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Properties

Label 18.0.27668797159...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $120.26$
Ramified primes $2, 3, 5, 7$
Class number $2102464$ (GRH)
Class group $[2, 2, 2, 2, 38, 3458]$ (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![95320292761, -36356971812, 54409682934, -16785090334, 14201203935, -3934442502, 2338835099, -530370468, 238717983, -42497460, 15170424, -2062338, 598896, -58926, 14145, -914, 183, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 183*x^16 - 914*x^15 + 14145*x^14 - 58926*x^13 + 598896*x^12 - 2062338*x^11 + 15170424*x^10 - 42497460*x^9 + 238717983*x^8 - 530370468*x^7 + 2338835099*x^6 - 3934442502*x^5 + 14201203935*x^4 - 16785090334*x^3 + 54409682934*x^2 - 36356971812*x + 95320292761)
 
gp: K = bnfinit(x^18 - 6*x^17 + 183*x^16 - 914*x^15 + 14145*x^14 - 58926*x^13 + 598896*x^12 - 2062338*x^11 + 15170424*x^10 - 42497460*x^9 + 238717983*x^8 - 530370468*x^7 + 2338835099*x^6 - 3934442502*x^5 + 14201203935*x^4 - 16785090334*x^3 + 54409682934*x^2 - 36356971812*x + 95320292761, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 183 x^{16} - 914 x^{15} + 14145 x^{14} - 58926 x^{13} + 598896 x^{12} - 2062338 x^{11} + 15170424 x^{10} - 42497460 x^{9} + 238717983 x^{8} - 530370468 x^{7} + 2338835099 x^{6} - 3934442502 x^{5} + 14201203935 x^{4} - 16785090334 x^{3} + 54409682934 x^{2} - 36356971812 x + 95320292761 \)

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:  \(-27668797159880354103659593728000000000=-\,2^{27}\cdot 3^{27}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.26$
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 Galois and abelian over $\Q$.
Conductor:  \(2520=2^{3}\cdot 3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{2520}(1,·)$, $\chi_{2520}(389,·)$, $\chi_{2520}(961,·)$, $\chi_{2520}(1801,·)$, $\chi_{2520}(1229,·)$, $\chi_{2520}(1681,·)$, $\chi_{2520}(2069,·)$, $\chi_{2520}(121,·)$, $\chi_{2520}(29,·)$, $\chi_{2520}(869,·)$, $\chi_{2520}(1829,·)$, $\chi_{2520}(361,·)$, $\chi_{2520}(1709,·)$, $\chi_{2520}(989,·)$, $\chi_{2520}(1201,·)$, $\chi_{2520}(841,·)$, $\chi_{2520}(2041,·)$, $\chi_{2520}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{3}{10} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{10}$, $\frac{1}{10} a^{7} + \frac{3}{10} a^{5} + \frac{2}{5} a^{3} + \frac{3}{10} a + \frac{1}{5}$, $\frac{1}{10} a^{8} - \frac{2}{5} a^{5} + \frac{3}{10} a^{4} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{5} + \frac{1}{10} a^{3} + \frac{1}{5} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{10} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{11} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{100} a^{12} - \frac{1}{25} a^{11} - \frac{1}{50} a^{10} + \frac{1}{25} a^{9} - \frac{1}{100} a^{8} - \frac{1}{50} a^{7} + \frac{1}{50} a^{6} + \frac{17}{50} a^{5} - \frac{11}{25} a^{3} - \frac{3}{10} a^{2} + \frac{11}{50} a - \frac{49}{100}$, $\frac{1}{100} a^{13} + \frac{1}{50} a^{11} - \frac{1}{25} a^{10} - \frac{1}{20} a^{9} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{1}{50} a^{6} + \frac{13}{50} a^{5} + \frac{23}{50} a^{4} + \frac{7}{50} a^{3} + \frac{8}{25} a^{2} + \frac{49}{100} a - \frac{13}{50}$, $\frac{1}{100} a^{14} + \frac{1}{25} a^{11} - \frac{1}{100} a^{10} - \frac{1}{25} a^{9} - \frac{1}{25} a^{8} - \frac{1}{25} a^{7} + \frac{1}{50} a^{6} + \frac{7}{25} a^{5} + \frac{11}{25} a^{4} + \frac{2}{5} a^{3} - \frac{1}{100} a^{2} + \frac{2}{5} a - \frac{3}{25}$, $\frac{1}{100} a^{15} - \frac{1}{20} a^{11} + \frac{1}{25} a^{10} - \frac{1}{50} a^{5} - \frac{2}{5} a^{4} + \frac{3}{20} a^{3} - \frac{1}{5} a^{2} - \frac{3}{10} a + \frac{9}{25}$, $\frac{1}{13420503954857471252315724100} a^{16} - \frac{32246359055469224035974721}{13420503954857471252315724100} a^{15} - \frac{37151142828879079652561017}{13420503954857471252315724100} a^{14} + \frac{6636961623589406911354611}{2684100790971494250463144820} a^{13} - \frac{2359513286220642031812951}{536820158194298850092628964} a^{12} - \frac{340228545542913681801655739}{13420503954857471252315724100} a^{11} + \frac{526682763078914085063998973}{13420503954857471252315724100} a^{10} - \frac{481603885906596864440899107}{13420503954857471252315724100} a^{9} - \frac{161382475566979588572473671}{6710251977428735626157862050} a^{8} - \frac{50189266026824469869142751}{6710251977428735626157862050} a^{7} + \frac{31509123846062094783619011}{3355125988714367813078931025} a^{6} - \frac{2116785119726730239765531277}{6710251977428735626157862050} a^{5} - \frac{4267740684871288590855921813}{13420503954857471252315724100} a^{4} + \frac{64333433240634610412757657}{536820158194298850092628964} a^{3} - \frac{1193440601386627742759192413}{13420503954857471252315724100} a^{2} - \frac{3178033677187033371816231479}{13420503954857471252315724100} a + \frac{1292553393922368841783956559}{6710251977428735626157862050}$, $\frac{1}{240533032263247648589055920816443759724536900} a^{17} - \frac{6379957445220123}{240533032263247648589055920816443759724536900} a^{16} - \frac{41667051662292801024800940034996209582493}{240533032263247648589055920816443759724536900} a^{15} + \frac{995550219157999751387280744236148973866173}{240533032263247648589055920816443759724536900} a^{14} - \frac{553539748046867680633852165643425835872961}{240533032263247648589055920816443759724536900} a^{13} - \frac{586024517609845638659024443682416358781459}{240533032263247648589055920816443759724536900} a^{12} - \frac{1915634139929677468290274037451957250365071}{48106606452649529717811184163288751944907380} a^{11} + \frac{2306211972870017071942655900967319329504347}{48106606452649529717811184163288751944907380} a^{10} - \frac{1822523221537111546658609786751945711412717}{120266516131623824294527960408221879862268450} a^{9} + \frac{1788775222917648341301452932143859701425881}{120266516131623824294527960408221879862268450} a^{8} - \frac{3481714871499429722329335206945841799820621}{120266516131623824294527960408221879862268450} a^{7} + \frac{1518217962607749160620958857070264933439646}{60133258065811912147263980204110939931134225} a^{6} - \frac{44867745607477856097604816972033645406437653}{240533032263247648589055920816443759724536900} a^{5} - \frac{33272218549388244073641145890259601239125849}{240533032263247648589055920816443759724536900} a^{4} + \frac{109642973937733492787083256584564310638302853}{240533032263247648589055920816443759724536900} a^{3} - \frac{102731734100214737076627902677011208579352539}{240533032263247648589055920816443759724536900} a^{2} + \frac{25164065772212173474305230469010337754927418}{60133258065811912147263980204110939931134225} a - \frac{30568547640677988721781982864284048824834823}{120266516131623824294527960408221879862268450}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{38}\times C_{3458}$, which has order $2102464$ (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

\(\Q(\sqrt{-30}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 3.3.3969.2, 6.0.1259712000.1, 6.0.3024568512000.19, 6.0.4148928000.4, 6.0.3024568512000.20, 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 R R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed