↑

Properties

Label 18.0.42156140023...0128.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 29^{9}$
Root discriminant $139.90$
Ramified primes $2, 3, 7, 29$
Class number $4898880$ (GRH)
Class group $[2, 2, 2, 18, 18, 1890]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![93654474472, -69223313904, 46526330856, -7029585108, 4016812476, -539106924, 372512932, -179179878, 84214434, -26756821, 10057503, -1932168, 579766, -72270, 15900, -1310, 204, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 204*x^16 - 1310*x^15 + 15900*x^14 - 72270*x^13 + 579766*x^12 - 1932168*x^11 + 10057503*x^10 - 26756821*x^9 + 84214434*x^8 - 179179878*x^7 + 372512932*x^6 - 539106924*x^5 + 4016812476*x^4 - 7029585108*x^3 + 46526330856*x^2 - 69223313904*x + 93654474472)
 
gp: K = bnfinit(x^18 - 9*x^17 + 204*x^16 - 1310*x^15 + 15900*x^14 - 72270*x^13 + 579766*x^12 - 1932168*x^11 + 10057503*x^10 - 26756821*x^9 + 84214434*x^8 - 179179878*x^7 + 372512932*x^6 - 539106924*x^5 + 4016812476*x^4 - 7029585108*x^3 + 46526330856*x^2 - 69223313904*x + 93654474472, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 204 x^{16} - 1310 x^{15} + 15900 x^{14} - 72270 x^{13} + 579766 x^{12} - 1932168 x^{11} + 10057503 x^{10} - 26756821 x^{9} + 84214434 x^{8} - 179179878 x^{7} + 372512932 x^{6} - 539106924 x^{5} + 4016812476 x^{4} - 7029585108 x^{3} + 46526330856 x^{2} - 69223313904 x + 93654474472 \)

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:  \(-421561400235330951371150243189110960128=-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $139.90$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{28} a^{15} + \frac{3}{28} a^{14} + \frac{3}{28} a^{13} - \frac{3}{28} a^{12} + \frac{1}{28} a^{11} + \frac{3}{28} a^{10} - \frac{3}{28} a^{9} + \frac{3}{28} a^{8} + \frac{1}{14} a^{7} - \frac{1}{7} a^{6} + \frac{1}{14} a^{5} - \frac{1}{14} a^{4} + \frac{5}{14} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{28} a^{16} + \frac{1}{28} a^{14} + \frac{1}{14} a^{13} + \frac{3}{28} a^{12} - \frac{5}{28} a^{10} - \frac{1}{14} a^{9} + \frac{1}{7} a^{7} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{252403043013987454662782507587218629084287668450956902349069765870889523390461021630724} a^{17} - \frac{3262797855740393586657822914922926175928618465424230475295698202544154977650780891929}{252403043013987454662782507587218629084287668450956902349069765870889523390461021630724} a^{16} + \frac{43463543423803804166338559683480216351078844153620825360052101075018820129134941290}{63100760753496863665695626896804657271071917112739225587267441467722380847615255407681} a^{15} + \frac{271691800437337842753306870929295582005488792690409516638982892325135907952965604897}{4853904673345912589668894376677281328543993624056863506712880112901721603662711954437} a^{14} - \frac{481355466688601226622448694283628140628557317861964566384912640222346520629865146395}{18028788786713389618770179113372759220306262032211207310647840419349251670747215830766} a^{13} - \frac{2586070153479078922133624421641980528022819742435397670229643811830361131248350749075}{36057577573426779237540358226745518440612524064422414621295680838698503341494431661532} a^{12} - \frac{2023043518065097706129088233613977256650846945199285298013352088638145387823040678575}{18028788786713389618770179113372759220306262032211207310647840419349251670747215830766} a^{11} + \frac{19572669998336256589128854500631223761517458680767744571595002694741576916192226171911}{252403043013987454662782507587218629084287668450956902349069765870889523390461021630724} a^{10} - \frac{48513979820160292335581771665661902178535597891250795400683863178042251257080155636457}{252403043013987454662782507587218629084287668450956902349069765870889523390461021630724} a^{9} - \frac{30214995960062230905557182421974901836084534525796168852980769318647521118632351517791}{126201521506993727331391253793609314542143834225478451174534882935444761695230510815362} a^{8} + \frac{439406213611966889301971757530855868924081020480439834879368873163816264669773088486}{4853904673345912589668894376677281328543993624056863506712880112901721603662711954437} a^{7} - \frac{3577885618454434960482401568189085614321759796697963733025594311468009181268455137607}{36057577573426779237540358226745518440612524064422414621295680838698503341494431661532} a^{6} + \frac{4335839538672450313629600915207417706889374016748993947747552727282436153816123260590}{9014394393356694809385089556686379610153131016105603655323920209674625835373607915383} a^{5} - \frac{1311244515658881203365826586462620858894724737572313104919732335187186853851818885171}{18028788786713389618770179113372759220306262032211207310647840419349251670747215830766} a^{4} - \frac{10381495310635678436726482548906148110421692495818115401625203335442826385396306342087}{63100760753496863665695626896804657271071917112739225587267441467722380847615255407681} a^{3} + \frac{20506429593582282308830505901724764958097533689857196197553970394640748874822477977696}{63100760753496863665695626896804657271071917112739225587267441467722380847615255407681} a^{2} + \frac{21280720033604770309656093682763166774395349824946441448489345222613383797666304953454}{63100760753496863665695626896804657271071917112739225587267441467722380847615255407681} a + \frac{900765432785292483468917814318163026453284445915994198006093785473471584876983233997}{4853904673345912589668894376677281328543993624056863506712880112901721603662711954437}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}\times C_{18}\times C_{1890}$, which has order $4898880$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-87}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.41817574512.4, 6.0.1581065703.2, 9.9.1037426999616.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$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}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$