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Properties

Label 18.0.76944553869...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{33}\cdot 5^{16}\cdot 7^{12}$
Root discriminant $212.33$
Ramified primes $2, 3, 5, 7$
Class number $19683$ (GRH)
Class group $[3, 3, 3, 3, 3, 9, 9]$ (GRH)
Galois group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6652087267, 4051301331, -3082794183, -1835112552, 666797838, 289673118, -77689590, -16434924, 4397733, -132499, -33261, 32004, -4710, -462, 234, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 234*x^14 - 462*x^13 - 4710*x^12 + 32004*x^11 - 33261*x^10 - 132499*x^9 + 4397733*x^8 - 16434924*x^7 - 77689590*x^6 + 289673118*x^5 + 666797838*x^4 - 1835112552*x^3 - 3082794183*x^2 + 4051301331*x + 6652087267)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 234*x^14 - 462*x^13 - 4710*x^12 + 32004*x^11 - 33261*x^10 - 132499*x^9 + 4397733*x^8 - 16434924*x^7 - 77689590*x^6 + 289673118*x^5 + 666797838*x^4 - 1835112552*x^3 - 3082794183*x^2 + 4051301331*x + 6652087267, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 234 x^{14} - 462 x^{13} - 4710 x^{12} + 32004 x^{11} - 33261 x^{10} - 132499 x^{9} + 4397733 x^{8} - 16434924 x^{7} - 77689590 x^{6} + 289673118 x^{5} + 666797838 x^{4} - 1835112552 x^{3} - 3082794183 x^{2} + 4051301331 x + 6652087267 \)

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:  \(-769445538694487691557611230000000000000000=-\,2^{16}\cdot 3^{33}\cdot 5^{16}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $212.33$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8}$, $\frac{1}{112} a^{12} - \frac{3}{56} a^{11} - \frac{3}{112} a^{10} - \frac{1}{8} a^{9} + \frac{3}{56} a^{8} - \frac{3}{56} a^{7} - \frac{1}{16} a^{6} + \frac{5}{56} a^{5} + \frac{3}{56} a^{4} + \frac{17}{56} a^{3} - \frac{5}{16} a^{2} + \frac{3}{8} a - \frac{1}{16}$, $\frac{1}{112} a^{13} + \frac{3}{112} a^{11} - \frac{1}{28} a^{10} + \frac{3}{56} a^{9} - \frac{3}{28} a^{8} - \frac{1}{112} a^{7} - \frac{1}{28} a^{6} - \frac{9}{56} a^{5} - \frac{1}{4} a^{4} + \frac{43}{112} a^{3} + \frac{1}{4} a^{2} + \frac{7}{16} a - \frac{1}{4}$, $\frac{1}{4256} a^{14} - \frac{1}{608} a^{13} + \frac{5}{2128} a^{12} + \frac{31}{4256} a^{11} - \frac{1}{4256} a^{10} - \frac{223}{2128} a^{9} + \frac{11}{224} a^{8} + \frac{339}{4256} a^{7} + \frac{115}{4256} a^{6} - \frac{61}{304} a^{5} - \frac{195}{4256} a^{4} + \frac{33}{608} a^{3} + \frac{123}{304} a^{2} + \frac{139}{608} a - \frac{263}{608}$, $\frac{1}{4256} a^{15} - \frac{1}{4256} a^{13} - \frac{13}{4256} a^{12} - \frac{25}{2128} a^{11} - \frac{263}{4256} a^{10} - \frac{25}{4256} a^{9} - \frac{201}{2128} a^{8} - \frac{29}{2128} a^{7} - \frac{467}{4256} a^{6} + \frac{515}{4256} a^{5} + \frac{155}{2128} a^{4} - \frac{1031}{4256} a^{3} + \frac{151}{608} a^{2} - \frac{3}{152} a - \frac{55}{608}$, $\frac{1}{2312567807848547616925134752} a^{16} - \frac{1}{289070975981068452115641844} a^{15} + \frac{259065109389230968083937}{2312567807848547616925134752} a^{14} - \frac{259065109389230968083917}{330366829692649659560733536} a^{13} - \frac{143912938427096238173325}{60857047574961779392766704} a^{12} + \frac{156196883423263048197103}{6406005007890713620291232} a^{11} + \frac{6631257876159457123397173}{136033400461679271583831456} a^{10} + \frac{16262384319734308069780897}{1156283903924273808462567376} a^{9} + \frac{4321737690476237842975914}{72267743995267113028910461} a^{8} - \frac{3175328874039559969152541}{136033400461679271583831456} a^{7} - \frac{18110331228208112579064033}{330366829692649659560733536} a^{6} - \frac{645834696892271721640831}{6683721988001582707876112} a^{5} + \frac{191477776563573810205496427}{2312567807848547616925134752} a^{4} + \frac{981725285043778585397690119}{2312567807848547616925134752} a^{3} - \frac{2374028813285532400484516}{10323963427895301861272923} a^{2} + \frac{83737037120726296861724863}{330366829692649659560733536} a + \frac{1035820310720966159615821}{9716671461548519398845104}$, $\frac{1}{60091179523202872437871876503570976} a^{17} + \frac{6496149}{30045589761601436218935938251785488} a^{16} - \frac{235288595724369331550412180581}{7511397440400359054733984562946372} a^{15} - \frac{3695583133476883311129131330601}{60091179523202872437871876503570976} a^{14} - \frac{3602113201970301243503971965647}{8584454217600410348267410929081568} a^{13} - \frac{1555168451008519166004884208441}{1581346829557970327312417802725552} a^{12} - \frac{2168295392353104985069255478804457}{60091179523202872437871876503570976} a^{11} + \frac{2718659469335162307394447040019557}{60091179523202872437871876503570976} a^{10} + \frac{18239483452071318397902519416607}{186040803477408273801460917967712} a^{9} + \frac{16894316858807353388891830445055}{8584454217600410348267410929081568} a^{8} + \frac{5734978901506459951571988440934115}{60091179523202872437871876503570976} a^{7} - \frac{2390005426619104126265598328941663}{60091179523202872437871876503570976} a^{6} + \frac{1103794645234922765938577855232901}{15022794880800718109467969125892744} a^{5} - \frac{1892975829043207925649472426218803}{8584454217600410348267410929081568} a^{4} - \frac{3510350230732745828834605885496335}{8584454217600410348267410929081568} a^{3} + \frac{663943652037265358799446059250675}{4292227108800205174133705464540784} a^{2} - \frac{1528896431471883002153099911274825}{4292227108800205174133705464540784} a - \frac{158197214314015476751810925973783}{504967895152965314603965348769504}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $19683$ (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:  \( \frac{154400772656218350}{95030597200850615538642583} a^{17} - \frac{77200386328109175}{5590035129461800914037799} a^{16} + \frac{6164620139821628776}{95030597200850615538642583} a^{15} - \frac{19986519697105096320}{95030597200850615538642583} a^{14} + \frac{3313646250098481900}{5590035129461800914037799} a^{13} - \frac{7507808438395344820}{5001610378992137659928557} a^{12} - \frac{489752160090578544480}{95030597200850615538642583} a^{11} + \frac{3846117489114246887376}{95030597200850615538642583} a^{10} - \frac{5134696624760595803830}{95030597200850615538642583} a^{9} - \frac{7227430561518996965550}{95030597200850615538642583} a^{8} + \frac{658701928215376305820320}{95030597200850615538642583} a^{7} - \frac{2242872110419695756368960}{95030597200850615538642583} a^{6} - \frac{9827364358304186009946324}{95030597200850615538642583} a^{5} + \frac{4305436277746674026473680}{13575799600121516505520369} a^{4} + \frac{67419472709259606646678280}{95030597200850615538642583} a^{3} - \frac{18911096425357220027857200}{13575799600121516505520369} a^{2} - \frac{24711060151724786395588830}{13575799600121516505520369} a + \frac{1320373143636905576042675}{798576447065971559148257} \) (order $6$)
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:  \( 30369487687.174183 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.2700.1 x3, 6.0.21870000.2, 9.1.506440367892900000000.1 x3

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

Sibling fields

Degree 9 sibling: 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$