Properties

Label 18.0.55850279261...1424.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 37^{6}$
Root discriminant $11.00$
Ramified primes $2, 3, 37$
Class number $1$
Class group Trivial
Galois Group 18T189

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 13, -22, 29, -31, 28, -24, 25, -28, 28, -24, 17, -13, 10, -7, 5, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*x + 1)
gp: K = bnfinit(x^18 - 2*x^17 + 5*x^16 - 7*x^15 + 10*x^14 - 13*x^13 + 17*x^12 - 24*x^11 + 28*x^10 - 28*x^9 + 25*x^8 - 24*x^7 + 28*x^6 - 31*x^5 + 29*x^4 - 22*x^3 + 13*x^2 - 5*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 2 x^{17} \) \(\mathstrut +\mathstrut 5 x^{16} \) \(\mathstrut -\mathstrut 7 x^{15} \) \(\mathstrut +\mathstrut 10 x^{14} \) \(\mathstrut -\mathstrut 13 x^{13} \) \(\mathstrut +\mathstrut 17 x^{12} \) \(\mathstrut -\mathstrut 24 x^{11} \) \(\mathstrut +\mathstrut 28 x^{10} \) \(\mathstrut -\mathstrut 28 x^{9} \) \(\mathstrut +\mathstrut 25 x^{8} \) \(\mathstrut -\mathstrut 24 x^{7} \) \(\mathstrut +\mathstrut 28 x^{6} \) \(\mathstrut -\mathstrut 31 x^{5} \) \(\mathstrut +\mathstrut 29 x^{4} \) \(\mathstrut -\mathstrut 22 x^{3} \) \(\mathstrut +\mathstrut 13 x^{2} \) \(\mathstrut -\mathstrut 5 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-5585027926119911424=-\,2^{12}\cdot 3^{12}\cdot 37^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.00$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2507} a^{17} - \frac{1001}{2507} a^{16} - \frac{289}{2507} a^{15} + \frac{399}{2507} a^{14} + \frac{22}{2507} a^{13} + \frac{572}{2507} a^{12} + \frac{185}{2507} a^{11} + \frac{679}{2507} a^{10} + \frac{48}{109} a^{9} + \frac{156}{2507} a^{8} - \frac{385}{2507} a^{7} + \frac{1020}{2507} a^{6} - \frac{1110}{2507} a^{5} + \frac{765}{2507} a^{4} + \frac{429}{2507} a^{3} + \frac{104}{2507} a^{2} - \frac{1096}{2507} a - \frac{660}{2507}$

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:  \( -\frac{82}{23} a^{17} + \frac{179}{23} a^{16} - \frac{383}{23} a^{15} + \frac{586}{23} a^{14} - \frac{723}{23} a^{13} + \frac{1028}{23} a^{12} - \frac{1278}{23} a^{11} + \frac{1845}{23} a^{10} - 94 a^{9} + \frac{1951}{23} a^{8} - \frac{1780}{23} a^{7} + \frac{1690}{23} a^{6} - \frac{2107}{23} a^{5} + \frac{2337}{23} a^{4} - \frac{2012}{23} a^{3} + \frac{1477}{23} a^{2} - \frac{771}{23} a + \frac{231}{23} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2562}{2507} a^{17} + \frac{99}{2507} a^{16} + \frac{6668}{2507} a^{15} + \frac{1889}{2507} a^{14} + \frac{6224}{2507} a^{13} - \frac{1131}{2507} a^{12} + \frac{5161}{2507} a^{11} - \frac{10288}{2507} a^{10} - \frac{194}{109} a^{9} + \frac{3566}{2507} a^{8} - \frac{6133}{2507} a^{7} + \frac{946}{2507} a^{6} + \frac{9146}{2507} a^{5} + \frac{4470}{2507} a^{4} - \frac{8996}{2507} a^{3} + \frac{13241}{2507} a^{2} - \frac{12647}{2507} a + \frac{6319}{2507} \),  \( \frac{6319}{2507} a^{17} - \frac{15200}{2507} a^{16} + \frac{31496}{2507} a^{15} - \frac{50901}{2507} a^{14} + \frac{61301}{2507} a^{13} - \frac{88371}{2507} a^{12} + \frac{108554}{2507} a^{11} - \frac{156817}{2507} a^{10} + \frac{8140}{109} a^{9} - \frac{172470}{2507} a^{8} + \frac{154409}{2507} a^{7} - \frac{145523}{2507} a^{6} + \frac{175986}{2507} a^{5} - \frac{205035}{2507} a^{4} + \frac{178781}{2507} a^{3} - \frac{130022}{2507} a^{2} + \frac{68906}{2507} a - \frac{18948}{2507} \),  \( \frac{1093}{2507} a^{17} - \frac{6055}{2507} a^{16} + \frac{7526}{2507} a^{15} - \frac{20167}{2507} a^{14} + \frac{16525}{2507} a^{13} - \frac{31638}{2507} a^{12} + \frac{34236}{2507} a^{11} - \frac{50065}{2507} a^{10} + \frac{2978}{109} a^{9} - \frac{55122}{2507} a^{8} + \frac{55525}{2507} a^{7} - \frac{45881}{2507} a^{6} + \frac{57819}{2507} a^{5} - \frac{73896}{2507} a^{4} + \frac{57749}{2507} a^{3} - \frac{46776}{2507} a^{2} + \frac{22981}{2507} a - \frac{6885}{2507} \),  \( \frac{4101}{2507} a^{17} + \frac{6379}{2507} a^{16} + \frac{5636}{2507} a^{15} + \frac{24298}{2507} a^{14} - \frac{5044}{2507} a^{13} + \frac{29304}{2507} a^{12} - \frac{26006}{2507} a^{11} + \frac{29386}{2507} a^{10} - \frac{3494}{109} a^{9} + \frac{63146}{2507} a^{8} - \frac{69671}{2507} a^{7} + \frac{46470}{2507} a^{6} - \frac{42017}{2507} a^{5} + \frac{83739}{2507} a^{4} - \frac{78302}{2507} a^{3} + \frac{73017}{2507} a^{2} - \frac{47278}{2507} a + \frac{20956}{2507} \),  \( \frac{3339}{2507} a^{17} - \frac{5522}{2507} a^{16} + \frac{15266}{2507} a^{15} - \frac{19012}{2507} a^{14} + \frac{28332}{2507} a^{13} - \frac{35524}{2507} a^{12} + \frac{46119}{2507} a^{11} - \frac{66836}{2507} a^{10} + \frac{3203}{109} a^{9} - \frac{73275}{2507} a^{8} + \frac{63251}{2507} a^{7} - \frac{58894}{2507} a^{6} + \frac{74266}{2507} a^{5} - \frac{80522}{2507} a^{4} + \frac{76144}{2507} a^{3} - \frac{53864}{2507} a^{2} + \frac{25746}{2507} a - \frac{7608}{2507} \),  \( \frac{8321}{2507} a^{17} - \frac{6081}{2507} a^{16} + \frac{27021}{2507} a^{15} - \frac{16738}{2507} a^{14} + \frac{37656}{2507} a^{13} - \frac{38786}{2507} a^{12} + \frac{55241}{2507} a^{11} - \frac{86057}{2507} a^{10} + \frac{2866}{109} a^{9} - \frac{60718}{2507} a^{8} + \frac{50501}{2507} a^{7} - \frac{61450}{2507} a^{6} + \frac{92237}{2507} a^{5} - \frac{72411}{2507} a^{4} + \frac{54895}{2507} a^{3} - \frac{24601}{2507} a^{2} + \frac{3157}{2507} a + \frac{5991}{2507} \),  \( \frac{1971}{2507} a^{17} - \frac{4976}{2507} a^{16} + \frac{9498}{2507} a^{15} - \frac{15811}{2507} a^{14} + \frac{18292}{2507} a^{13} - \frac{25808}{2507} a^{12} + \frac{33711}{2507} a^{11} - \frac{45555}{2507} a^{10} + \frac{2503}{109} a^{9} - \frac{48518}{2507} a^{8} + \frac{43405}{2507} a^{7} - \frac{42813}{2507} a^{6} + \frac{50941}{2507} a^{5} - \frac{61567}{2507} a^{4} + \frac{50840}{2507} a^{3} - \frac{33181}{2507} a^{2} + \frac{18367}{2507} a - \frac{4741}{2507} \),  \( \frac{1548}{2507} a^{17} + \frac{4792}{2507} a^{16} - \frac{1126}{2507} a^{15} + \frac{18479}{2507} a^{14} - \frac{11070}{2507} a^{13} + \frac{25555}{2507} a^{12} - \frac{26995}{2507} a^{11} + \frac{33250}{2507} a^{10} - \frac{2868}{109} a^{9} + \frac{55970}{2507} a^{8} - \frac{56975}{2507} a^{7} + \frac{39662}{2507} a^{6} - \frac{41097}{2507} a^{5} + \frac{71112}{2507} a^{4} - \frac{65445}{2507} a^{3} + \frac{58205}{2507} a^{2} - \frac{34467}{2507} a + \frac{13711}{2507} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 172.576549312 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T189:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 648
The 54 conjugacy class representatives for t18n189 are not computed
Character table for t18n189 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 6.0.36963.1, 6.0.1774224.1

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

Sibling fields

Degree 12 siblings: 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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.18$x^{12} + 80 x^{10} + 81 x^{8} - 160 x^{6} - 117 x^{4} + 80 x^{2} + 227$$2$$6$$12$$D_4 \times C_3$$[2, 2]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.9.2$x^{12} - 9 x^{4} + 27$$4$$3$$9$$D_4 \times C_3$$[\ ]_{4}^{6}$
$37$37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$