Properties

Label 18.0.12827848719...6283.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 1297^{2}$
Root discriminant $11.52$
Ramified primes $3, 1297$
Class number $1$
Class group Trivial
Galois Group 18T286

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 39, -106, 198, -270, 274, -186, 63, 43, -87, 90, -47, 18, 3, -9, 6, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 3*x^14 + 18*x^13 - 47*x^12 + 90*x^11 - 87*x^10 + 43*x^9 + 63*x^8 - 186*x^7 + 274*x^6 - 270*x^5 + 198*x^4 - 106*x^3 + 39*x^2 - 9*x + 1)
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 - 9*x^15 + 3*x^14 + 18*x^13 - 47*x^12 + 90*x^11 - 87*x^10 + 43*x^9 + 63*x^8 - 186*x^7 + 274*x^6 - 270*x^5 + 198*x^4 - 106*x^3 + 39*x^2 - 9*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 9 x^{15} \) \(\mathstrut +\mathstrut 3 x^{14} \) \(\mathstrut +\mathstrut 18 x^{13} \) \(\mathstrut -\mathstrut 47 x^{12} \) \(\mathstrut +\mathstrut 90 x^{11} \) \(\mathstrut -\mathstrut 87 x^{10} \) \(\mathstrut +\mathstrut 43 x^{9} \) \(\mathstrut +\mathstrut 63 x^{8} \) \(\mathstrut -\mathstrut 186 x^{7} \) \(\mathstrut +\mathstrut 274 x^{6} \) \(\mathstrut -\mathstrut 270 x^{5} \) \(\mathstrut +\mathstrut 198 x^{4} \) \(\mathstrut -\mathstrut 106 x^{3} \) \(\mathstrut +\mathstrut 39 x^{2} \) \(\mathstrut -\mathstrut 9 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-12827848719622496283=-\,3^{27}\cdot 1297^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.52$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{11331399151} a^{17} - \frac{5012802240}{11331399151} a^{16} - \frac{332300229}{11331399151} a^{15} - \frac{3807216236}{11331399151} a^{14} + \frac{1648854709}{11331399151} a^{13} - \frac{76066867}{11331399151} a^{12} - \frac{3914875299}{11331399151} a^{11} + \frac{3602980441}{11331399151} a^{10} + \frac{1405698519}{11331399151} a^{9} + \frac{204808519}{11331399151} a^{8} + \frac{2080157976}{11331399151} a^{7} + \frac{1759198270}{11331399151} a^{6} + \frac{4074136241}{11331399151} a^{5} - \frac{1683044718}{11331399151} a^{4} + \frac{1573467686}{11331399151} a^{3} + \frac{2111109130}{11331399151} a^{2} - \frac{2166427783}{11331399151} a + \frac{2225142671}{11331399151}$

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:  \( -\frac{35352706938}{11331399151} a^{17} + \frac{90854094538}{11331399151} a^{16} - \frac{172876998924}{11331399151} a^{15} + \frac{242261203095}{11331399151} a^{14} + \frac{1668131236}{11331399151} a^{13} - \frac{641319898088}{11331399151} a^{12} + \frac{1391708984120}{11331399151} a^{11} - \frac{2577536726340}{11331399151} a^{10} + \frac{1939234074618}{11331399151} a^{9} - \frac{636677436344}{11331399151} a^{8} - \frac{2576010774093}{11331399151} a^{7} + \frac{5507921744460}{11331399151} a^{6} - \frac{7298219065135}{11331399151} a^{5} + \frac{6299662986481}{11331399151} a^{4} - \frac{4104563025992}{11331399151} a^{3} + \frac{1784458218203}{11331399151} a^{2} - \frac{470870344327}{11331399151} a + \frac{52125926226}{11331399151} \) (order $18$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{2884678815}{11331399151} a^{17} - \frac{9893467934}{11331399151} a^{16} + \frac{19740715892}{11331399151} a^{15} - \frac{31773519793}{11331399151} a^{14} + \frac{15586843113}{11331399151} a^{13} + \frac{53485492941}{11331399151} a^{12} - \frac{153869065958}{11331399151} a^{11} + \frac{303806366951}{11331399151} a^{10} - \frac{334641987141}{11331399151} a^{9} + \frac{165353746840}{11331399151} a^{8} + \frac{137171206611}{11331399151} a^{7} - \frac{653180032275}{11331399151} a^{6} + \frac{934723938823}{11331399151} a^{5} - \frac{1023628925252}{11331399151} a^{4} + \frac{755289526404}{11331399151} a^{3} - \frac{441706773487}{11331399151} a^{2} + \frac{148222558103}{11331399151} a - \frac{37271628153}{11331399151} \),  \( \frac{926557419}{11331399151} a^{17} - \frac{3853869677}{11331399151} a^{16} + \frac{5476737664}{11331399151} a^{15} - \frac{7374235789}{11331399151} a^{14} - \frac{1266738565}{11331399151} a^{13} + \frac{30922755043}{11331399151} a^{12} - \frac{55192567214}{11331399151} a^{11} + \frac{74456980901}{11331399151} a^{10} - \frac{65734171837}{11331399151} a^{9} - \frac{66165146237}{11331399151} a^{8} + \frac{112542532672}{11331399151} a^{7} - \frac{250622831263}{11331399151} a^{6} + \frac{195307182463}{11331399151} a^{5} - \frac{118569214583}{11331399151} a^{4} - \frac{36309219882}{11331399151} a^{3} + \frac{73697934999}{11331399151} a^{2} - \frac{75031697454}{11331399151} a + \frac{17580354862}{11331399151} \),  \( \frac{10331380776}{11331399151} a^{17} - \frac{11401177908}{11331399151} a^{16} + \frac{12473210074}{11331399151} a^{15} - \frac{715052709}{11331399151} a^{14} - \frac{96626640346}{11331399151} a^{13} + \frac{174297648689}{11331399151} a^{12} - \frac{123883705620}{11331399151} a^{11} + \frac{177964529087}{11331399151} a^{10} + \frac{474917411340}{11331399151} a^{9} - \frac{528807965415}{11331399151} a^{8} + \frac{878852757781}{11331399151} a^{7} - \frac{452268740815}{11331399151} a^{6} - \frac{170265055513}{11331399151} a^{5} + \frac{1013122705216}{11331399151} a^{4} - \frac{1107331774314}{11331399151} a^{3} + \frac{838912062078}{11331399151} a^{2} - \frac{357540247841}{11331399151} a + \frac{63675535370}{11331399151} \),  \( \frac{40568306535}{11331399151} a^{17} - \frac{115203308789}{11331399151} a^{16} + \frac{222621880022}{11331399151} a^{15} - \frac{322554740751}{11331399151} a^{14} + \frac{57734424392}{11331399151} a^{13} + \frac{757611495694}{11331399151} a^{12} - \frac{1787712923361}{11331399151} a^{11} + \frac{3316804498838}{11331399151} a^{10} - \frac{2894419813789}{11331399151} a^{9} + \frac{1094450917116}{11331399151} a^{8} + \frac{2897409457765}{11331399151} a^{7} - \frac{7107210053382}{11331399151} a^{6} + \frac{9804165713946}{11331399151} a^{5} - \frac{8949870625084}{11331399151} a^{4} + \frac{6043288337032}{11331399151} a^{3} - \frac{2865170362118}{11331399151} a^{2} + \frac{831285531925}{11331399151} a - \frac{111401040113}{11331399151} \),  \( \frac{4686630}{2200699} a^{17} - \frac{12275212}{2200699} a^{16} + \frac{23159619}{2200699} a^{15} - \frac{32705400}{2200699} a^{14} + \frac{537399}{2200699} a^{13} + \frac{85864311}{2200699} a^{12} - \frac{186602208}{2200699} a^{11} + \frac{345316472}{2200699} a^{10} - \frac{267463767}{2200699} a^{9} + \frac{84839946}{2200699} a^{8} + \frac{333627691}{2200699} a^{7} - \frac{741124713}{2200699} a^{6} + \frac{982067958}{2200699} a^{5} - \frac{858337855}{2200699} a^{4} + \frac{567026214}{2200699} a^{3} - \frac{256890042}{2200699} a^{2} + \frac{72914194}{2200699} a - \frac{9764586}{2200699} \),  \( \frac{8309329333}{11331399151} a^{17} - \frac{15692101247}{11331399151} a^{16} + \frac{30310005074}{11331399151} a^{15} - \frac{38997221259}{11331399151} a^{14} - \frac{21443469049}{11331399151} a^{13} + \frac{126857720380}{11331399151} a^{12} - \frac{232758003228}{11331399151} a^{11} + \frac{459889069528}{11331399151} a^{10} - \frac{188565601542}{11331399151} a^{9} + \frac{103001387940}{11331399151} a^{8} + \frac{560323989682}{11331399151} a^{7} - \frac{872668733584}{11331399151} a^{6} + \frac{1153464672395}{11331399151} a^{5} - \frac{894747065052}{11331399151} a^{4} + \frac{631930928299}{11331399151} a^{3} - \frac{285167200881}{11331399151} a^{2} + \frac{112594578105}{11331399151} a - \frac{25889684195}{11331399151} \),  \( \frac{17256348296}{11331399151} a^{17} - \frac{42056093950}{11331399151} a^{16} + \frac{78090640669}{11331399151} a^{15} - \frac{106042171169}{11331399151} a^{14} - \frac{17900986928}{11331399151} a^{13} + \frac{314986577820}{11331399151} a^{12} - \frac{636805141402}{11331399151} a^{11} + \frac{1159776843668}{11331399151} a^{10} - \frac{766141995261}{11331399151} a^{9} + \frac{161852403423}{11331399151} a^{8} + \frac{1312487809641}{11331399151} a^{7} - \frac{2509543822519}{11331399151} a^{6} + \frac{3173289725485}{11331399151} a^{5} - \frac{2533417485169}{11331399151} a^{4} + \frac{1544221430006}{11331399151} a^{3} - \frac{576192420550}{11331399151} a^{2} + \frac{127671266030}{11331399151} a - \frac{13341546634}{11331399151} \),  \( \frac{9192593121}{11331399151} a^{17} - \frac{21708130539}{11331399151} a^{16} + \frac{38943384825}{11331399151} a^{15} - \frac{51941783631}{11331399151} a^{14} - \frac{16241148649}{11331399151} a^{13} + \frac{169442040974}{11331399151} a^{12} - \frac{320180547555}{11331399151} a^{11} + \frac{577002186941}{11331399151} a^{10} - \frac{343218742054}{11331399151} a^{9} + \frac{20213051620}{11331399151} a^{8} + \frac{690226745514}{11331399151} a^{7} - \frac{1266969982199}{11331399151} a^{6} + \frac{1530266756767}{11331399151} a^{5} - \frac{1155468291712}{11331399151} a^{4} + \frac{676469049422}{11331399151} a^{3} - \frac{232668337995}{11331399151} a^{2} + \frac{22250594749}{11331399151} a + \frac{10410621222}{11331399151} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 784.727413524 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

18T286:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.5.689278977.1

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 $18$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
1297Data not computed