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Properties

Label 18.0.15312014746...0303.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 29^{9}$
Root discriminant $102.40$
Ramified primes $3, 7, 29$
Class number $830088$ (GRH)
Class group $[6, 6, 23058]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10029275353, -814254078, 6188670615, -405210528, 1803174465, -48374466, 286497301, -10998372, 33312717, -2959990, 3000297, -363096, 184456, -20664, 6714, -560, 129, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 129*x^16 - 560*x^15 + 6714*x^14 - 20664*x^13 + 184456*x^12 - 363096*x^11 + 3000297*x^10 - 2959990*x^9 + 33312717*x^8 - 10998372*x^7 + 286497301*x^6 - 48374466*x^5 + 1803174465*x^4 - 405210528*x^3 + 6188670615*x^2 - 814254078*x + 10029275353)
 
gp: K = bnfinit(x^18 - 6*x^17 + 129*x^16 - 560*x^15 + 6714*x^14 - 20664*x^13 + 184456*x^12 - 363096*x^11 + 3000297*x^10 - 2959990*x^9 + 33312717*x^8 - 10998372*x^7 + 286497301*x^6 - 48374466*x^5 + 1803174465*x^4 - 405210528*x^3 + 6188670615*x^2 - 814254078*x + 10029275353, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 129 x^{16} - 560 x^{15} + 6714 x^{14} - 20664 x^{13} + 184456 x^{12} - 363096 x^{11} + 3000297 x^{10} - 2959990 x^{9} + 33312717 x^{8} - 10998372 x^{7} + 286497301 x^{6} - 48374466 x^{5} + 1803174465 x^{4} - 405210528 x^{3} + 6188670615 x^{2} - 814254078 x + 10029275353 \)

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:  \(-1531201474666953640931762831258280303=-\,3^{27}\cdot 7^{12}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $102.40$
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:  $3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1827=3^{2}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1827}(1,·)$, $\chi_{1827}(1219,·)$, $\chi_{1827}(260,·)$, $\chi_{1827}(1478,·)$, $\chi_{1827}(1304,·)$, $\chi_{1827}(1045,·)$, $\chi_{1827}(86,·)$, $\chi_{1827}(88,·)$, $\chi_{1827}(1306,·)$, $\chi_{1827}(347,·)$, $\chi_{1827}(1565,·)$, $\chi_{1827}(610,·)$, $\chi_{1827}(869,·)$, $\chi_{1827}(436,·)$, $\chi_{1827}(1654,·)$, $\chi_{1827}(695,·)$, $\chi_{1827}(697,·)$, $\chi_{1827}(956,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} - \frac{7}{16} a^{3} - \frac{1}{16} a^{2} + \frac{1}{16} a - \frac{3}{16}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} + \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{7}{16} a^{3} + \frac{7}{16} a^{2} - \frac{7}{16}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{1}{32} a^{5} + \frac{7}{32} a^{4} + \frac{15}{32} a^{3} - \frac{1}{8} a^{2} - \frac{13}{32} a + \frac{7}{32}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} + \frac{1}{32} a^{6} - \frac{5}{32} a^{5} - \frac{5}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{32} a^{2} - \frac{5}{32} a + \frac{1}{8}$, $\frac{1}{26464} a^{14} + \frac{139}{13232} a^{13} + \frac{259}{26464} a^{12} - \frac{749}{26464} a^{11} + \frac{167}{13232} a^{10} + \frac{71}{3308} a^{9} - \frac{1091}{13232} a^{8} - \frac{1217}{26464} a^{7} + \frac{2037}{26464} a^{6} - \frac{1287}{26464} a^{5} + \frac{303}{13232} a^{4} - \frac{1093}{26464} a^{3} + \frac{3825}{26464} a^{2} + \frac{135}{6616} a - \frac{285}{3308}$, $\frac{1}{26464} a^{15} - \frac{57}{13232} a^{13} + \frac{25}{26464} a^{12} - \frac{675}{26464} a^{11} - \frac{487}{26464} a^{10} + \frac{11}{827} a^{9} - \frac{3293}{26464} a^{8} - \frac{2015}{26464} a^{7} + \frac{1115}{13232} a^{6} - \frac{4661}{26464} a^{5} - \frac{13}{13232} a^{4} + \frac{11613}{26464} a^{3} - \frac{3423}{26464} a^{2} - \frac{6021}{26464} a - \frac{3129}{13232}$, $\frac{1}{52928} a^{16} - \frac{67}{6616} a^{13} - \frac{47}{26464} a^{12} - \frac{173}{13232} a^{11} + \frac{193}{26464} a^{10} + \frac{17}{827} a^{9} + \frac{2299}{52928} a^{8} - \frac{2701}{13232} a^{7} + \frac{5855}{26464} a^{6} + \frac{529}{13232} a^{5} + \frac{2959}{52928} a^{4} + \frac{2521}{13232} a^{3} - \frac{419}{26464} a^{2} - \frac{391}{1654} a - \frac{19263}{52928}$, $\frac{1}{108809009162877462660216500013146207148225456066709893258368} a^{17} - \frac{173956497605488305581835106191578027628065108340196817}{108809009162877462660216500013146207148225456066709893258368} a^{16} + \frac{163302759913821915601477823657450052632039986515738657}{27202252290719365665054125003286551787056364016677473314592} a^{15} + \frac{9327734455970214501337815823308262580490625655512059}{13601126145359682832527062501643275893528182008338736657296} a^{14} - \frac{299459717398468805997772255382474939135439746869867594525}{54404504581438731330108250006573103574112728033354946629184} a^{13} + \frac{496067969870034190579536463602927248090351479097250164299}{54404504581438731330108250006573103574112728033354946629184} a^{12} - \frac{330596123139054241735655054448919769125947557688188410887}{54404504581438731330108250006573103574112728033354946629184} a^{11} - \frac{648500068156849113823816843803048508133840898825849180627}{54404504581438731330108250006573103574112728033354946629184} a^{10} - \frac{2906765199720049447244294799815163347373736997458623055021}{108809009162877462660216500013146207148225456066709893258368} a^{9} + \frac{4248766179002158530228681137557918287541085734821155094189}{108809009162877462660216500013146207148225456066709893258368} a^{8} - \frac{1824758370341167151188176644965068256644433745638187392939}{54404504581438731330108250006573103574112728033354946629184} a^{7} + \frac{10889149805208984639570499590344912596833804061598479987735}{54404504581438731330108250006573103574112728033354946629184} a^{6} - \frac{16538705417879514717314649771588716490638660066626686658921}{108809009162877462660216500013146207148225456066709893258368} a^{5} + \frac{1906356410272476542196892814197200652452449674006923558773}{108809009162877462660216500013146207148225456066709893258368} a^{4} - \frac{25329172883822757070652170794724697611693826543752253061729}{54404504581438731330108250006573103574112728033354946629184} a^{3} - \frac{19578307498106097212708205710669724993250697413536139883327}{54404504581438731330108250006573103574112728033354946629184} a^{2} + \frac{37647715507237930671343528362017946290380764000626356268817}{108809009162877462660216500013146207148225456066709893258368} a - \frac{39256826314490495753206918943244260440051496043046859915177}{108809009162877462660216500013146207148225456066709893258368}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{23058}$, which has order $830088$ (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:  \( 54408.48888868202 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-87}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.480048687.1, 6.0.1152596897487.9, 6.0.1581065703.2, 6.0.1152596897487.8, 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ 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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$