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Properties

Label 18.0.44083395769...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{18}\cdot 5^{9}\cdot 7^{14}$
Root discriminant $108.59$
Ramified primes $2, 3, 5, 7$
Class number $407592$ (GRH)
Class group $[6, 6, 11322]$ (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![17040056384, 3796985088, 6833853696, 885349568, 1241215776, 90910848, 151261200, 6632928, 13801632, 298064, 940608, 4800, 49888, 0, 1980, -16, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 54*x^16 - 16*x^15 + 1980*x^14 + 49888*x^12 + 4800*x^11 + 940608*x^10 + 298064*x^9 + 13801632*x^8 + 6632928*x^7 + 151261200*x^6 + 90910848*x^5 + 1241215776*x^4 + 885349568*x^3 + 6833853696*x^2 + 3796985088*x + 17040056384)
 
gp: K = bnfinit(x^18 + 54*x^16 - 16*x^15 + 1980*x^14 + 49888*x^12 + 4800*x^11 + 940608*x^10 + 298064*x^9 + 13801632*x^8 + 6632928*x^7 + 151261200*x^6 + 90910848*x^5 + 1241215776*x^4 + 885349568*x^3 + 6833853696*x^2 + 3796985088*x + 17040056384, 1)
 

Normalized defining polynomial

\( x^{18} + 54 x^{16} - 16 x^{15} + 1980 x^{14} + 49888 x^{12} + 4800 x^{11} + 940608 x^{10} + 298064 x^{9} + 13801632 x^{8} + 6632928 x^{7} + 151261200 x^{6} + 90910848 x^{5} + 1241215776 x^{4} + 885349568 x^{3} + 6833853696 x^{2} + 3796985088 x + 17040056384 \)

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:  \(-4408339576964120838747979776000000000=-\,2^{33}\cdot 3^{18}\cdot 5^{9}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.59$
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 not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{8} a^{10}$, $\frac{1}{8} a^{11}$, $\frac{1}{16} a^{12}$, $\frac{1}{16} a^{13}$, $\frac{1}{16} a^{14}$, $\frac{1}{32} a^{15}$, $\frac{1}{416} a^{16} + \frac{5}{416} a^{15} - \frac{3}{104} a^{14} + \frac{1}{208} a^{13} - \frac{3}{104} a^{12} - \frac{1}{52} a^{11} - \frac{5}{104} a^{10} + \frac{5}{104} a^{9} - \frac{3}{52} a^{8} - \frac{1}{26} a^{7} + \frac{3}{52} a^{6} + \frac{1}{13} a^{5} + \frac{1}{26} a^{4} - \frac{3}{26} a^{3} - \frac{4}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{581870358363040840820703573873703375142709278953105696} a^{17} - \frac{101434144320984785004163378516785934007196111603713}{581870358363040840820703573873703375142709278953105696} a^{16} - \frac{3648529444148413165350944837708255944934184921671907}{290935179181520420410351786936851687571354639476552848} a^{15} + \frac{797483615334785147295549118619259203009803417401181}{36366897397690052551293973367106460946419329934569106} a^{14} - \frac{419003300437581101901011553946706517914943438763006}{18183448698845026275646986683553230473209664967284553} a^{13} - \frac{8924566690342344701922953771728772068606316189229983}{290935179181520420410351786936851687571354639476552848} a^{12} - \frac{41386113596101768219631257925307406058261817663171}{1398726822988078944280537437196402344093051151329581} a^{11} - \frac{1643244073779972411029267613112934775627438082119813}{145467589590760210205175893468425843785677319738276424} a^{10} - \frac{7467673328702749225893745825137425714635616428309313}{145467589590760210205175893468425843785677319738276424} a^{9} - \frac{5368361305485908786423256271180406342574233000224827}{72733794795380105102587946734212921892838659869138212} a^{8} - \frac{7838891404832973848843544782006308461221773741674711}{72733794795380105102587946734212921892838659869138212} a^{7} - \frac{13654311774300044865379044513774206319987647766887}{172354963970095035788123096526570904959333317225446} a^{6} + \frac{903829993753386433651871624582724714346844544520787}{36366897397690052551293973367106460946419329934569106} a^{5} + \frac{6746831047156058584414106133988472480216780612083017}{36366897397690052551293973367106460946419329934569106} a^{4} + \frac{2334558500987217759755200906947782165290596704868831}{18183448698845026275646986683553230473209664967284553} a^{3} - \frac{3646224711475022206166947807001680072043729980400245}{18183448698845026275646986683553230473209664967284553} a^{2} - \frac{4239478823625516984050119876930342948161588128285182}{18183448698845026275646986683553230473209664967284553} a - \frac{104599723995408627667978255078394527508698670853054}{1398726822988078944280537437196402344093051151329581}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{11322}$, which has order $407592$ (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

\(\Q(\sqrt{-10}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.9144576000.34, 6.0.153664000.1, 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$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}$