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Properties

Label 18.0.17249251837...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 5^{9}\cdot 13^{12}$
Root discriminant $151.29$
Ramified primes $2, 3, 5, 13$
Class number $2667168$ (GRH)
Class group $[2, 2, 2, 126, 2646]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21442432211, -4973847282, 9908381091, -2009879192, 2043736503, -377964774, 254826777, -44268360, 21853467, -3584354, 1383909, -209784, 66460, -8964, 2382, -284, 63, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 63*x^16 - 284*x^15 + 2382*x^14 - 8964*x^13 + 66460*x^12 - 209784*x^11 + 1383909*x^10 - 3584354*x^9 + 21853467*x^8 - 44268360*x^7 + 254826777*x^6 - 377964774*x^5 + 2043736503*x^4 - 2009879192*x^3 + 9908381091*x^2 - 4973847282*x + 21442432211)
 
gp: K = bnfinit(x^18 - 6*x^17 + 63*x^16 - 284*x^15 + 2382*x^14 - 8964*x^13 + 66460*x^12 - 209784*x^11 + 1383909*x^10 - 3584354*x^9 + 21853467*x^8 - 44268360*x^7 + 254826777*x^6 - 377964774*x^5 + 2043736503*x^4 - 2009879192*x^3 + 9908381091*x^2 - 4973847282*x + 21442432211, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 63 x^{16} - 284 x^{15} + 2382 x^{14} - 8964 x^{13} + 66460 x^{12} - 209784 x^{11} + 1383909 x^{10} - 3584354 x^{9} + 21853467 x^{8} - 44268360 x^{7} + 254826777 x^{6} - 377964774 x^{5} + 2043736503 x^{4} - 2009879192 x^{3} + 9908381091 x^{2} - 4973847282 x + 21442432211 \)

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:  \(-1724925183796757382490845609984000000000=-\,2^{27}\cdot 3^{24}\cdot 5^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $151.29$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4680=2^{3}\cdot 3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{4680}(1,·)$, $\chi_{4680}(3979,·)$, $\chi_{4680}(3721,·)$, $\chi_{4680}(139,·)$, $\chi_{4680}(4579,·)$, $\chi_{4680}(1561,·)$, $\chi_{4680}(601,·)$, $\chi_{4680}(859,·)$, $\chi_{4680}(2401,·)$, $\chi_{4680}(1699,·)$, $\chi_{4680}(2161,·)$, $\chi_{4680}(3019,·)$, $\chi_{4680}(3121,·)$, $\chi_{4680}(1459,·)$, $\chi_{4680}(841,·)$, $\chi_{4680}(3961,·)$, $\chi_{4680}(2419,·)$, $\chi_{4680}(3259,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{2} a^{7} - \frac{5}{16} a^{6} - \frac{1}{4} a^{4} + \frac{7}{16} a^{3} + \frac{5}{16} a^{2} - \frac{1}{4} a - \frac{3}{16}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{5}{16} a^{7} - \frac{1}{16} a^{6} - \frac{5}{16} a^{4} + \frac{1}{4} a^{3} + \frac{3}{16} a^{2} + \frac{5}{16} a + \frac{1}{16}$, $\frac{1}{173062400144} a^{15} - \frac{571615383}{86531200072} a^{14} + \frac{2820391411}{173062400144} a^{13} + \frac{2004640911}{86531200072} a^{12} + \frac{741903309}{21632800018} a^{11} + \frac{5555783197}{173062400144} a^{10} + \frac{14113704913}{173062400144} a^{9} - \frac{20556845551}{173062400144} a^{8} - \frac{35568452893}{173062400144} a^{7} + \frac{61045671467}{173062400144} a^{6} - \frac{11237240625}{173062400144} a^{5} + \frac{626187583}{86531200072} a^{4} + \frac{3394259718}{10816400009} a^{3} - \frac{4596232710}{10816400009} a^{2} + \frac{56056331625}{173062400144} a + \frac{57881253141}{173062400144}$, $\frac{1}{149352851324272} a^{16} + \frac{111}{149352851324272} a^{15} - \frac{1653994038625}{74676425662136} a^{14} + \frac{3033411990357}{149352851324272} a^{13} + \frac{2357539713667}{74676425662136} a^{12} - \frac{699642478741}{37338212831068} a^{11} - \frac{8306353855391}{149352851324272} a^{10} + \frac{3159142923507}{149352851324272} a^{9} - \frac{8521238144461}{74676425662136} a^{8} - \frac{5672687494019}{149352851324272} a^{7} + \frac{36726216309157}{149352851324272} a^{6} + \frac{48173830238773}{149352851324272} a^{5} - \frac{63616889216167}{149352851324272} a^{4} - \frac{1736617406405}{18669106415534} a^{3} - \frac{13283562547817}{37338212831068} a^{2} - \frac{1945043031653}{149352851324272} a + \frac{13502394318049}{74676425662136}$, $\frac{1}{43075833729524728716612234235234594450064} a^{17} + \frac{40616975491863088403064323}{43075833729524728716612234235234594450064} a^{16} + \frac{20228256393867199613818646573}{10768958432381182179153058558808648612516} a^{15} - \frac{14570462014827334328214080270903952025}{2267149143659196248242749170275504971056} a^{14} + \frac{811697727472816966638785906921718024729}{43075833729524728716612234235234594450064} a^{13} + \frac{1347001776849996413413234796891650964013}{43075833729524728716612234235234594450064} a^{12} - \frac{473984097948814640991745799369134882483}{21537916864762364358306117117617297225032} a^{11} - \frac{1563418849055912274629928429539783950233}{43075833729524728716612234235234594450064} a^{10} + \frac{3880311514816530201456325832612510994819}{43075833729524728716612234235234594450064} a^{9} + \frac{2123821950836864071991497817840591402773}{43075833729524728716612234235234594450064} a^{8} + \frac{21162230688857578673512452964206847677177}{43075833729524728716612234235234594450064} a^{7} - \frac{136531932877038818208075711424031396761}{1133574571829598124121374585137752485528} a^{6} + \frac{433272750368498761703307589907054554699}{43075833729524728716612234235234594450064} a^{5} - \frac{202785227647048452514461100654399074995}{5384479216190591089576529279404324306258} a^{4} + \frac{9743546352537553754576339685901118435791}{43075833729524728716612234235234594450064} a^{3} - \frac{2975329713009444690646338918231090736189}{21537916864762364358306117117617297225032} a^{2} - \frac{3255238577689668161084634968487497983805}{21537916864762364358306117117617297225032} a + \frac{16025294136737907178393650671261568919967}{43075833729524728716612234235234594450064}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{126}\times C_{2646}$, which has order $2667168$ (assuming GRH)

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 400417.1364448253 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-10}) \), 3.3.13689.2, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 6.0.11992878144000.7, 6.0.419904000.3, 6.0.11992878144000.6, 6.0.1827904000.2, 9.9.2565164201769.1

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

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$