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Properties

Label 18.0.67804051110...9728.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{20}\cdot 7^{15}$
Root discriminant $27.23$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 0, 0, -44, 0, 0, 130, 0, 0, -99, 0, 0, 65, 0, 0, -11, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 11*x^15 + 65*x^12 - 99*x^9 + 130*x^6 - 44*x^3 + 8)
 
gp: K = bnfinit(x^18 - 11*x^15 + 65*x^12 - 99*x^9 + 130*x^6 - 44*x^3 + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 11 x^{15} + 65 x^{12} - 99 x^{9} + 130 x^{6} - 44 x^{3} + 8 \)

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:  \(-67804051110532132041289728=-\,2^{12}\cdot 3^{20}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.23$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{30} a^{12} - \frac{1}{10} a^{9} + \frac{3}{10} a^{6} + \frac{3}{10} a^{3} + \frac{2}{15}$, $\frac{1}{30} a^{13} - \frac{1}{10} a^{10} + \frac{3}{10} a^{7} + \frac{3}{10} a^{4} + \frac{2}{15} a$, $\frac{1}{60} a^{14} - \frac{1}{20} a^{11} + \frac{3}{20} a^{8} + \frac{3}{20} a^{5} - \frac{13}{30} a^{2}$, $\frac{1}{2820} a^{15} + \frac{29}{2820} a^{12} - \frac{469}{940} a^{9} - \frac{181}{940} a^{6} - \frac{79}{1410} a^{3} + \frac{242}{705}$, $\frac{1}{16920} a^{16} + \frac{1}{8460} a^{15} - \frac{1}{180} a^{14} - \frac{13}{3384} a^{13} + \frac{29}{8460} a^{12} + \frac{7}{20} a^{11} + \frac{163}{376} a^{10} + \frac{157}{940} a^{9} - \frac{1}{20} a^{8} - \frac{781}{1880} a^{7} + \frac{253}{940} a^{6} - \frac{1}{20} a^{5} + \frac{932}{2115} a^{4} - \frac{79}{4230} a^{3} + \frac{43}{90} a^{2} + \frac{853}{4230} a + \frac{947}{2115}$, $\frac{1}{16920} a^{17} - \frac{1}{8460} a^{15} + \frac{41}{5640} a^{14} + \frac{1}{90} a^{13} + \frac{13}{1692} a^{12} - \frac{501}{1880} a^{11} + \frac{3}{10} a^{10} + \frac{25}{188} a^{9} - \frac{593}{1880} a^{8} + \frac{1}{10} a^{7} - \frac{159}{940} a^{6} - \frac{1943}{4230} a^{5} + \frac{1}{10} a^{4} + \frac{251}{2115} a^{3} + \frac{347}{1410} a^{2} + \frac{2}{45} a - \frac{853}{2115}$

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:  \( 818382.7735614459 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.5292.1, 3.1.1323.1, 3.1.108.1, 3.1.588.1, 6.0.196036848.1, 6.0.4000752.4, 6.0.12252303.1, 6.0.2420208.1, 9.1.444611571264.1

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

Sibling fields

Galois closure: data not computed
Degree 18 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{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$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
7Data not computed