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Properties

Label 18.0.14753918082...4432.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{20}\cdot 7^{9}$
Root discriminant $19.37$
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![4, 24, 72, 84, 66, 6, -31, 36, 12, 26, -3, -24, 16, -6, 21, -28, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 28*x^15 + 21*x^14 - 6*x^13 + 16*x^12 - 24*x^11 - 3*x^10 + 26*x^9 + 12*x^8 + 36*x^7 - 31*x^6 + 6*x^5 + 66*x^4 + 84*x^3 + 72*x^2 + 24*x + 4)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 28*x^15 + 21*x^14 - 6*x^13 + 16*x^12 - 24*x^11 - 3*x^10 + 26*x^9 + 12*x^8 + 36*x^7 - 31*x^6 + 6*x^5 + 66*x^4 + 84*x^3 + 72*x^2 + 24*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 28 x^{15} + 21 x^{14} - 6 x^{13} + 16 x^{12} - 24 x^{11} - 3 x^{10} + 26 x^{9} + 12 x^{8} + 36 x^{7} - 31 x^{6} + 6 x^{5} + 66 x^{4} + 84 x^{3} + 72 x^{2} + 24 x + 4 \)

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:  \(-147539180820034388754432=-\,2^{20}\cdot 3^{20}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{12} a^{15} - \frac{1}{4} a^{14} - \frac{1}{6} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{24} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{13} + \frac{1}{6} a^{10} + \frac{3}{8} a^{8} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{92396545968} a^{17} + \frac{535291205}{30798848656} a^{16} - \frac{96826285}{30798848656} a^{15} - \frac{7298450069}{30798848656} a^{14} + \frac{65920817}{290555176} a^{13} + \frac{484632649}{3849856082} a^{12} + \frac{2148464873}{11549568246} a^{11} + \frac{722715863}{3849856082} a^{10} - \frac{46648925}{581110352} a^{9} + \frac{11659169057}{30798848656} a^{8} - \frac{1181504539}{30798848656} a^{7} + \frac{6782591533}{30798848656} a^{6} - \frac{5204262329}{11549568246} a^{5} + \frac{4618827837}{15399424328} a^{4} + \frac{230736345}{3849856082} a^{3} - \frac{9623745337}{23099136492} a^{2} + \frac{2403606955}{7699712164} a - \frac{2855574207}{7699712164}$

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:  \( -1 \) (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:  \( 158670.5761863574 \)
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

\(\Q(\sqrt{-7}) \), 3.1.756.1, 3.1.108.1, 6.0.4000752.2, 6.0.4000752.4, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[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$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$