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Properties

Label 18.0.27277478847...9616.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 11^{9}$
Root discriminant $22.78$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![676, -2340, 3528, -4584, 5418, -2772, 1667, -1326, 1305, -382, 198, -198, 155, -24, 18, -12, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676)
 
gp: K = bnfinit(x^18 + 3*x^16 - 12*x^15 + 18*x^14 - 24*x^13 + 155*x^12 - 198*x^11 + 198*x^10 - 382*x^9 + 1305*x^8 - 1326*x^7 + 1667*x^6 - 2772*x^5 + 5418*x^4 - 4584*x^3 + 3528*x^2 - 2340*x + 676, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 12 x^{15} + 18 x^{14} - 24 x^{13} + 155 x^{12} - 198 x^{11} + 198 x^{10} - 382 x^{9} + 1305 x^{8} - 1326 x^{7} + 1667 x^{6} - 2772 x^{5} + 5418 x^{4} - 4584 x^{3} + 3528 x^{2} - 2340 x + 676 \)

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:  \(-2727747884710191986159616=-\,2^{12}\cdot 3^{24}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.78$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{12} - \frac{2}{11} a^{11} - \frac{2}{11} a^{10} - \frac{5}{22} a^{9} - \frac{2}{11} a^{8} + \frac{1}{11} a^{7} + \frac{5}{22} a^{6} + \frac{4}{11} a^{5} - \frac{4}{11} a^{4} + \frac{7}{22} a^{3} - \frac{1}{11} a^{2} - \frac{4}{11} a - \frac{5}{11}$, $\frac{1}{22} a^{13} + \frac{1}{11} a^{11} + \frac{1}{22} a^{10} - \frac{1}{11} a^{9} + \frac{4}{11} a^{8} - \frac{9}{22} a^{7} + \frac{3}{11} a^{6} + \frac{1}{11} a^{5} - \frac{3}{22} a^{4} + \frac{2}{11} a^{3} + \frac{3}{11} a^{2} + \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{22} a^{14} - \frac{1}{11} a^{11} - \frac{5}{22} a^{10} - \frac{2}{11} a^{9} + \frac{5}{11} a^{8} + \frac{1}{11} a^{7} + \frac{3}{22} a^{6} - \frac{4}{11} a^{5} - \frac{1}{11} a^{4} - \frac{4}{11} a^{3} + \frac{3}{11} a^{2} - \frac{1}{11} a - \frac{1}{11}$, $\frac{1}{44044} a^{15} - \frac{255}{44044} a^{14} + \frac{101}{6292} a^{13} - \frac{97}{11011} a^{12} - \frac{4897}{44044} a^{11} + \frac{8725}{44044} a^{10} + \frac{6001}{44044} a^{9} + \frac{3525}{22022} a^{8} + \frac{13775}{44044} a^{7} - \frac{18855}{44044} a^{6} - \frac{707}{6292} a^{5} - \frac{444}{11011} a^{4} + \frac{2831}{11011} a^{3} + \frac{10519}{22022} a^{2} - \frac{4649}{11011} a + \frac{281}{847}$, $\frac{1}{1893892} a^{16} - \frac{1}{172172} a^{15} + \frac{32581}{1893892} a^{14} - \frac{463}{36421} a^{13} + \frac{16547}{1893892} a^{12} - \frac{7287}{270556} a^{11} + \frac{277045}{1893892} a^{10} - \frac{1915}{43043} a^{9} - \frac{151909}{1893892} a^{8} - \frac{197291}{1893892} a^{7} + \frac{629661}{1893892} a^{6} - \frac{107169}{946946} a^{5} + \frac{152353}{946946} a^{4} - \frac{195367}{473473} a^{3} + \frac{31373}{67639} a^{2} + \frac{4780}{67639} a - \frac{6357}{36421}$, $\frac{1}{66918034876568876} a^{17} - \frac{168617940}{1520864429012929} a^{16} + \frac{13815551780}{1286885286087863} a^{15} - \frac{1477813828587553}{66918034876568876} a^{14} + \frac{14822403372625}{869065388007388} a^{13} + \frac{285322963437}{116581942293674} a^{12} + \frac{2297232892914457}{16729508719142219} a^{11} + \frac{238614552788281}{5147541144351452} a^{10} + \frac{1599505113754741}{66918034876568876} a^{9} - \frac{1439692519714853}{33459017438284438} a^{8} + \frac{2943332335012981}{16729508719142219} a^{7} - \frac{18845764474165257}{66918034876568876} a^{6} - \frac{77350181361316}{183840755155409} a^{5} - \frac{5597244763484553}{16729508719142219} a^{4} - \frac{2205163098609369}{16729508719142219} a^{3} - \frac{328848656218828}{16729508719142219} a^{2} + \frac{124777276995494}{389058342305633} a - \frac{69791010415649}{1286885286087863}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 59552.7806315 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.3564.1 x3, 3.1.44.1 x3, 3.1.891.1 x3, 3.1.3564.2 x3, 6.0.139723056.1, 6.0.21296.1, 6.0.8732691.5, 6.0.139723056.2, 9.1.497972971584.1 x9

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

Sibling fields

Degree 9 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$