Properties

Label 18.0.96511468169...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 5^{6}$
Root discriminant $11.34$
Ramified primes $3, 5$
Class number $1$
Class group Trivial
Galois Group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, -9, 0, 36, -27, -57, 81, 45, -156, 72, 90, -111, 0, 84, -78, 36, -9, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3)
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 9 x^{17} \) \(\mathstrut +\mathstrut 36 x^{16} \) \(\mathstrut -\mathstrut 78 x^{15} \) \(\mathstrut +\mathstrut 84 x^{14} \) \(\mathstrut -\mathstrut 111 x^{12} \) \(\mathstrut +\mathstrut 90 x^{11} \) \(\mathstrut +\mathstrut 72 x^{10} \) \(\mathstrut -\mathstrut 156 x^{9} \) \(\mathstrut +\mathstrut 45 x^{8} \) \(\mathstrut +\mathstrut 81 x^{7} \) \(\mathstrut -\mathstrut 57 x^{6} \) \(\mathstrut -\mathstrut 27 x^{5} \) \(\mathstrut +\mathstrut 36 x^{4} \) \(\mathstrut -\mathstrut 9 x^{2} \) \(\mathstrut +\mathstrut 3 \)

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:  \(-9651146816936671875=-\,3^{31}\cdot 5^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.34$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{78986} a^{17} - \frac{5056}{39493} a^{16} - \frac{6819}{78986} a^{15} + \frac{16487}{78986} a^{14} + \frac{13397}{78986} a^{13} - \frac{3690}{39493} a^{12} - \frac{2755}{78986} a^{11} - \frac{4355}{39493} a^{10} + \frac{3399}{39493} a^{9} - \frac{2023}{78986} a^{8} - \frac{9480}{39493} a^{7} - \frac{13791}{39493} a^{6} - \frac{1719}{78986} a^{5} + \frac{29603}{78986} a^{4} - \frac{38077}{78986} a^{3} + \frac{30111}{78986} a^{2} + \frac{3137}{78986} a - \frac{19725}{78986}$

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:  \( \frac{91558}{39493} a^{17} - \frac{710971}{39493} a^{16} + \frac{2419908}{39493} a^{15} - \frac{4170958}{39493} a^{14} + \frac{2635470}{39493} a^{13} + \frac{5859851}{78986} a^{12} - \frac{11651433}{78986} a^{11} + \frac{90955}{39493} a^{10} + \frac{7189330}{39493} a^{9} - \frac{4817810}{39493} a^{8} - \frac{5776215}{78986} a^{7} + \frac{7992351}{78986} a^{6} + \frac{1365061}{78986} a^{5} - \frac{2382696}{39493} a^{4} + \frac{188074}{39493} a^{3} + \frac{607482}{39493} a^{2} - \frac{227751}{78986} a - \frac{446729}{78986} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{57180}{39493} a^{17} - \frac{882633}{78986} a^{16} + \frac{1504703}{39493} a^{15} - \frac{5274055}{78986} a^{14} + \frac{3662327}{78986} a^{13} + \frac{1534532}{39493} a^{12} - \frac{3745158}{39493} a^{11} + \frac{1035241}{39493} a^{10} + \frac{3376439}{39493} a^{9} - \frac{3199076}{39493} a^{8} - \frac{523866}{39493} a^{7} + \frac{2028338}{39493} a^{6} - \frac{389273}{39493} a^{5} - \frac{839286}{39493} a^{4} + \frac{525869}{78986} a^{3} + \frac{296755}{78986} a^{2} - \frac{122025}{39493} a - \frac{187291}{78986} \),  \( \frac{36879}{78986} a^{17} - \frac{132250}{39493} a^{16} + \frac{421438}{39493} a^{15} - \frac{696205}{39493} a^{14} + \frac{498929}{39493} a^{13} + \frac{453159}{78986} a^{12} - \frac{1526383}{78986} a^{11} + \frac{1086283}{78986} a^{10} + \frac{278329}{78986} a^{9} - \frac{1465181}{78986} a^{8} + \frac{1656431}{78986} a^{7} - \frac{54363}{78986} a^{6} - \frac{1548963}{78986} a^{5} + \frac{406949}{39493} a^{4} + \frac{281910}{39493} a^{3} - \frac{335993}{39493} a^{2} - \frac{71773}{39493} a + \frac{180757}{78986} \),  \( \frac{166339}{78986} a^{17} - \frac{1316367}{78986} a^{16} + \frac{4554507}{78986} a^{15} - \frac{3989953}{39493} a^{14} + \frac{2553081}{39493} a^{13} + \frac{5903049}{78986} a^{12} - \frac{6055762}{39493} a^{11} + \frac{527163}{39493} a^{10} + \frac{6994734}{39493} a^{9} - \frac{10133645}{78986} a^{8} - \frac{5208015}{78986} a^{7} + \frac{7956591}{78986} a^{6} + \frac{450459}{39493} a^{5} - \frac{4277039}{78986} a^{4} + \frac{197351}{39493} a^{3} + \frac{559347}{39493} a^{2} - \frac{47276}{39493} a - \frac{432251}{78986} \),  \( \frac{87879}{78986} a^{17} - \frac{335918}{39493} a^{16} + \frac{2231489}{78986} a^{15} - \frac{1825987}{39493} a^{14} + \frac{902909}{39493} a^{13} + \frac{1741205}{39493} a^{12} - \frac{5227631}{78986} a^{11} - \frac{815735}{39493} a^{10} + \frac{8521319}{78986} a^{9} - \frac{2182727}{39493} a^{8} - \frac{4359893}{78986} a^{7} + \frac{2233503}{39493} a^{6} + \frac{1615937}{78986} a^{5} - \frac{3239285}{78986} a^{4} + \frac{95843}{39493} a^{3} + \frac{382475}{39493} a^{2} + \frac{15283}{78986} a - \frac{381449}{78986} \),  \( \frac{39235}{39493} a^{17} - \frac{587679}{78986} a^{16} + \frac{1899391}{78986} a^{15} - \frac{1489136}{39493} a^{14} + \frac{650846}{39493} a^{13} + \frac{2741769}{78986} a^{12} - \frac{1619297}{39493} a^{11} - \frac{2574887}{78986} a^{10} + \frac{6483961}{78986} a^{9} - \frac{741842}{39493} a^{8} - \frac{4394627}{78986} a^{7} + \frac{2265933}{78986} a^{6} + \frac{1233362}{39493} a^{5} - \frac{2281951}{78986} a^{4} - \frac{246849}{39493} a^{3} + \frac{366920}{39493} a^{2} + \frac{79507}{78986} a - \frac{287545}{78986} \),  \( \frac{39749}{39493} a^{17} - \frac{298078}{39493} a^{16} + \frac{979353}{39493} a^{15} - \frac{1624292}{39493} a^{14} + \frac{2001625}{78986} a^{13} + \frac{954216}{39493} a^{12} - \frac{3503689}{78986} a^{11} - \frac{255110}{39493} a^{10} + \frac{2293190}{39493} a^{9} - \frac{2497017}{78986} a^{8} - \frac{825474}{39493} a^{7} + \frac{877101}{39493} a^{6} + \frac{502127}{78986} a^{5} - \frac{557190}{39493} a^{4} + \frac{7059}{39493} a^{3} + \frac{133041}{78986} a^{2} + \frac{13212}{39493} a - \frac{107471}{78986} \),  \( \frac{7488}{39493} a^{17} - \frac{50068}{39493} a^{16} + \frac{284005}{78986} a^{15} - \frac{356361}{78986} a^{14} + \frac{4516}{39493} a^{13} + \frac{265718}{39493} a^{12} - \frac{290545}{39493} a^{11} + \frac{241377}{78986} a^{10} - \frac{282557}{78986} a^{9} + \frac{389613}{78986} a^{8} + \frac{123334}{39493} a^{7} - \frac{301570}{39493} a^{6} - \frac{76140}{39493} a^{5} + \frac{657491}{78986} a^{4} - \frac{238683}{78986} a^{3} - \frac{73355}{39493} a^{2} - \frac{8479}{39493} a + \frac{3020}{39493} \),  \( \frac{67518}{39493} a^{17} - \frac{1040375}{78986} a^{16} + \frac{1741844}{39493} a^{15} - \frac{5766335}{78986} a^{14} + \frac{1452215}{39493} a^{13} + \frac{5806153}{78986} a^{12} - \frac{4897192}{39493} a^{11} + \frac{126962}{39493} a^{10} + \frac{5686710}{39493} a^{9} - \frac{7824361}{78986} a^{8} - \frac{4256107}{78986} a^{7} + \frac{6617209}{78986} a^{6} + \frac{45978}{39493} a^{5} - \frac{1624589}{39493} a^{4} + \frac{657251}{78986} a^{3} + \frac{329788}{39493} a^{2} - \frac{191451}{78986} a - \frac{216673}{78986} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 868.676171318 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.135.1, \(\Q(\zeta_{9})^+\), 6.0.54675.1, \(\Q(\zeta_{9})\), 9.3.1793613375.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$