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Properties

Label 18.0.89104281868...3328.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 41^{8}$
Root discriminant $52.41$
Ramified primes $2, 3, 7, 41$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12769, -32205, 148460, 27195, 492669, -198907, 521839, -70936, 245183, -40497, 70988, -5628, 10614, -1299, 942, -56, 37, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 37*x^16 - 56*x^15 + 942*x^14 - 1299*x^13 + 10614*x^12 - 5628*x^11 + 70988*x^10 - 40497*x^9 + 245183*x^8 - 70936*x^7 + 521839*x^6 - 198907*x^5 + 492669*x^4 + 27195*x^3 + 148460*x^2 - 32205*x + 12769)
 
gp: K = bnfinit(x^18 - 2*x^17 + 37*x^16 - 56*x^15 + 942*x^14 - 1299*x^13 + 10614*x^12 - 5628*x^11 + 70988*x^10 - 40497*x^9 + 245183*x^8 - 70936*x^7 + 521839*x^6 - 198907*x^5 + 492669*x^4 + 27195*x^3 + 148460*x^2 - 32205*x + 12769, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 37 x^{16} - 56 x^{15} + 942 x^{14} - 1299 x^{13} + 10614 x^{12} - 5628 x^{11} + 70988 x^{10} - 40497 x^{9} + 245183 x^{8} - 70936 x^{7} + 521839 x^{6} - 198907 x^{5} + 492669 x^{4} + 27195 x^{3} + 148460 x^{2} - 32205 x + 12769 \)

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:  \(-8910428186811131012489454563328=-\,2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.41$
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:  $2, 3, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{883601634} a^{16} + \frac{15082742}{441800817} a^{15} - \frac{60330935}{883601634} a^{14} + \frac{17453369}{883601634} a^{13} + \frac{6972088}{441800817} a^{12} + \frac{120095737}{441800817} a^{11} - \frac{215612167}{883601634} a^{10} - \frac{269998363}{883601634} a^{9} + \frac{6231117}{147266939} a^{8} - \frac{21227574}{147266939} a^{7} + \frac{42981802}{147266939} a^{6} - \frac{440663057}{883601634} a^{5} + \frac{307161739}{883601634} a^{4} + \frac{126974771}{883601634} a^{3} + \frac{45747842}{441800817} a^{2} + \frac{21377815}{883601634} a + \frac{171665627}{883601634}$, $\frac{1}{236124131409619738039416712579419178956938586} a^{17} - \frac{82343944454976512148715853495593421}{236124131409619738039416712579419178956938586} a^{16} - \frac{7506729341842465238840421236078572759744165}{236124131409619738039416712579419178956938586} a^{15} - \frac{1952282660106730482202705623254644738039519}{39354021901603289673236118763236529826156431} a^{14} - \frac{13769016940685593930299952502054124984556277}{236124131409619738039416712579419178956938586} a^{13} + \frac{11940559591942827508281993422125063197101969}{236124131409619738039416712579419178956938586} a^{12} - \frac{47772781264648186221896942984313536227502935}{236124131409619738039416712579419178956938586} a^{11} - \frac{55511849040044334719936720851119977160539776}{118062065704809869019708356289709589478469293} a^{10} + \frac{34696663852704162983111845523885393657884011}{78708043803206579346472237526473059652312862} a^{9} + \frac{30002647543635042131187734891269766792832131}{236124131409619738039416712579419178956938586} a^{8} - \frac{15589599562475742962233429343979168566138242}{118062065704809869019708356289709589478469293} a^{7} + \frac{1809889165667275160613136750271534555929287}{78708043803206579346472237526473059652312862} a^{6} + \frac{6234728816636287315743430382505660252387711}{39354021901603289673236118763236529826156431} a^{5} + \frac{53676423395754311943913306419103529597118685}{236124131409619738039416712579419178956938586} a^{4} + \frac{8071174169540290569763630510856311975840473}{78708043803206579346472237526473059652312862} a^{3} - \frac{29963305468902586534659458093453098763629168}{118062065704809869019708356289709589478469293} a^{2} + \frac{18448001370859757296024491696320512544100275}{39354021901603289673236118763236529826156431} a - \frac{357767244983415357451692321791999930286319}{1044797041635485566546091648581500791844861}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  \( -\frac{5980176101770349354851223765022}{267229170164277602546190756115576829} a^{17} + \frac{31935775754609316363216140276129}{801687510492832807638572268346730487} a^{16} - \frac{652693870568586726811064117179825}{801687510492832807638572268346730487} a^{15} + \frac{851777001379400401315558600617466}{801687510492832807638572268346730487} a^{14} - \frac{5520517826220042400944470052309615}{267229170164277602546190756115576829} a^{13} + \frac{6465038793562295456212576306152135}{267229170164277602546190756115576829} a^{12} - \frac{182326758504202744431109235484100355}{801687510492832807638572268346730487} a^{11} + \frac{18195431285457097759861191873223499}{267229170164277602546190756115576829} a^{10} - \frac{1218775383587301379831784961592208933}{801687510492832807638572268346730487} a^{9} + \frac{426124113219327832773110545725212735}{801687510492832807638572268346730487} a^{8} - \frac{4030736783378976332083334205330599213}{801687510492832807638572268346730487} a^{7} + \frac{148143849723047111179251685049969326}{801687510492832807638572268346730487} a^{6} - \frac{8415420212681381539029774553426728350}{801687510492832807638572268346730487} a^{5} + \frac{1233269138087728656719089024556322184}{801687510492832807638572268346730487} a^{4} - \frac{2252313089244893042095879952018829552}{267229170164277602546190756115576829} a^{3} - \frac{3456577715917098931944177173038665355}{801687510492832807638572268346730487} a^{2} - \frac{1856013106355198027207701370710997747}{801687510492832807638572268346730487} a + \frac{1124239640883438813457284887808858}{2364859912958208872090183682438733} \) (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:  \( 8375514.85964 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{7})^+\), 6.0.64827.1, 9.9.574470067776192.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$