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)

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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)

Normalized defining 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 Abelian group, 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:  \( \frac{44552}{42907} a^{17} - \frac{347127}{42907} a^{16} + \frac{1371712}{42907} a^{15} - \frac{3590437}{42907} a^{14} + \frac{6856553}{42907} a^{13} - \frac{10000699}{42907} a^{12} + \frac{11008669}{42907} a^{11} - \frac{8377199}{42907} a^{10} + \frac{3122579}{42907} a^{9} + \frac{1567297}{42907} a^{8} - \frac{2711090}{42907} a^{7} + \frac{817479}{42907} a^{6} + \frac{798600}{42907} a^{5} - \frac{844118}{42907} a^{4} + \frac{288684}{42907} a^{3} + \frac{44915}{42907} a^{2} + \frac{5002}{42907} a + \frac{11048}{42907} \),  \( \frac{103056}{42907} a^{17} - \frac{785240}{42907} a^{16} + \frac{3028250}{42907} a^{15} - \frac{7706213}{42907} a^{14} + \frac{14245108}{42907} a^{13} - \frac{20013687}{42907} a^{12} + \frac{20973622}{42907} a^{11} - \frac{14655840}{42907} a^{10} + \frac{4072874}{42907} a^{9} + \frac{3981656}{42907} a^{8} - \frac{4492848}{42907} a^{7} + \frac{212087}{42907} a^{6} + \frac{1860819}{42907} a^{5} - \frac{865909}{42907} a^{4} + \frac{155497}{42907} a^{3} - \frac{128075}{42907} a^{2} + \frac{172621}{42907} a + \frac{40309}{42907} \),  \( \frac{16775}{42907} a^{17} - \frac{99841}{42907} a^{16} + \frac{251103}{42907} a^{15} - \frac{205045}{42907} a^{14} - \frac{680366}{42907} a^{13} + \frac{2999797}{42907} a^{12} - \frac{6605929}{42907} a^{11} + \frac{10023305}{42907} a^{10} - \frac{10742221}{42907} a^{9} + \frac{7437037}{42907} a^{8} - \frac{1793494}{42907} a^{7} - \frac{2219586}{42907} a^{6} + \frac{2201511}{42907} a^{5} - \frac{221386}{42907} a^{4} - \frac{722634}{42907} a^{3} + \frac{596005}{42907} a^{2} - \frac{224641}{42907} a + \frac{88858}{42907} \),  \( \frac{126460}{42907} a^{17} - \frac{1099092}{42907} a^{16} + \frac{4802020}{42907} a^{15} - \frac{13850019}{42907} a^{14} + \frac{29223591}{42907} a^{13} - \frac{47450368}{42907} a^{12} + \frac{59867761}{42907} a^{11} - \frac{56720689}{42907} a^{10} + \frac{36180863}{42907} a^{9} - \frac{9044905}{42907} a^{8} - \frac{8141678}{42907} a^{7} + \frac{8367716}{42907} a^{6} - \frac{748705}{42907} a^{5} - \frac{3243252}{42907} a^{4} + \frac{2635400}{42907} a^{3} - \frac{1159898}{42907} a^{2} + \frac{497843}{42907} a - \frac{153089}{42907} \),  \( \frac{113472}{42907} a^{17} - \frac{893143}{42907} a^{16} + \frac{3534623}{42907} a^{15} - \frac{9194599}{42907} a^{14} + \frac{17315405}{42907} a^{13} - \frac{24668927}{42907} a^{12} + \frac{26127804}{42907} a^{11} - \frac{18273027}{42907} a^{10} + \frac{4490017}{42907} a^{9} + \frac{6352550}{42907} a^{8} - \frac{7180286}{42907} a^{7} + \frac{775524}{42907} a^{6} + \frac{3190271}{42907} a^{5} - \frac{2011005}{42907} a^{4} + \frac{220931}{42907} a^{3} + \frac{169188}{42907} a^{2} + \frac{74790}{42907} a + \frac{26111}{42907} \),  \( \frac{128105}{42907} a^{17} - \frac{1117729}{42907} a^{16} + \frac{4875929}{42907} a^{15} - \frac{14002867}{42907} a^{14} + \frac{29343745}{42907} a^{13} - \frac{47175108}{42907} a^{12} + \frac{58660989}{42907} a^{11} - \frac{54175007}{42907} a^{10} + \frac{32564582}{42907} a^{9} - \frac{5478206}{42907} a^{8} - \frac{10344732}{42907} a^{7} + \frac{8636927}{42907} a^{6} + \frac{31491}{42907} a^{5} - \frac{3864061}{42907} a^{4} + \frac{2665572}{42907} a^{3} - \frac{946458}{42907} a^{2} + \frac{366527}{42907} a - \frac{113472}{42907} \),  \( \frac{3808}{42907} a^{17} - \frac{55694}{42907} a^{16} + \frac{319704}{42907} a^{15} - \frac{1117729}{42907} a^{14} + \frac{2755247}{42907} a^{13} - \frac{5120354}{42907} a^{12} + \frac{7402952}{42907} a^{11} - \frac{8219522}{42907} a^{10} + \frac{6573697}{42907} a^{9} - \frac{3039795}{42907} a^{8} - \frac{364654}{42907} a^{7} + \frac{1599191}{42907} a^{6} - \frac{818861}{42907} a^{5} - \frac{223368}{42907} a^{4} + \frac{510863}{42907} a^{3} - \frac{232430}{42907} a^{2} + \frac{46671}{42907} a - \frac{71067}{42907} \),  \( \frac{3808}{42907} a^{17} - \frac{9042}{42907} a^{16} - \frac{53512}{42907} a^{15} + \frac{416009}{42907} a^{14} - \frac{1449639}{42907} a^{13} + \frac{3349338}{42907} a^{12} - \frac{5733010}{42907} a^{11} + \frac{7444957}{42907} a^{10} - \frac{7077256}{42907} a^{9} + \frac{4294520}{42907} a^{8} - \frac{546982}{42907} a^{7} - \frac{1692450}{42907} a^{6} + \frac{1372820}{42907} a^{5} + \frac{42420}{42907} a^{4} - \frac{708295}{42907} a^{3} + \frac{419735}{42907} a^{2} - \frac{53695}{42907} a + \frac{8434}{42907} \)
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

$C_3\wr C_3:C_2$ (as 18T88):

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

\(\Q(\sqrt{-11}) \), 3.1.44.1 x3, 6.0.21296.1, 9.1.1083199744.1 x3

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type 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$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$