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Properties

Label 18.0.36542633964...0208.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 31^{8}$
Root discriminant $26.31$
Ramified primes $2, 3, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 198, -345, 6063, 741, -10499, 2976, 7767, -4481, -2700, 2376, 346, -579, 12, 68, -3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 3*x^16 + 68*x^15 + 12*x^14 - 579*x^13 + 346*x^12 + 2376*x^11 - 2700*x^10 - 4481*x^9 + 7767*x^8 + 2976*x^7 - 10499*x^6 + 741*x^5 + 6063*x^4 - 345*x^3 + 198*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 - 3*x^16 + 68*x^15 + 12*x^14 - 579*x^13 + 346*x^12 + 2376*x^11 - 2700*x^10 - 4481*x^9 + 7767*x^8 + 2976*x^7 - 10499*x^6 + 741*x^5 + 6063*x^4 - 345*x^3 + 198*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 3 x^{16} + 68 x^{15} + 12 x^{14} - 579 x^{13} + 346 x^{12} + 2376 x^{11} - 2700 x^{10} - 4481 x^{9} + 7767 x^{8} + 2976 x^{7} - 10499 x^{6} + 741 x^{5} + 6063 x^{4} - 345 x^{3} + 198 x^{2} - 3 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-36542633964773200992350208=-\,2^{12}\cdot 3^{21}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.31$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1525018018320474019231037} a^{17} - \frac{312230638545564957862933}{1525018018320474019231037} a^{16} + \frac{662503618725064814772505}{1525018018320474019231037} a^{15} + \frac{670324246153728308364958}{1525018018320474019231037} a^{14} - \frac{333140082343998958952953}{1525018018320474019231037} a^{13} - \frac{469830432968822938508622}{1525018018320474019231037} a^{12} - \frac{716012646512301726341344}{1525018018320474019231037} a^{11} - \frac{547850005572961563348}{1525018018320474019231037} a^{10} + \frac{314123954475013776905437}{1525018018320474019231037} a^{9} - \frac{712464327573734964106647}{1525018018320474019231037} a^{8} - \frac{137337529524342610167217}{1525018018320474019231037} a^{7} + \frac{566186010339201639796520}{1525018018320474019231037} a^{6} + \frac{291969096873254744162740}{1525018018320474019231037} a^{5} - \frac{656295867055772714656410}{1525018018320474019231037} a^{4} + \frac{58670094976079271701224}{1525018018320474019231037} a^{3} - \frac{623912025144299171795651}{1525018018320474019231037} a^{2} - \frac{727338682051990066958306}{1525018018320474019231037} a + \frac{171656612131699192318256}{1525018018320474019231037}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{80175057070766}{1670596077643049} a^{17} - \frac{481458729828347}{1670596077643049} a^{16} - \frac{235977749504265}{1670596077643049} a^{15} + \frac{5437008077308020}{1670596077643049} a^{14} + \frac{945901418525125}{1670596077643049} a^{13} - \frac{46271956574817727}{1670596077643049} a^{12} + \frac{27842348351612035}{1670596077643049} a^{11} + \frac{189062360878011087}{1670596077643049} a^{10} - \frac{215431835349005225}{1670596077643049} a^{9} - \frac{354177145409251907}{1670596077643049} a^{8} + \frac{615023617419548543}{1670596077643049} a^{7} + \frac{232690104456450962}{1670596077643049} a^{6} - \frac{824197668732660532}{1670596077643049} a^{5} + \frac{57112094832835476}{1670596077643049} a^{4} + \frac{469833133234570858}{1670596077643049} a^{3} - \frac{21609610762143921}{1670596077643049} a^{2} + \frac{21290036629892721}{1670596077643049} a + \frac{655812656932034}{1670596077643049} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 396277.1598571161 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.1163370485952.1

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

Sibling fields

Degree 9 siblings: data not computed
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$