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Properties

Label 18.6.11360989554...1641.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{32}\cdot 19^{10}$
Root discriminant $36.19$
Ramified primes $3, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_6^2:C_3$ (as 18T48)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -192, -1344, 168, -696, 10734, -11115, -2055, 6243, -974, -510, -627, 930, -627, 309, -117, 30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64)
 
gp: K = bnfinit(x^18 - 6*x^17 + 30*x^16 - 117*x^15 + 309*x^14 - 627*x^13 + 930*x^12 - 627*x^11 - 510*x^10 - 974*x^9 + 6243*x^8 - 2055*x^7 - 11115*x^6 + 10734*x^5 - 696*x^4 + 168*x^3 - 1344*x^2 - 192*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 30 x^{16} - 117 x^{15} + 309 x^{14} - 627 x^{13} + 930 x^{12} - 627 x^{11} - 510 x^{10} - 974 x^{9} + 6243 x^{8} - 2055 x^{7} - 11115 x^{6} + 10734 x^{5} - 696 x^{4} + 168 x^{3} - 1344 x^{2} - 192 x + 64 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11360989554893559098699461641=3^{32}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.19$
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:  $3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{12} a^{4} + \frac{1}{3} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{13} + \frac{1}{36} a^{12} + \frac{1}{18} a^{11} - \frac{1}{36} a^{10} + \frac{1}{18} a^{9} + \frac{1}{6} a^{8} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{17}{36} a^{5} - \frac{5}{18} a^{4} - \frac{1}{36} a^{3} + \frac{5}{18} a^{2} - \frac{7}{18} a - \frac{2}{9}$, $\frac{1}{72} a^{15} + \frac{1}{72} a^{12} - \frac{1}{24} a^{11} + \frac{1}{24} a^{10} - \frac{1}{36} a^{9} + \frac{1}{8} a^{8} + \frac{5}{18} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{11}{72} a^{3} - \frac{1}{12} a^{2} - \frac{1}{6} a + \frac{4}{9}$, $\frac{1}{31392} a^{16} + \frac{65}{15696} a^{15} + \frac{181}{15696} a^{14} - \frac{93}{3488} a^{13} - \frac{289}{10464} a^{12} + \frac{2401}{31392} a^{11} - \frac{845}{15696} a^{10} + \frac{17}{31392} a^{9} - \frac{325}{5232} a^{8} - \frac{1505}{15696} a^{7} + \frac{11315}{31392} a^{6} + \frac{13777}{31392} a^{5} + \frac{4867}{10464} a^{4} - \frac{1589}{5232} a^{3} - \frac{61}{7848} a^{2} + \frac{559}{1962} a - \frac{455}{1962}$, $\frac{1}{108398747346848691451776} a^{17} - \frac{967386342095642}{94096134849695044663} a^{16} - \frac{147761256393466843777}{54199373673424345725888} a^{15} - \frac{307930675493450166529}{108398747346848691451776} a^{14} + \frac{1148241248941107467215}{108398747346848691451776} a^{13} - \frac{8019156166635475094425}{108398747346848691451776} a^{12} + \frac{249816720244754477227}{3011076315190241429216} a^{11} - \frac{7952329272019608862219}{108398747346848691451776} a^{10} + \frac{26978911784583877755}{376384539398780178652} a^{9} - \frac{10518356943565831124951}{54199373673424345725888} a^{8} + \frac{1297109966684372763461}{36132915782282897150592} a^{7} - \frac{37873886179164135246733}{108398747346848691451776} a^{6} + \frac{53010021600771331413319}{108398747346848691451776} a^{5} - \frac{3206995385909881925933}{13549843418356086431472} a^{4} + \frac{4218132370763044303919}{13549843418356086431472} a^{3} + \frac{1582605470131647433057}{4516614472785362143824} a^{2} - \frac{2403746966314023180851}{6774921709178043215736} a - \frac{446816550359665954817}{1129153618196340535956}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 22453155.9774 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2:C_3$ (as 18T48):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_6^2:C_3$
Character table for $C_6^2:C_3$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.2368521.1, 9.9.5609891727441.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.9.16.1$x^{9} + 3 x^{8} + 3 x^{6} + 3$$9$$1$$16$$C_3^2:C_3$$[2, 2]^{3}$
3.9.16.1$x^{9} + 3 x^{8} + 3 x^{6} + 3$$9$$1$$16$$C_3^2:C_3$$[2, 2]^{3}$
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.6.5.1$x^{6} - 304$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$