# Properties

 Label 18.0.12906538539...0896.1 Degree $18$ Signature $[0, 9]$ Discriminant $-\,2^{16}\cdot 11^{9}\cdot 17^{4}$ Root discriminant $11.53$ Ramified primes $2, 11, 17$ Class number $1$ Class group Trivial Galois Group $C_3\wr C_3:C_2$ (as 18T88)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 9, -22, 36, -18, -48, 98, -15, -231, 517, -678, 646, -476, 276, -124, 41, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*x^2 - 3*x + 1)
gp: K = bnfinit(x^18 - 9*x^17 + 41*x^16 - 124*x^15 + 276*x^14 - 476*x^13 + 646*x^12 - 678*x^11 + 517*x^10 - 231*x^9 - 15*x^8 + 98*x^7 - 48*x^6 - 18*x^5 + 36*x^4 - 22*x^3 + 9*x^2 - 3*x + 1, 1)

## Normalizeddefining polynomial

$$x^{18}$$ $$\mathstrut -\mathstrut 9 x^{17}$$ $$\mathstrut +\mathstrut 41 x^{16}$$ $$\mathstrut -\mathstrut 124 x^{15}$$ $$\mathstrut +\mathstrut 276 x^{14}$$ $$\mathstrut -\mathstrut 476 x^{13}$$ $$\mathstrut +\mathstrut 646 x^{12}$$ $$\mathstrut -\mathstrut 678 x^{11}$$ $$\mathstrut +\mathstrut 517 x^{10}$$ $$\mathstrut -\mathstrut 231 x^{9}$$ $$\mathstrut -\mathstrut 15 x^{8}$$ $$\mathstrut +\mathstrut 98 x^{7}$$ $$\mathstrut -\mathstrut 48 x^{6}$$ $$\mathstrut -\mathstrut 18 x^{5}$$ $$\mathstrut +\mathstrut 36 x^{4}$$ $$\mathstrut -\mathstrut 22 x^{3}$$ $$\mathstrut +\mathstrut 9 x^{2}$$ $$\mathstrut -\mathstrut 3 x$$ $$\mathstrut +\mathstrut 1$$

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: $$-12906538539418320896=-\,2^{16}\cdot 11^{9}\cdot 17^{4}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.53$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 11, 17$ magma: PrimeDivisors(Discriminant(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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{401} a^{16} - \frac{8}{401} a^{15} + \frac{20}{401} a^{14} + \frac{16}{401} a^{12} - \frac{59}{401} a^{11} - \frac{22}{401} a^{10} + \frac{67}{401} a^{9} + \frac{68}{401} a^{8} + \frac{169}{401} a^{7} + \frac{72}{401} a^{6} - \frac{22}{401} a^{5} + \frac{197}{401} a^{4} + \frac{64}{401} a^{3} - \frac{55}{401} a^{2} - \frac{107}{401} a - \frac{185}{401}$, $\frac{1}{42907} a^{17} + \frac{45}{42907} a^{16} - \frac{20855}{42907} a^{15} + \frac{4268}{42907} a^{14} - \frac{21237}{42907} a^{13} + \frac{11215}{42907} a^{12} + \frac{18505}{42907} a^{11} + \frac{14941}{42907} a^{10} - \frac{9614}{42907} a^{9} + \frac{20214}{42907} a^{8} + \frac{20257}{42907} a^{7} - \frac{16256}{42907} a^{6} + \frac{17076}{42907} a^{5} + \frac{19327}{42907} a^{4} + \frac{9753}{42907} a^{3} + \frac{20637}{42907} a^{2} - \frac{5455}{42907} a + \frac{13052}{42907}$

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: $$113.075873428$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 162 The 13 conjugacy class representatives for $C_3\wr C_3:C_2$ Character table for $C_3\wr C_3:C_2$

## Intermediate fields

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

## Sibling fields

 Degree 9 siblings: data not computed Degree 18 siblings: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.2x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2} 11.6.3.2x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 1717.6.4.1x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6} 17.6.0.1x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$