Properties

Label 18.0.11982410022...8867.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,31^{6}\cdot 67^{5}$
Root discriminant $10.10$
Ramified primes $31, 67$
Class number $1$
Class group Trivial
Galois Group 18T284

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 22, -41, 49, -39, 29, -40, 62, -65, 45, -23, 15, -16, 17, -14, 9, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1)
gp: K = bnfinit(x^18 - 4*x^17 + 9*x^16 - 14*x^15 + 17*x^14 - 16*x^13 + 15*x^12 - 23*x^11 + 45*x^10 - 65*x^9 + 62*x^8 - 40*x^7 + 29*x^6 - 39*x^5 + 49*x^4 - 41*x^3 + 22*x^2 - 7*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 9 x^{16} \) \(\mathstrut -\mathstrut 14 x^{15} \) \(\mathstrut +\mathstrut 17 x^{14} \) \(\mathstrut -\mathstrut 16 x^{13} \) \(\mathstrut +\mathstrut 15 x^{12} \) \(\mathstrut -\mathstrut 23 x^{11} \) \(\mathstrut +\mathstrut 45 x^{10} \) \(\mathstrut -\mathstrut 65 x^{9} \) \(\mathstrut +\mathstrut 62 x^{8} \) \(\mathstrut -\mathstrut 40 x^{7} \) \(\mathstrut +\mathstrut 29 x^{6} \) \(\mathstrut -\mathstrut 39 x^{5} \) \(\mathstrut +\mathstrut 49 x^{4} \) \(\mathstrut -\mathstrut 41 x^{3} \) \(\mathstrut +\mathstrut 22 x^{2} \) \(\mathstrut -\mathstrut 7 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-1198241002273018867=-\,31^{6}\cdot 67^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.10$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $31, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{6457} a^{17} + \frac{2237}{6457} a^{16} + \frac{2494}{6457} a^{15} - \frac{2722}{6457} a^{14} + \frac{1880}{6457} a^{13} + \frac{3100}{6457} a^{12} - \frac{617}{6457} a^{11} - \frac{922}{6457} a^{10} + \frac{83}{6457} a^{9} - \frac{1315}{6457} a^{8} - \frac{2461}{6457} a^{7} - \frac{863}{6457} a^{6} + \frac{286}{587} a^{5} - \frac{897}{6457} a^{4} - \frac{2001}{6457} a^{3} - \frac{284}{587} a^{2} - \frac{134}{587} a + \frac{2743}{6457}$

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:  \( \frac{80980}{6457} a^{17} - \frac{270069}{6457} a^{16} + \frac{544462}{6457} a^{15} - \frac{760420}{6457} a^{14} + \frac{851578}{6457} a^{13} - \frac{707516}{6457} a^{12} + \frac{722790}{6457} a^{11} - \frac{1370153}{6457} a^{10} + \frac{2711543}{6457} a^{9} - \frac{3409152}{6457} a^{8} + \frac{2657352}{6457} a^{7} - \frac{1389884}{6457} a^{6} + \frac{126400}{587} a^{5} - \frac{2225475}{6457} a^{4} + \frac{2438181}{6457} a^{3} - \frac{148171}{587} a^{2} + \frac{59249}{587} a - \frac{121800}{6457} \),  \( a \),  \( \frac{86349}{6457} a^{17} - \frac{282550}{6457} a^{16} + \frac{562301}{6457} a^{15} - \frac{775561}{6457} a^{14} + \frac{859464}{6457} a^{13} - \frac{703305}{6457} a^{12} + \frac{729015}{6457} a^{11} - \frac{1419508}{6457} a^{10} + \frac{2802035}{6457} a^{9} - \frac{3457085}{6457} a^{8} + \frac{2616523}{6457} a^{7} - \frac{1329092}{6457} a^{6} + \frac{128690}{587} a^{5} - \frac{2295573}{6457} a^{4} + \frac{2452174}{6457} a^{3} - \frac{143832}{587} a^{2} + \frac{54769}{587} a - \frac{103679}{6457} \),  \( \frac{23316}{6457} a^{17} - \frac{92152}{6457} a^{16} + \frac{204986}{6457} a^{15} - \frac{310235}{6457} a^{14} + \frac{365556}{6457} a^{13} - \frac{329365}{6457} a^{12} + \frac{303703}{6457} a^{11} - \frac{492731}{6457} a^{10} + \frac{1005420}{6457} a^{9} - \frac{1436158}{6457} a^{8} + \frac{1300540}{6457} a^{7} - \frac{737794}{6457} a^{6} + \frac{47603}{587} a^{5} - \frac{833182}{6457} a^{4} + \frac{1081285}{6457} a^{3} - \frac{75520}{587} a^{2} + \frac{33716}{587} a - \frac{78281}{6457} \),  \( \frac{63887}{6457} a^{17} - \frac{217100}{6457} a^{16} + \frac{446779}{6457} a^{15} - \frac{633276}{6457} a^{14} + \frac{717630}{6457} a^{13} - \frac{606362}{6457} a^{12} + \frac{608664}{6457} a^{11} - \frac{1113664}{6457} a^{10} + \frac{2216175}{6457} a^{9} - \frac{2853372}{6457} a^{8} + \frac{2307192}{6457} a^{7} - \frac{1231445}{6457} a^{6} + \frac{103445}{587} a^{5} - \frac{1815181}{6457} a^{4} + \frac{2064039}{6457} a^{3} - \frac{129465}{587} a^{2} + \frac{52780}{587} a - \frac{110708}{6457} \),  \( \frac{77169}{6457} a^{17} - \frac{278493}{6457} a^{16} + \frac{576817}{6457} a^{15} - \frac{827847}{6457} a^{14} + \frac{938109}{6457} a^{13} - \frac{802161}{6457} a^{12} + \frac{775485}{6457} a^{11} - \frac{1420675}{6457} a^{10} + \frac{2860134}{6457} a^{9} - \frac{3769911}{6457} a^{8} + \frac{3060993}{6457} a^{7} - \frac{1607142}{6457} a^{6} + \frac{130035}{587} a^{5} - \frac{2377729}{6457} a^{4} + \frac{2741754}{6457} a^{3} - \frac{172342}{587} a^{2} + \frac{69799}{587} a - \frac{140861}{6457} \),  \( \frac{172}{587} a^{17} + \frac{279}{587} a^{16} - \frac{129}{587} a^{15} + \frac{242}{587} a^{14} + \frac{510}{587} a^{13} - \frac{383}{587} a^{12} + \frac{1884}{587} a^{11} - \frac{94}{587} a^{10} - \frac{986}{587} a^{9} + \frac{2750}{587} a^{8} + \frac{4044}{587} a^{7} - \frac{6382}{587} a^{6} + \frac{3420}{587} a^{5} + \frac{4206}{587} a^{4} - \frac{190}{587} a^{3} - \frac{3745}{587} a^{2} + \frac{4165}{587} a - \frac{1326}{587} \),  \( \frac{74523}{6457} a^{17} - \frac{244241}{6457} a^{16} + \frac{486349}{6457} a^{15} - \frac{670022}{6457} a^{14} + \frac{741809}{6457} a^{13} - \frac{604204}{6457} a^{12} + \frac{625935}{6457} a^{11} - \frac{1221642}{6457} a^{10} + \frac{2420978}{6457} a^{9} - \frac{2989447}{6457} a^{8} + \frac{2257018}{6457} a^{7} - \frac{1131604}{6457} a^{6} + \frac{109377}{587} a^{5} - \frac{1973652}{6457} a^{4} + \frac{2121788}{6457} a^{3} - \frac{124104}{587} a^{2} + \frac{46335}{587} a - \frac{83058}{6457} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 22.3834819773 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T284:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 98 conjugacy class representatives for t18n284 are not computed
Character table for t18n284 is not computed

Intermediate fields

3.1.31.1, 6.0.64387.1, 9.3.133731799.1

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

Sibling fields

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 $18$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ R ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.5.1$x^{6} - 67$$6$$1$$5$$C_6$$[\ ]_{6}$