# Properties

 Label 18.0.16481672572...4375.1 Degree $18$ Signature $[0, 9]$ Discriminant $-\,5^{4}\cdot 11^{4}\cdot 23^{9}$ Root discriminant $11.68$ Ramified primes $5, 11, 23$ 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, 14, -37, 52, -57, 46, -17, -8, 19, -11, 10, -5, 0, -5, -1, 2, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1)

gp: K = bnfinit(x^18 + 2*x^16 - x^15 - 5*x^14 - 5*x^12 + 10*x^11 - 11*x^10 + 19*x^9 - 8*x^8 - 17*x^7 + 46*x^6 - 57*x^5 + 52*x^4 - 37*x^3 + 14*x^2 - 3*x + 1, 1)

## Normalizeddefining polynomial

$$x^{18} + 2 x^{16} - x^{15} - 5 x^{14} - 5 x^{12} + 10 x^{11} - 11 x^{10} + 19 x^{9} - 8 x^{8} - 17 x^{7} + 46 x^{6} - 57 x^{5} + 52 x^{4} - 37 x^{3} + 14 x^{2} - 3 x + 1$$

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: $$-16481672572799864375=-\,5^{4}\cdot 11^{4}\cdot 23^{9}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $11.68$ 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: $5, 11, 23$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{2}{5}$, $\frac{1}{3029771545} a^{17} - \frac{292269546}{3029771545} a^{16} + \frac{113494807}{3029771545} a^{15} - \frac{676258248}{3029771545} a^{14} - \frac{624457397}{3029771545} a^{13} - \frac{1001474408}{3029771545} a^{12} - \frac{1233781428}{3029771545} a^{11} + \frac{701788097}{3029771545} a^{10} - \frac{217782827}{3029771545} a^{9} - \frac{804115889}{3029771545} a^{8} + \frac{257956867}{3029771545} a^{7} - \frac{138295421}{605954309} a^{6} + \frac{160462277}{605954309} a^{5} + \frac{490679776}{3029771545} a^{4} + \frac{23624492}{605954309} a^{3} - \frac{1190301754}{3029771545} a^{2} + \frac{1455677013}{3029771545} a + \frac{1387349104}{3029771545}$

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: $$102.074554739$$ 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2} 5.6.0.1x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6} 1111.6.0.1x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2} 11.6.0.1x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2} 23.2.1.2x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2} 23.6.3.2x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$