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Properties

Label 18.0.84728860944...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 5^{12}$
Root discriminant $21.35$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 9, 3, -3, -60, -6, 175, -450, 805, -680, 130, 156, -69, -43, 24, 10, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 2*x^16 + 10*x^15 + 24*x^14 - 43*x^13 - 69*x^12 + 156*x^11 + 130*x^10 - 680*x^9 + 805*x^8 - 450*x^7 + 175*x^6 - 6*x^5 - 60*x^4 - 3*x^3 + 3*x^2 + 9*x + 3)
 
gp: K = bnfinit(x^18 - 2*x^17 - 2*x^16 + 10*x^15 + 24*x^14 - 43*x^13 - 69*x^12 + 156*x^11 + 130*x^10 - 680*x^9 + 805*x^8 - 450*x^7 + 175*x^6 - 6*x^5 - 60*x^4 - 3*x^3 + 3*x^2 + 9*x + 3, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 2 x^{16} + 10 x^{15} + 24 x^{14} - 43 x^{13} - 69 x^{12} + 156 x^{11} + 130 x^{10} - 680 x^{9} + 805 x^{8} - 450 x^{7} + 175 x^{6} - 6 x^{5} - 60 x^{4} - 3 x^{3} + 3 x^{2} + 9 x + 3 \)

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:  \(-847288609443000000000000=-\,2^{12}\cdot 3^{25}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.35$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{218458256091775424489} a^{17} + \frac{35437208218052967402}{218458256091775424489} a^{16} - \frac{68104202272546478409}{218458256091775424489} a^{15} - \frac{106706387629582875899}{218458256091775424489} a^{14} - \frac{219980036848180923}{506863703229177319} a^{13} + \frac{11935570018206651480}{218458256091775424489} a^{12} + \frac{53470385960747108474}{218458256091775424489} a^{11} - \frac{11265827534764990455}{218458256091775424489} a^{10} + \frac{22764094189980431631}{218458256091775424489} a^{9} - \frac{37126219967119587224}{218458256091775424489} a^{8} - \frac{13197265672834334750}{218458256091775424489} a^{7} + \frac{80895362089155170585}{218458256091775424489} a^{6} - \frac{76335472171288650967}{218458256091775424489} a^{5} - \frac{52953520179728259006}{218458256091775424489} a^{4} - \frac{33183095781611480056}{218458256091775424489} a^{3} + \frac{8493542578492448786}{218458256091775424489} a^{2} - \frac{27512226492239381864}{218458256091775424489} a + \frac{101917389434281753693}{218458256091775424489}$

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:  \( -\frac{4499672331136}{9898264731071} a^{17} + \frac{6966026961587}{9898264731071} a^{16} + \frac{10581090326983}{9898264731071} a^{15} - \frac{38157782026021}{9898264731071} a^{14} - \frac{280825496280}{22965811441} a^{13} + \frac{126554046684003}{9898264731071} a^{12} + \frac{322436177093746}{9898264731071} a^{11} - \frac{522777760434579}{9898264731071} a^{10} - \frac{699250409001240}{9898264731071} a^{9} + \frac{2592547297272655}{9898264731071} a^{8} - \frac{2734724375293126}{9898264731071} a^{7} + \frac{1621095042901538}{9898264731071} a^{6} - \frac{792661021770507}{9898264731071} a^{5} + \frac{59591092618152}{9898264731071} a^{4} + \frac{68583162072937}{9898264731071} a^{3} + \frac{58461386493279}{9898264731071} a^{2} + \frac{22685023915530}{9898264731071} a + \frac{1235379096877}{9898264731071} \) (order $6$)
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:  \( 160070.62698640698 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 6.0.270000.1, 9.3.177147000000.3

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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.11.10$x^{6} + 18 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[5/2]_{2}^{3}$
3.6.11.10$x^{6} + 18 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[5/2]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$