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Properties

Label 18.0.28746584380...8928.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{14}\cdot 43^{9}$
Root discriminant $295.07$
Ramified primes $2, 3, 7, 43$
Class number $163396116$ (GRH)
Class group $[6, 27232686]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36321165287, -24485234526, 23602542069, -10346104119, 5595461043, -1766926269, 688224225, -163367103, 51165147, -9084590, 2446248, -287883, 72951, -3471, 1581, -30, 51, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 51*x^16 - 30*x^15 + 1581*x^14 - 3471*x^13 + 72951*x^12 - 287883*x^11 + 2446248*x^10 - 9084590*x^9 + 51165147*x^8 - 163367103*x^7 + 688224225*x^6 - 1766926269*x^5 + 5595461043*x^4 - 10346104119*x^3 + 23602542069*x^2 - 24485234526*x + 36321165287)
 
gp: K = bnfinit(x^18 - 3*x^17 + 51*x^16 - 30*x^15 + 1581*x^14 - 3471*x^13 + 72951*x^12 - 287883*x^11 + 2446248*x^10 - 9084590*x^9 + 51165147*x^8 - 163367103*x^7 + 688224225*x^6 - 1766926269*x^5 + 5595461043*x^4 - 10346104119*x^3 + 23602542069*x^2 - 24485234526*x + 36321165287, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 51 x^{16} - 30 x^{15} + 1581 x^{14} - 3471 x^{13} + 72951 x^{12} - 287883 x^{11} + 2446248 x^{10} - 9084590 x^{9} + 51165147 x^{8} - 163367103 x^{7} + 688224225 x^{6} - 1766926269 x^{5} + 5595461043 x^{4} - 10346104119 x^{3} + 23602542069 x^{2} - 24485234526 x + 36321165287 \)

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:  \(-287465843806740094425446907511025310072188928=-\,2^{12}\cdot 3^{30}\cdot 7^{14}\cdot 43^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $295.07$
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:  $2, 3, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{33} a^{16} + \frac{2}{33} a^{15} + \frac{1}{11} a^{14} + \frac{1}{33} a^{13} + \frac{4}{33} a^{12} - \frac{2}{33} a^{10} - \frac{4}{11} a^{9} + \frac{14}{33} a^{8} - \frac{2}{33} a^{7} + \frac{14}{33} a^{6} - \frac{10}{33} a^{5} + \frac{1}{11} a^{4} - \frac{8}{33} a^{3} + \frac{13}{33} a^{2} + \frac{13}{33} a + \frac{1}{3}$, $\frac{1}{40356656028564859650090761881825216660139778784631475885349323} a^{17} - \frac{40944108845885412432832120924003123473481804081968936346121}{13452218676188286550030253960608405553379926261543825295116441} a^{16} + \frac{6446621245015499633800404048232423553052248287022859964024940}{40356656028564859650090761881825216660139778784631475885349323} a^{15} + \frac{3161765658222111554815310438474770966795036597871089978451123}{40356656028564859650090761881825216660139778784631475885349323} a^{14} + \frac{778871704857468510691106465038761503224962330300474835001867}{40356656028564859650090761881825216660139778784631475885349323} a^{13} - \frac{246234539939756520389479022910650316816123849015093057327661}{13452218676188286550030253960608405553379926261543825295116441} a^{12} - \frac{247669172991207549297060987552042977109800305727098453387348}{13452218676188286550030253960608405553379926261543825295116441} a^{11} + \frac{1046852082917447065915816431369770132370150505501453375303159}{40356656028564859650090761881825216660139778784631475885349323} a^{10} + \frac{74950274387858877135714315396242792565691371066682123328961}{1222928970562571504548204905509855050307266023776711390465131} a^{9} - \frac{15103756847975185681643981618516673915889229789774204812058812}{40356656028564859650090761881825216660139778784631475885349323} a^{8} + \frac{6123921906698651911416225688153983613197762852303434637610190}{13452218676188286550030253960608405553379926261543825295116441} a^{7} + \frac{17223655537796448603711042085951612142476186055716204405220399}{40356656028564859650090761881825216660139778784631475885349323} a^{6} + \frac{375436462394577369019544585241388547563143349707127036799521}{13452218676188286550030253960608405553379926261543825295116441} a^{5} + \frac{1386901402030193198739906650155479246517009173793263994995268}{40356656028564859650090761881825216660139778784631475885349323} a^{4} + \frac{475717660284475845054459531127186876299405282823825099139957}{3668786911687714513644614716529565150921798071330134171395393} a^{3} - \frac{1221626558638268300569390892811046564550249723884655045786090}{13452218676188286550030253960608405553379926261543825295116441} a^{2} - \frac{4730170529319776026207884530699426650656607615352640793085831}{13452218676188286550030253960608405553379926261543825295116441} a + \frac{831754248006497054788155166021295790085640206567069095924494}{3668786911687714513644614716529565150921798071330134171395393}$

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

Class group and class number

$C_{6}\times C_{27232686}$, which has order $163396116$ (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:  \( 4695974.091249611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-43}) \), 3.3.3969.2, 3.3.756.1, 6.0.1252470670227.3, 6.0.45441112752.2, 9.9.756284282720064.2

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 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$43$43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43.6.3.2$x^{6} - 1849 x^{2} + 795070$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$