{238328215556667695632621} a^{14} - \frac{29335402475256592277268}{238328215556667695632621} a^{13} - \frac{66497732363310737364821}{238328215556667695632621} a^{12} - \frac{3824197030795775589064}{34046887936666813661803} a^{11} + \frac{5220820059683351328385}{238328215556667695632621} a^{10} - \frac{39874344346276337337051}{238328215556667695632621} a^{9} - \frac{106609183916534468879606}{238328215556667695632621} a^{8} - \frac{89508778933741767550099}{238328215556667695632621} a^{7} - \frac{73015673660315033886039}{238328215556667695632621} a^{6} + \frac{12881407189788204321609}{34046887936666813661803} a^{5} + \frac{13626997406868834678181}{238328215556667695632621} a^{4} + \frac{52401389190287560818821}{238328215556667695632621} a^{3} - \frac{20955077419787007068384}{238328215556667695632621} a^{2} + \frac{62649461943082698727331}{238328215556667695632621} a - \frac{22951822335570408418906}{238328215556667695632621}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{15212591720258}{1522227414307243} a^{17} + \frac{52921042723180}{1522227414307243} a^{16} - \frac{83368169960277}{1522227414307243} a^{15} - \frac{101472791851520}{1522227414307243} a^{14} + \frac{581510075493933}{1522227414307243} a^{13} - \frac{786004073191664}{1522227414307243} a^{12} - \frac{51630999663319}{217461059186749} a^{11} + \frac{1294337404048091}{1522227414307243} a^{10} + \frac{1724033872292780}{1522227414307243} a^{9} - \frac{2057384270800276}{1522227414307243} a^{8} - \frac{11785243952055857}{1522227414307243} a^{7} + \frac{27190137317764072}{1522227414307243} a^{6} - \frac{1738136939298293}{217461059186749} a^{5} - \frac{5088326971002902}{1522227414307243} a^{4} - \frac{22065821193515804}{1522227414307243} a^{3} + \frac{63627057463250654}{1522227414307243} a^{2} - \frac{52646853618291816}{1522227414307243} a + \frac{16234308813897989}{1522227414307243} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 164073.537426 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3\wr S_3$ (as 18T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 324
The 44 conjugacy class representatives for $C_2\times C_3\wr S_3$
Character table for $C_2\times C_3\wr S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1929.1, 6.0.11163123.4, 9.9.351716516361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R $18$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ $18$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
643Data not computed