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Properties

Label 18.0.48015245256...6816.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 17^{12}\cdot 23^{9}\cdot 313^{9}$
Root discriminant $2671.43$
Ramified primes $2, 3, 17, 23, 313$
Class number $68931440640$ (GRH)
Class group $[2, 6, 6, 6, 6, 6, 36, 36, 3420]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63555755265792, -19444819046400, 2031770407680, -3040844951808, 847107279360, -99026005248, 55401409440, -13129615872, 1752021792, -445448336, 77297904, -11920464, 1291103, -71064, 13683, -360, 141, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 141*x^16 - 360*x^15 + 13683*x^14 - 71064*x^13 + 1291103*x^12 - 11920464*x^11 + 77297904*x^10 - 445448336*x^9 + 1752021792*x^8 - 13129615872*x^7 + 55401409440*x^6 - 99026005248*x^5 + 847107279360*x^4 - 3040844951808*x^3 + 2031770407680*x^2 - 19444819046400*x + 63555755265792)
 
gp: K = bnfinit(x^18 + 141*x^16 - 360*x^15 + 13683*x^14 - 71064*x^13 + 1291103*x^12 - 11920464*x^11 + 77297904*x^10 - 445448336*x^9 + 1752021792*x^8 - 13129615872*x^7 + 55401409440*x^6 - 99026005248*x^5 + 847107279360*x^4 - 3040844951808*x^3 + 2031770407680*x^2 - 19444819046400*x + 63555755265792, 1)
 

Normalized defining polynomial

\( x^{18} + 141 x^{16} - 360 x^{15} + 13683 x^{14} - 71064 x^{13} + 1291103 x^{12} - 11920464 x^{11} + 77297904 x^{10} - 445448336 x^{9} + 1752021792 x^{8} - 13129615872 x^{7} + 55401409440 x^{6} - 99026005248 x^{5} + 847107279360 x^{4} - 3040844951808 x^{3} + 2031770407680 x^{2} - 19444819046400 x + 63555755265792 \)

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:  \(-48015245256971127697547575538546724520114168666540072247586816=-\,2^{12}\cdot 3^{18}\cdot 17^{12}\cdot 23^{9}\cdot 313^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2671.43$
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:  $2, 3, 17, 23, 313$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{1152} a^{12} + \frac{1}{96} a^{11} - \frac{3}{128} a^{10} + \frac{5}{96} a^{9} - \frac{23}{384} a^{8} + \frac{1}{32} a^{7} + \frac{71}{1152} a^{6} - \frac{19}{96} a^{5} - \frac{11}{48} a^{4} - \frac{25}{144} a^{3} - \frac{1}{3} a^{2} + \frac{5}{12} a + \frac{5}{24}$, $\frac{1}{2304} a^{13} + \frac{5}{256} a^{11} - \frac{1}{48} a^{10} + \frac{1}{768} a^{9} - \frac{1}{2304} a^{7} + \frac{3}{32} a^{6} + \frac{11}{48} a^{5} - \frac{25}{288} a^{4} + \frac{3}{8} a^{3} - \frac{1}{6} a^{2} + \frac{5}{48} a - \frac{1}{4}$, $\frac{1}{4608} a^{14} + \frac{1}{4608} a^{12} + \frac{13}{1536} a^{10} + \frac{5}{96} a^{9} - \frac{133}{4608} a^{8} - \frac{3}{64} a^{7} + \frac{71}{1152} a^{6} + \frac{5}{576} a^{5} - \frac{11}{48} a^{4} - \frac{7}{144} a^{3} + \frac{7}{32} a^{2} + \frac{1}{24} a + \frac{5}{24}$, $\frac{1}{32514048} a^{15} - \frac{1}{55296} a^{14} + \frac{1615}{10838016} a^{13} + \frac{437}{1354752} a^{12} - \frac{291791}{10838016} a^{11} + \frac{6233}{225792} a^{10} + \frac{284975}{32514048} a^{9} + \frac{17989}{677376} a^{8} - \frac{5209}{338688} a^{7} - \frac{89045}{1161216} a^{6} + \frac{165941}{677376} a^{5} + \frac{433}{84672} a^{4} + \frac{278335}{677376} a^{3} + \frac{11899}{28224} a^{2} + \frac{3931}{14112} a + \frac{21943}{56448}$, $\frac{1}{6795436032} a^{16} - \frac{1}{283143168} a^{15} - \frac{94817}{2265145344} a^{14} + \frac{1621}{566286336} a^{13} - \frac{74639}{2265145344} a^{12} - \frac{291077}{188762112} a^{11} - \frac{103853809}{6795436032} a^{10} - \frac{4963477}{80898048} a^{9} - \frac{556723}{17696448} a^{8} - \frac{90518699}{1698859008} a^{7} - \frac{1061141}{8848224} a^{6} - \frac{717097}{35392896} a^{5} + \frac{26845711}{141571584} a^{4} - \frac{1643065}{11797632} a^{3} - \frac{71237}{2949408} a^{2} + \frac{835855}{11797632} a - \frac{58439}{983136}$, $\frac{1}{2229713156783281414541773229805852205190859814751976334678253582106624} a^{17} - \frac{936869551446160218864333741453923758019743417660987393935}{13763661461625193916924526109912667933276912436740594658507738161152} a^{16} + \frac{400687641735532876946895062765981910400127961917053320385541}{202701196071207401321979384527804745926441801341088757698023052918784} a^{15} - \frac{208998399611621300136589024368296506379497173868004610871217541}{13763661461625193916924526109912667933276912436740594658507738161152} a^{14} + \frac{107201614289333154694822476898087871168780958425362025301819409679}{743237718927760471513924409935284068396953271583992111559417860702208} a^{13} + \frac{51417965427812337968041897059505331909468795594567846977550535863}{371618859463880235756962204967642034198476635791996055779708930351104} a^{12} - \frac{9649660227104942645144256152235487378857344397701158397420127333115}{2229713156783281414541773229805852205190859814751976334678253582106624} a^{11} + \frac{343245086078902889487175024897975436456214247393349395161788279567}{17696136164946677893188676427030573057070315990095050275224234778624} a^{10} + \frac{35046777184137459228511872725886940288697395739286781886097232622743}{1114856578391640707270886614902926102595429907375988167339126791053312} a^{9} - \frac{21343238596992281700302907466637004663909870593597875708211982973451}{557428289195820353635443307451463051297714953687994083669563395526656} a^{8} - \frac{4086484938872843066278886526827830386597958675714635163668362532901}{92904714865970058939240551241910508549619158947999013944927232587776} a^{7} - \frac{823000725788281193371430977935855953175170508042801720504822492721}{39816306371130025259674521960818789378408210977713863119254528251904} a^{6} + \frac{3438319238238249784879066701433363435133971832397482241613255737059}{15484119144328343156540091873651751424936526491333168990821205431296} a^{5} - \frac{111612566676417606516143221933348474231302354748464140695767192731}{23226178716492514734810137810477627137404789736999753486231808146944} a^{4} + \frac{458589854803072355238733871391097202202161387914096426976770834919}{23226178716492514734810137810477627137404789736999753486231808146944} a^{3} + \frac{30553094046435019056796976277842174069913443448923001309529316949}{61444917239398187129127348704967267559271930521163369011195259648} a^{2} - \frac{386742150925569707253198368743467141596089801470607263854065044067}{1935514893041042894567511484206468928117065811416646123852650678912} a + \frac{581298740951320712379360759745321287928571904380657974530712583}{1520435893983537230610771000947736785637915012896029948038217344}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{6}\times C_{6}\times C_{36}\times C_{36}\times C_{3420}$, which has order $68931440640$ (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:  \( 342971481898070.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-7199}) \), 3.1.31212.1, 3.1.7199.1 x3, Deg 6, 6.0.373092501599.2, 9.1.11344394363669665966884672.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$17$17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
313Data not computed