↑

Properties

Label 18.0.13486081239...3631.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 37^{17}$
Root discriminant $471.94$
Ramified primes $3, 37$
Class number $14802753096$ (GRH)
Class group $[14802753096]$ (GRH)
Galois group $C_{18}$ (as 18T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![33270400000, 65283984000, 41395363200, -24669932040, 9112996548, 1801890306, -459144729, 21004974, 77298624, -2521772, -2869461, 267732, 504828, -132534, 10989, -666, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 666*x^15 + 10989*x^14 - 132534*x^13 + 504828*x^12 + 267732*x^11 - 2869461*x^10 - 2521772*x^9 + 77298624*x^8 + 21004974*x^7 - 459144729*x^6 + 1801890306*x^5 + 9112996548*x^4 - 24669932040*x^3 + 41395363200*x^2 + 65283984000*x + 33270400000)
 
gp: K = bnfinit(x^18 - 666*x^15 + 10989*x^14 - 132534*x^13 + 504828*x^12 + 267732*x^11 - 2869461*x^10 - 2521772*x^9 + 77298624*x^8 + 21004974*x^7 - 459144729*x^6 + 1801890306*x^5 + 9112996548*x^4 - 24669932040*x^3 + 41395363200*x^2 + 65283984000*x + 33270400000, 1)
 

Normalized defining polynomial

\( x^{18} - 666 x^{15} + 10989 x^{14} - 132534 x^{13} + 504828 x^{12} + 267732 x^{11} - 2869461 x^{10} - 2521772 x^{9} + 77298624 x^{8} + 21004974 x^{7} - 459144729 x^{6} + 1801890306 x^{5} + 9112996548 x^{4} - 24669932040 x^{3} + 41395363200 x^{2} + 65283984000 x + 33270400000 \)

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:  \(-1348608123969580736044287667264238937751275533631=-\,3^{45}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $471.94$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(999=3^{3}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{999}(1,·)$, $\chi_{999}(899,·)$, $\chi_{999}(10,·)$, $\chi_{999}(268,·)$, $\chi_{999}(589,·)$, $\chi_{999}(410,·)$, $\chi_{999}(731,·)$, $\chi_{999}(989,·)$, $\chi_{999}(100,·)$, $\chi_{999}(998,·)$, $\chi_{999}(104,·)$, $\chi_{999}(41,·)$, $\chi_{999}(682,·)$, $\chi_{999}(173,·)$, $\chi_{999}(826,·)$, $\chi_{999}(317,·)$, $\chi_{999}(958,·)$, $\chi_{999}(895,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{40} a^{5} - \frac{1}{8} a^{3} + \frac{1}{10} a$, $\frac{1}{80} a^{6} - \frac{1}{80} a^{5} - \frac{1}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{20} a^{2} - \frac{1}{20} a$, $\frac{1}{80} a^{7} + \frac{19}{80} a^{3} - \frac{1}{4} a$, $\frac{1}{640} a^{8} - \frac{1}{160} a^{7} + \frac{1}{320} a^{6} - \frac{1}{80} a^{5} - \frac{31}{640} a^{4} + \frac{31}{160} a^{3} + \frac{7}{160} a^{2} - \frac{7}{40} a$, $\frac{1}{640} a^{9} + \frac{1}{320} a^{7} + \frac{1}{640} a^{5} - \frac{33}{160} a^{3} + \frac{1}{5} a$, $\frac{1}{6400} a^{10} - \frac{1}{1280} a^{9} + \frac{3}{640} a^{7} + \frac{13}{6400} a^{6} - \frac{1}{1280} a^{5} - \frac{3}{128} a^{4} - \frac{9}{320} a^{3} + \frac{17}{800} a^{2} + \frac{1}{40} a$, $\frac{1}{6400} a^{11} - \frac{1}{1280} a^{9} + \frac{3}{6400} a^{7} - \frac{3}{256} a^{5} + \frac{99}{1600} a^{3} - \frac{1}{20} a$, $\frac{1}{25600} a^{12} - \frac{1}{12800} a^{11} + \frac{1}{25600} a^{10} - \frac{1}{2560} a^{9} - \frac{17}{25600} a^{8} - \frac{23}{12800} a^{7} - \frac{37}{25600} a^{6} - \frac{19}{2560} a^{5} + \frac{29}{6400} a^{4} + \frac{431}{3200} a^{3} - \frac{1}{400} a^{2} - \frac{1}{8} a$, $\frac{1}{204800} a^{13} - \frac{3}{204800} a^{12} - \frac{1}{40960} a^{11} - \frac{11}{204800} a^{10} - \frac{127}{204800} a^{9} - \frac{109}{204800} a^{8} - \frac{3}{40960} a^{7} + \frac{967}{204800} a^{6} + \frac{693}{102400} a^{5} - \frac{147}{51200} a^{4} + \frac{1073}{5120} a^{3} - \frac{101}{800} a^{2} - \frac{149}{320} a + \frac{3}{8}$, $\frac{1}{4096000} a^{14} - \frac{1}{2048000} a^{13} - \frac{7}{512000} a^{12} - \frac{13}{256000} a^{11} - \frac{61}{2048000} a^{10} + \frac{21}{1024000} a^{9} + \frac{413}{1024000} a^{8} + \frac{1967}{512000} a^{7} + \frac{12001}{4096000} a^{6} - \frac{4241}{2048000} a^{5} + \frac{63351}{1024000} a^{4} - \frac{83691}{512000} a^{3} + \frac{1383}{6400} a^{2} - \frac{433}{1280} a + \frac{7}{32}$, $\frac{1}{163840000} a^{15} - \frac{7}{163840000} a^{14} + \frac{37}{81920000} a^{13} + \frac{81}{5120000} a^{12} + \frac{5279}{81920000} a^{11} + \frac{6247}{81920000} a^{10} - \frac{11351}{20480000} a^{9} + \frac{15159}{40960000} a^{8} + \frac{218241}{163840000} a^{7} - \frac{821887}{163840000} a^{6} + \frac{499867}{81920000} a^{5} + \frac{2272943}{40960000} a^{4} - \frac{12169}{819200} a^{3} - \frac{751}{6400} a^{2} - \frac{4399}{10240} a - \frac{127}{256}$, $\frac{1}{328990720000} a^{16} + \frac{7}{10280960000} a^{15} + \frac{21657}{328990720000} a^{14} + \frac{315443}{164495360000} a^{13} - \frac{185629}{32899072000} a^{12} - \frac{28159}{642560000} a^{11} - \frac{6661547}{164495360000} a^{10} + \frac{26795397}{82247680000} a^{9} + \frac{120252357}{328990720000} a^{8} + \frac{85461173}{41123840000} a^{7} + \frac{1760487037}{328990720000} a^{6} - \frac{764131237}{164495360000} a^{5} - \frac{656889017}{82247680000} a^{4} - \frac{296437311}{1644953600} a^{3} - \frac{12019283}{102809600} a^{2} + \frac{1155731}{4112384} a + \frac{145799}{514048}$, $\frac{1}{1808549339269637964276614264080826368000000} a^{17} - \frac{825003731387721772639142685013}{1808549339269637964276614264080826368000000} a^{16} - \frac{3551347779467063385594622358718631}{1808549339269637964276614264080826368000000} a^{15} + \frac{132005888188911570618756243449099737}{1808549339269637964276614264080826368000000} a^{14} - \frac{12398000086514828693716924048231957}{7064645856522023297955524469065728000000} a^{13} + \frac{14250822438624442711890832967346872581}{904274669634818982138307132040413184000000} a^{12} - \frac{24483358819385820065430108813218753739}{904274669634818982138307132040413184000000} a^{11} + \frac{9419330803135107599795929926318952273}{904274669634818982138307132040413184000000} a^{10} + \frac{1163776682839811284182912828284842798241}{1808549339269637964276614264080826368000000} a^{9} - \frac{183908897132719608651058804115890021101}{361709867853927592855322852816165273600000} a^{8} + \frac{5466699125537358728097860148828484715989}{1808549339269637964276614264080826368000000} a^{7} - \frac{8770676775357315022858216646169842044683}{1808549339269637964276614264080826368000000} a^{6} - \frac{38830450890170006909537037749003316701}{7234197357078551857106457056323305472000} a^{5} + \frac{24387151541909952740723562631950613583839}{452137334817409491069153566020206592000000} a^{4} - \frac{10405375065035063844449450379434359761497}{45213733481740949106915356602020659200000} a^{3} + \frac{50867176359399037970016528426450309061}{282585834260880931918220978762629120000} a^{2} + \frac{28089609520868439631889471517939900173}{113034333704352372767288391505051648000} a + \frac{217513379830183524115096008481762569}{565171668521761863836441957525258240}$

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

Class group and class number

$C_{14802753096}$, which has order $14802753096$ (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:  \( 23654456315312.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-111}) \), 3.3.110889.2, 6.0.1364897105631.1, 9.9.110225327118882669776889.7

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/5.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
37Data not computed