Properties

Label 18.0.13415474159...1623.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 41^{4}$
Root discriminant $11.55$
Ramified primes $7, 41$
Class number $1$
Class group Trivial
Galois Group $C_3^3:C_6$ (as 18T85)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 12, -26, 25, -85, 220, -298, 300, -308, 302, -240, 152, -78, 37, -23, 15, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*x^2 + 1)
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 23*x^15 + 37*x^14 - 78*x^13 + 152*x^12 - 240*x^11 + 302*x^10 - 308*x^9 + 300*x^8 - 298*x^7 + 220*x^6 - 85*x^5 + 25*x^4 - 26*x^3 + 12*x^2 + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 6 x^{17} \) \(\mathstrut +\mathstrut 15 x^{16} \) \(\mathstrut -\mathstrut 23 x^{15} \) \(\mathstrut +\mathstrut 37 x^{14} \) \(\mathstrut -\mathstrut 78 x^{13} \) \(\mathstrut +\mathstrut 152 x^{12} \) \(\mathstrut -\mathstrut 240 x^{11} \) \(\mathstrut +\mathstrut 302 x^{10} \) \(\mathstrut -\mathstrut 308 x^{9} \) \(\mathstrut +\mathstrut 300 x^{8} \) \(\mathstrut -\mathstrut 298 x^{7} \) \(\mathstrut +\mathstrut 220 x^{6} \) \(\mathstrut -\mathstrut 85 x^{5} \) \(\mathstrut +\mathstrut 25 x^{4} \) \(\mathstrut -\mathstrut 26 x^{3} \) \(\mathstrut +\mathstrut 12 x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(-13415474159898041623=-\,7^{15}\cdot 41^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.55$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{7} - \frac{1}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{8} - \frac{1}{7} a$, $\frac{1}{91} a^{16} - \frac{2}{91} a^{15} - \frac{4}{91} a^{14} - \frac{4}{91} a^{13} + \frac{31}{91} a^{11} + \frac{29}{91} a^{10} + \frac{3}{91} a^{9} - \frac{5}{91} a^{8} + \frac{16}{91} a^{7} - \frac{36}{91} a^{6} + \frac{45}{91} a^{5} + \frac{6}{13} a^{4} + \frac{17}{91} a^{3} + \frac{3}{13} a^{2} - \frac{25}{91} a + \frac{45}{91}$, $\frac{1}{294931} a^{17} - \frac{899}{294931} a^{16} - \frac{17151}{294931} a^{15} + \frac{18300}{294931} a^{14} - \frac{57}{22687} a^{13} - \frac{18975}{294931} a^{12} - \frac{119038}{294931} a^{11} - \frac{140228}{294931} a^{10} - \frac{298}{637} a^{9} - \frac{129022}{294931} a^{8} - \frac{3845}{294931} a^{7} + \frac{88575}{294931} a^{6} + \frac{54447}{294931} a^{5} - \frac{116450}{294931} a^{4} - \frac{10574}{294931} a^{3} + \frac{11246}{294931} a^{2} + \frac{40085}{294931} a - \frac{6659}{22687}$

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:  \( -\frac{5238}{6019} a^{17} + \frac{25717}{6019} a^{16} - \frac{49882}{6019} a^{15} + \frac{450633}{42133} a^{14} - \frac{866070}{42133} a^{13} + \frac{276177}{6019} a^{12} - \frac{3453501}{42133} a^{11} + \frac{5005620}{42133} a^{10} - \frac{12169}{91} a^{9} + \frac{5292185}{42133} a^{8} - \frac{801380}{6019} a^{7} + \frac{5328067}{42133} a^{6} - \frac{2711816}{42133} a^{5} + \frac{112612}{6019} a^{4} - \frac{803035}{42133} a^{3} + \frac{465427}{42133} a^{2} - \frac{58488}{42133} a + \frac{70982}{42133} \) (order $14$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{156918}{294931} a^{17} - \frac{876586}{294931} a^{16} + \frac{1934731}{294931} a^{15} - \frac{2652281}{294931} a^{14} + \frac{4664918}{294931} a^{13} - \frac{10386791}{294931} a^{12} + \frac{19164477}{294931} a^{11} - \frac{28939879}{294931} a^{10} + \frac{75149}{637} a^{9} - \frac{34123126}{294931} a^{8} + \frac{35055988}{294931} a^{7} - \frac{34996763}{294931} a^{6} + \frac{21316868}{294931} a^{5} - \frac{7807123}{294931} a^{4} + \frac{5741516}{294931} a^{3} - \frac{3281877}{294931} a^{2} + \frac{80798}{294931} a - \frac{649672}{294931} \),  \( \frac{115119}{294931} a^{17} - \frac{690702}{294931} a^{16} + \frac{1680346}{294931} a^{15} - \frac{188091}{22687} a^{14} + \frac{3947479}{294931} a^{13} - \frac{8634905}{294931} a^{12} + \frac{16638393}{294931} a^{11} - \frac{1979864}{22687} a^{10} + \frac{68493}{637} a^{9} - \frac{31236407}{294931} a^{8} + \frac{2337301}{22687} a^{7} - \frac{30750851}{294931} a^{6} + \frac{20973369}{294931} a^{5} - \frac{6396347}{294931} a^{4} + \frac{2300548}{294931} a^{3} - \frac{2251394}{294931} a^{2} + \frac{430645}{294931} a - \frac{60227}{294931} \),  \( \frac{28540}{294931} a^{17} - \frac{27632}{294931} a^{16} - \frac{20544}{22687} a^{15} + \frac{581471}{294931} a^{14} - \frac{302028}{294931} a^{13} + \frac{620944}{294931} a^{12} - \frac{2497491}{294931} a^{11} + \frac{4822205}{294931} a^{10} - \frac{15810}{637} a^{9} + \frac{572675}{22687} a^{8} - \frac{4701972}{294931} a^{7} + \frac{514503}{22687} a^{6} - \frac{8085811}{294931} a^{5} + \frac{723193}{294931} a^{4} + \frac{177048}{22687} a^{3} + \frac{1485747}{294931} a^{2} - \frac{925411}{294931} a - \frac{370872}{294931} \),  \( \frac{15674}{294931} a^{17} - \frac{177313}{294931} a^{16} + \frac{596315}{294931} a^{15} - \frac{888816}{294931} a^{14} + \frac{1028707}{294931} a^{13} - \frac{2398884}{294931} a^{12} + \frac{5373533}{294931} a^{11} - \frac{8547424}{294931} a^{10} + \frac{22536}{637} a^{9} - \frac{9014038}{294931} a^{8} + \frac{6585577}{294931} a^{7} - \frac{7554704}{294931} a^{6} + \frac{5618257}{294931} a^{5} + \frac{2057946}{294931} a^{4} - \frac{3528067}{294931} a^{3} - \frac{400347}{294931} a^{2} + \frac{730978}{294931} a + \frac{479642}{294931} \),  \( \frac{12927}{42133} a^{17} - \frac{57485}{42133} a^{16} + \frac{98433}{42133} a^{15} - \frac{16365}{6019} a^{14} + \frac{33218}{6019} a^{13} - \frac{527190}{42133} a^{12} + \frac{923109}{42133} a^{11} - \frac{1244949}{42133} a^{10} + \frac{366}{13} a^{9} - \frac{864137}{42133} a^{8} + \frac{115900}{6019} a^{7} - \frac{583078}{42133} a^{6} - \frac{26527}{6019} a^{5} + \frac{596779}{42133} a^{4} - \frac{49739}{6019} a^{3} + \frac{173760}{42133} a^{2} - \frac{91830}{42133} a + \frac{3186}{42133} \),  \( \frac{136770}{294931} a^{17} - \frac{582552}{294931} a^{16} + \frac{1026938}{294931} a^{15} - \frac{1445905}{294931} a^{14} + \frac{2981220}{294931} a^{13} - \frac{6095767}{294931} a^{12} + \frac{10752814}{294931} a^{11} - \frac{15699231}{294931} a^{10} + \frac{38148}{637} a^{9} - \frac{17525507}{294931} a^{8} + \frac{18578181}{294931} a^{7} - \frac{15418947}{294931} a^{6} + \frac{8546647}{294931} a^{5} - \frac{3989068}{294931} a^{4} + \frac{1519551}{294931} a^{3} + \frac{79424}{294931} a^{2} + \frac{436000}{294931} a - \frac{152515}{294931} \),  \( \frac{159195}{294931} a^{17} - \frac{810477}{294931} a^{16} + \frac{1462768}{294931} a^{15} - \frac{1414656}{294931} a^{14} + \frac{2825334}{294931} a^{13} - \frac{7541497}{294931} a^{12} + \frac{12953266}{294931} a^{11} - \frac{16675012}{294931} a^{10} + \frac{33281}{637} a^{9} - \frac{9708081}{294931} a^{8} + \frac{10646893}{294931} a^{7} - \frac{11598898}{294931} a^{6} - \frac{2597384}{294931} a^{5} + \frac{9304640}{294931} a^{4} + \frac{439631}{294931} a^{3} - \frac{145499}{22687} a^{2} - \frac{1921637}{294931} a - \frac{372867}{294931} \),  \( \frac{77771}{294931} a^{17} - \frac{351305}{294931} a^{16} + \frac{621606}{294931} a^{15} - \frac{814798}{294931} a^{14} + \frac{1766255}{294931} a^{13} - \frac{3788370}{294931} a^{12} + \frac{6473814}{294931} a^{11} - \frac{9436270}{294931} a^{10} + \frac{1796}{49} a^{9} - \frac{10956132}{294931} a^{8} + \frac{12722295}{294931} a^{7} - \frac{11587495}{294931} a^{6} + \frac{503005}{22687} a^{5} - \frac{4358842}{294931} a^{4} + \frac{3511343}{294931} a^{3} - \frac{1011545}{294931} a^{2} + \frac{496569}{294931} a - \frac{203439}{294931} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 765.05787444 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3^3:C_6$ (as 18T85):

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 $C_3^3:C_6$
Character table for $C_3^3:C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.3.1384375783.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$