# Properties

 Label 18.0.11982410022...8867.1 Degree $18$ Signature $[0, 9]$ Discriminant $-\,31^{6}\cdot 67^{5}$ Root discriminant $10.10$ Ramified primes $31, 67$ Class number $1$ Class group Trivial Galois Group 18T284

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 22, -41, 49, -39, 29, -40, 62, -65, 45, -23, 15, -16, 17, -14, 9, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1)
gp: K = bnfinit(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1, 1)

## Normalizeddefining polynomial

$$x^{18}$$ $$\mathstrut -\mathstrut 4 x^{17}$$ $$\mathstrut +\mathstrut 9 x^{16}$$ $$\mathstrut -\mathstrut 14 x^{15}$$ $$\mathstrut +\mathstrut 17 x^{14}$$ $$\mathstrut -\mathstrut 16 x^{13}$$ $$\mathstrut +\mathstrut 15 x^{12}$$ $$\mathstrut -\mathstrut 23 x^{11}$$ $$\mathstrut +\mathstrut 45 x^{10}$$ $$\mathstrut -\mathstrut 65 x^{9}$$ $$\mathstrut +\mathstrut 62 x^{8}$$ $$\mathstrut -\mathstrut 40 x^{7}$$ $$\mathstrut +\mathstrut 29 x^{6}$$ $$\mathstrut -\mathstrut 39 x^{5}$$ $$\mathstrut +\mathstrut 49 x^{4}$$ $$\mathstrut -\mathstrut 41 x^{3}$$ $$\mathstrut +\mathstrut 22 x^{2}$$ $$\mathstrut -\mathstrut 7 x$$ $$\mathstrut +\mathstrut 1$$

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: $$-1198241002273018867=-\,31^{6}\cdot 67^{5}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $10.10$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $31, 67$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6457} a^{17} + \frac{2237}{6457} a^{16} + \frac{2494}{6457} a^{15} - \frac{2722}{6457} a^{14} + \frac{1880}{6457} a^{13} + \frac{3100}{6457} a^{12} - \frac{617}{6457} a^{11} - \frac{922}{6457} a^{10} + \frac{83}{6457} a^{9} - \frac{1315}{6457} a^{8} - \frac{2461}{6457} a^{7} - \frac{863}{6457} a^{6} + \frac{286}{587} a^{5} - \frac{897}{6457} a^{4} - \frac{2001}{6457} a^{3} - \frac{284}{587} a^{2} - \frac{134}{587} a + \frac{2743}{6457}$

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: $$22.3834819773$$ 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 1296 The 98 conjugacy class representatives for t18n284 are not computed Character table for t18n284 is not computed

## Intermediate fields

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

## Sibling fields

 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 $18$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ R ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 31.4.2.1x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 31.6.0.1x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$67$$\Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] \Q_{67}$$x + 4$$1$$1$$0Trivial[\ ] 67.2.0.1x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2} 67.2.0.1x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.5.1$x^{6} - 67$$6$$1$$5$$C_6$$[\ ]_{6}$