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Properties

Label 18.0.63502608684...5248.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 31^{12}$
Root discriminant $27.13$
Ramified primes $2, 3, 31$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 10, 10, -28, -36, 187, 518, 818, 896, 818, 518, 187, -36, -28, 10, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 10*x^16 + 10*x^15 - 28*x^14 - 36*x^13 + 187*x^12 + 518*x^11 + 818*x^10 + 896*x^9 + 818*x^8 + 518*x^7 + 187*x^6 - 36*x^5 - 28*x^4 + 10*x^3 + 10*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 10*x^16 + 10*x^15 - 28*x^14 - 36*x^13 + 187*x^12 + 518*x^11 + 818*x^10 + 896*x^9 + 818*x^8 + 518*x^7 + 187*x^6 - 36*x^5 - 28*x^4 + 10*x^3 + 10*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 10 x^{16} + 10 x^{15} - 28 x^{14} - 36 x^{13} + 187 x^{12} + 518 x^{11} + 818 x^{10} + 896 x^{9} + 818 x^{8} + 518 x^{7} + 187 x^{6} - 36 x^{5} - 28 x^{4} + 10 x^{3} + 10 x^{2} - 6 x + 1 \)

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:  \(-63502608684277862177845248=-\,2^{12}\cdot 3^{9}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.13$
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 Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{732} a^{14} + \frac{23}{732} a^{13} + \frac{7}{366} a^{12} - \frac{41}{183} a^{11} + \frac{71}{366} a^{10} - \frac{20}{183} a^{9} + \frac{25}{122} a^{8} - \frac{5}{122} a^{7} + \frac{25}{122} a^{6} + \frac{143}{366} a^{5} - \frac{56}{183} a^{4} - \frac{41}{183} a^{3} - \frac{169}{732} a^{2} - \frac{343}{732} a + \frac{46}{183}$, $\frac{1}{22692} a^{15} - \frac{1}{7564} a^{14} - \frac{512}{5673} a^{13} + \frac{705}{3782} a^{12} + \frac{1376}{5673} a^{11} + \frac{1070}{5673} a^{10} - \frac{1813}{11346} a^{9} + \frac{313}{1891} a^{8} + \frac{521}{3782} a^{7} - \frac{3271}{11346} a^{6} - \frac{2915}{11346} a^{5} - \frac{415}{5673} a^{4} + \frac{2341}{7564} a^{3} - \frac{8027}{22692} a^{2} + \frac{419}{3782} a - \frac{5503}{11346}$, $\frac{1}{136152} a^{16} + \frac{1}{136152} a^{15} - \frac{7}{68076} a^{14} + \frac{14731}{136152} a^{13} + \frac{22703}{136152} a^{12} + \frac{1055}{17019} a^{11} + \frac{283}{1891} a^{10} + \frac{9227}{68076} a^{9} + \frac{575}{1891} a^{8} + \frac{17675}{68076} a^{7} - \frac{4511}{11346} a^{6} - \frac{13127}{34038} a^{5} - \frac{49465}{136152} a^{4} + \frac{47593}{136152} a^{3} - \frac{11821}{68076} a^{2} - \frac{50333}{136152} a - \frac{24959}{136152}$, $\frac{1}{136152} a^{17} - \frac{1}{45384} a^{15} + \frac{5}{45384} a^{14} - \frac{7093}{68076} a^{13} - \frac{33067}{136152} a^{12} - \frac{7753}{34038} a^{11} + \frac{2585}{68076} a^{10} - \frac{1169}{68076} a^{9} - \frac{27361}{68076} a^{8} - \frac{9803}{68076} a^{7} - \frac{139}{549} a^{6} + \frac{19387}{136152} a^{5} - \frac{17603}{68076} a^{4} + \frac{13399}{45384} a^{3} - \frac{7463}{45384} a^{2} - \frac{281}{3782} a - \frac{19693}{136152}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( -\frac{7949}{22692} a^{17} + \frac{44117}{22692} a^{16} - \frac{58097}{22692} a^{15} - \frac{116239}{22692} a^{14} + \frac{195353}{22692} a^{13} + \frac{122031}{7564} a^{12} - \frac{679631}{11346} a^{11} - \frac{2368553}{11346} a^{10} - \frac{696053}{1891} a^{9} - \frac{2589845}{5673} a^{8} - \frac{2576669}{5673} a^{7} - \frac{1308775}{3782} a^{6} - \frac{4125737}{22692} a^{5} - \frac{1043663}{22692} a^{4} + \frac{1103}{732} a^{3} - \frac{7397}{22692} a^{2} - \frac{47425}{22692} a + \frac{6875}{7564} \) (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:  \( 508194.463043 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.11532.1 x3, 3.3.961.1, 6.0.398961072.1, 6.0.24935067.1, 6.0.415152.1 x2, 9.3.1533606360768.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.415152.1
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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$$[\ ]_{3}^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$