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Properties

Label 18.0.68822941490...4784.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{20}\cdot 7^{6}$
Root discriminant $16.34$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 24, 36, 48, 18, -6, 45, 42, 36, -4, -21, 12, 18, 18, 3, -6, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^15 + 3*x^14 + 18*x^13 + 18*x^12 + 12*x^11 - 21*x^10 - 4*x^9 + 36*x^8 + 42*x^7 + 45*x^6 - 6*x^5 + 18*x^4 + 48*x^3 + 36*x^2 + 24*x + 8)
 
gp: K = bnfinit(x^18 - 6*x^15 + 3*x^14 + 18*x^13 + 18*x^12 + 12*x^11 - 21*x^10 - 4*x^9 + 36*x^8 + 42*x^7 + 45*x^6 - 6*x^5 + 18*x^4 + 48*x^3 + 36*x^2 + 24*x + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{15} + 3 x^{14} + 18 x^{13} + 18 x^{12} + 12 x^{11} - 21 x^{10} - 4 x^{9} + 36 x^{8} + 42 x^{7} + 45 x^{6} - 6 x^{5} + 18 x^{4} + 48 x^{3} + 36 x^{2} + 24 x + 8 \)

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:  \(-6882294149039505014784=-\,2^{24}\cdot 3^{20}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.34$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{1}{4} a^{5} + \frac{3}{8} a^{4} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{24} a^{13} - \frac{1}{12} a^{10} - \frac{1}{4} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{5}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{14} - \frac{1}{12} a^{11} + \frac{1}{12} a^{8} + \frac{1}{8} a^{6} - \frac{1}{6} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{15} - \frac{1}{12} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{16} - \frac{1}{12} a^{10} + \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{5691408} a^{17} + \frac{19463}{1897136} a^{16} - \frac{115213}{5691408} a^{15} + \frac{24131}{1897136} a^{14} - \frac{50635}{2845704} a^{13} + \frac{40879}{2845704} a^{12} + \frac{168281}{2845704} a^{11} + \frac{17687}{2845704} a^{10} + \frac{419929}{5691408} a^{9} + \frac{58143}{1897136} a^{8} - \frac{276841}{5691408} a^{7} + \frac{78181}{5691408} a^{6} - \frac{314689}{1422852} a^{5} - \frac{218003}{711426} a^{4} - \frac{345631}{1422852} a^{3} + \frac{447179}{1422852} a^{2} + \frac{131539}{355713} a + \frac{17811}{118571}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{205839}{948568} a^{17} + \frac{97473}{948568} a^{16} - \frac{57003}{948568} a^{15} + \frac{161676}{118571} a^{14} - \frac{1244835}{948568} a^{13} - \frac{762159}{237142} a^{12} - \frac{309846}{118571} a^{11} - \frac{165816}{118571} a^{10} + \frac{5236409}{948568} a^{9} - \frac{1466415}{948568} a^{8} - \frac{6239409}{948568} a^{7} - \frac{1572465}{237142} a^{6} - \frac{7031463}{948568} a^{5} + \frac{2365407}{474284} a^{4} - \frac{731004}{118571} a^{3} - \frac{757581}{118571} a^{2} - \frac{1266339}{237142} a - \frac{372938}{118571} \) (order $4$)
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:  \( 14270.669184852277 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.756.1, 3.1.108.1, 6.0.186624.1, 6.0.2286144.1, 9.1.20739898368.1

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

Sibling fields

Degree 12 siblings: data not computed
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{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}$
$7$7.6.0.1$x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$