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Properties

Label 18.18.3934777906...2304.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{35}\cdot 3^{45}\cdot 53^{16}$
Root discriminant $2045.64$
Ramified primes $2, 3, 53$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $S_3\times D_9$ (as 18T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7334325786624, 24196212919296, -26694357563520, 9205791789312, 2974331897280, -2602305600192, 191094022800, 211649296032, -38883795984, -6956743536, 1885287168, 84108456, -36510216, -286200, 296217, 0, -954, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 954*x^16 + 296217*x^14 - 286200*x^13 - 36510216*x^12 + 84108456*x^11 + 1885287168*x^10 - 6956743536*x^9 - 38883795984*x^8 + 211649296032*x^7 + 191094022800*x^6 - 2602305600192*x^5 + 2974331897280*x^4 + 9205791789312*x^3 - 26694357563520*x^2 + 24196212919296*x - 7334325786624)
 
gp: K = bnfinit(x^18 - 954*x^16 + 296217*x^14 - 286200*x^13 - 36510216*x^12 + 84108456*x^11 + 1885287168*x^10 - 6956743536*x^9 - 38883795984*x^8 + 211649296032*x^7 + 191094022800*x^6 - 2602305600192*x^5 + 2974331897280*x^4 + 9205791789312*x^3 - 26694357563520*x^2 + 24196212919296*x - 7334325786624, 1)
 

Normalized defining polynomial

\( x^{18} - 954 x^{16} + 296217 x^{14} - 286200 x^{13} - 36510216 x^{12} + 84108456 x^{11} + 1885287168 x^{10} - 6956743536 x^{9} - 38883795984 x^{8} + 211649296032 x^{7} + 191094022800 x^{6} - 2602305600192 x^{5} + 2974331897280 x^{4} + 9205791789312 x^{3} - 26694357563520 x^{2} + 24196212919296 x - 7334325786624 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(393477790691475867882573798675110334618493697489618079842304=2^{35}\cdot 3^{45}\cdot 53^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2045.64$
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:  $2, 3, 53$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{424} a^{9} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{424} a^{10} + \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{848} a^{11} + \frac{1}{16} a^{7} + \frac{1}{8} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{1696} a^{12} - \frac{1}{848} a^{10} + \frac{1}{32} a^{8} + \frac{1}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{3392} a^{13} - \frac{1}{1696} a^{11} - \frac{1}{848} a^{10} - \frac{3}{3392} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a$, $\frac{1}{33920} a^{14} + \frac{3}{16960} a^{12} - \frac{1}{8480} a^{11} - \frac{11}{33920} a^{10} - \frac{3}{4240} a^{9} + \frac{1}{10} a^{8} + \frac{1}{160} a^{7} - \frac{9}{160} a^{6} + \frac{17}{80} a^{5} + \frac{9}{40} a^{4} + \frac{3}{20} a^{3} - \frac{3}{20} a^{2} + \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{67840} a^{15} - \frac{1}{16960} a^{13} + \frac{1}{4240} a^{12} - \frac{31}{67840} a^{11} - \frac{3}{8480} a^{10} - \frac{9}{33920} a^{9} - \frac{7}{160} a^{8} + \frac{1}{320} a^{7} - \frac{7}{40} a^{6} - \frac{7}{160} a^{5} + \frac{3}{40} a^{4} + \frac{3}{10} a^{3} + \frac{2}{5} a^{2} - \frac{9}{20} a$, $\frac{1}{1085440} a^{16} + \frac{3}{271360} a^{14} - \frac{1}{16960} a^{13} + \frac{45}{217088} a^{12} - \frac{31}{135680} a^{11} - \frac{257}{542720} a^{10} + \frac{51}{135680} a^{9} - \frac{343}{5120} a^{8} - \frac{7}{128} a^{7} + \frac{481}{2560} a^{6} - \frac{89}{640} a^{5} + \frac{21}{160} a^{4} + \frac{51}{160} a^{3} + \frac{7}{320} a^{2} + \frac{11}{40} a + \frac{1}{20}$, $\frac{1}{35925540127542284243861651208478103504896221417931129803646376131686400} a^{17} - \frac{2975125162300515537991210337857079002663786082299038851826787929}{8981385031885571060965412802119525876224055354482782450911594032921600} a^{16} - \frac{43044727387754139063824534280194355453852570692370106381663209597}{8981385031885571060965412802119525876224055354482782450911594032921600} a^{15} + \frac{5854327585011540790494532830890260833177473146639435395538998143}{2245346257971392765241353200529881469056013838620695612727898508230400} a^{14} + \frac{4248283896846153959253805748907661706045994585993610928076618717089}{35925540127542284243861651208478103504896221417931129803646376131686400} a^{13} + \frac{2553427793835705960798964945166726628339818537897955043066913399929}{8981385031885571060965412802119525876224055354482782450911594032921600} a^{12} - \frac{2346476855645366938785893840681002517821419435079145222836409041041}{17962770063771142121930825604239051752448110708965564901823188065843200} a^{11} - \frac{603284014431394391057997946264153647785219624827899809555555480013}{561336564492848191310338300132470367264003459655173903181974627057600} a^{10} - \frac{407563263785812978328461601449002970161274779216904276344910702127}{1796277006377114212193082560423905175244811070896556490182318806584320} a^{9} + \frac{5106652154771869279221925241004118542373994440422572942283376406273}{42365023735309297457384022651507197529358751672088596466564122796800} a^{8} - \frac{9612405272516500925234737248925939102386169975573855940792376319647}{84730047470618594914768045303014395058717503344177192933128245593600} a^{7} - \frac{341154923318995591097690741298174795515774945366665574720600748119}{2118251186765464872869201132575359876467937583604429823328206139840} a^{6} + \frac{67000067167560764727845190912443302049158573784596882798278170381}{529562796691366218217300283143839969116984395901107455832051534960} a^{5} - \frac{945751872785903208374730694345487417214579929491484013210958628773}{5295627966913662182173002831438399691169843959011074558320515349600} a^{4} - \frac{3002299474029207709611045173117177565023997279503412720512779034449}{10591255933827324364346005662876799382339687918022149116641030699200} a^{3} + \frac{173162649588417121233286696661246502182415957462888845777947817943}{529562796691366218217300283143839969116984395901107455832051534960} a^{2} - \frac{7739125129180207726529827628494350783494142743284858607845289353}{26478139834568310910865014157191998455849219795055372791602576748} a + \frac{55289214053489140797664597897277973683501036081514386030244098502}{165488373966051943192906338482449990349057623719096079947516104675}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
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:  \( 35072364445100000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{6}) \), 3.3.1820232.1, 6.6.318071475247104.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{9}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
3Data not computed
$53$53.9.8.1$x^{9} - 53$$9$$1$$8$$D_{9}$$[\ ]_{9}^{2}$
53.9.8.1$x^{9} - 53$$9$$1$$8$$D_{9}$$[\ ]_{9}^{2}$