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Properties

Label 18.0.50809695180...3536.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 17^{9}$
Root discriminant $23.58$
Ramified primes $2, 3, 17$
Class number $6$
Class group $[6]$
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![51, -459, 1815, -4230, 6900, -9186, 10882, -11094, 8883, -5295, 2775, -1638, 988, -504, 240, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 240*x^14 - 504*x^13 + 988*x^12 - 1638*x^11 + 2775*x^10 - 5295*x^9 + 8883*x^8 - 11094*x^7 + 10882*x^6 - 9186*x^5 + 6900*x^4 - 4230*x^3 + 1815*x^2 - 459*x + 51, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 240 x^{14} - 504 x^{13} + 988 x^{12} - 1638 x^{11} + 2775 x^{10} - 5295 x^{9} + 8883 x^{8} - 11094 x^{7} + 10882 x^{6} - 9186 x^{5} + 6900 x^{4} - 4230 x^{3} + 1815 x^{2} - 459 x + 51 \)

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:  \(-5080969518089676267073536=-\,2^{12}\cdot 3^{21}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.58$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{408} a^{14} - \frac{7}{408} a^{13} - \frac{1}{204} a^{12} + \frac{1}{408} a^{11} + \frac{47}{408} a^{10} + \frac{41}{204} a^{9} + \frac{43}{408} a^{8} + \frac{41}{408} a^{7} + \frac{67}{408} a^{6} + \frac{11}{68} a^{5} + \frac{29}{136} a^{4} - \frac{13}{136} a^{3} + \frac{29}{68} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{408} a^{15} - \frac{1}{8} a^{13} - \frac{13}{408} a^{12} - \frac{2}{17} a^{11} + \frac{1}{136} a^{10} + \frac{5}{408} a^{9} + \frac{3}{34} a^{8} + \frac{2}{17} a^{7} - \frac{77}{408} a^{6} + \frac{47}{136} a^{5} - \frac{6}{17} a^{4} + \frac{1}{136} a^{3} - \frac{19}{136} a^{2} + \frac{3}{8}$, $\frac{1}{105142824} a^{16} - \frac{1}{13142853} a^{15} - \frac{42307}{35047608} a^{14} + \frac{888587}{105142824} a^{13} - \frac{3129253}{26285706} a^{12} + \frac{3659555}{35047608} a^{11} - \frac{5383447}{105142824} a^{10} - \frac{4366037}{52571412} a^{9} + \frac{2570183}{17523804} a^{8} + \frac{22595113}{105142824} a^{7} + \frac{3325021}{105142824} a^{6} + \frac{318311}{17523804} a^{5} - \frac{792827}{11682536} a^{4} + \frac{1909421}{11682536} a^{3} + \frac{5182351}{17523804} a^{2} + \frac{22543}{66504} a - \frac{453541}{1030812}$, $\frac{1}{5362284024} a^{17} + \frac{1}{315428472} a^{16} - \frac{1157933}{5362284024} a^{15} + \frac{1361663}{1340571006} a^{14} + \frac{507579859}{5362284024} a^{13} - \frac{251436847}{5362284024} a^{12} - \frac{86567585}{2681142012} a^{11} + \frac{558664723}{5362284024} a^{10} - \frac{36756923}{2681142012} a^{9} + \frac{361736023}{5362284024} a^{8} + \frac{509849251}{2681142012} a^{7} - \frac{868465709}{5362284024} a^{6} - \frac{224065097}{1787428008} a^{5} - \frac{3844219}{99301556} a^{4} - \frac{867963415}{1787428008} a^{3} + \frac{677653045}{1787428008} a^{2} - \frac{43738759}{105142824} a - \frac{18554209}{52571412}$

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

Class group and class number

$C_{6}$, which has order $6$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 88275.1610738 \)
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

\(\Q(\sqrt{-51}) \), 3.1.1836.2 x3, 3.1.204.1 x3, 3.1.1836.1 x3, 3.1.459.1 x3, 6.0.171915696.1, 6.0.2122416.1, 6.0.171915696.2, 6.0.10744731.1, 9.1.315637217856.1 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$