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Properties

Label 18.0.13016000727...0704.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{44}\cdot 19^{9}$
Root discriminant $101.48$
Ramified primes $2, 3, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4065589, -9066771, 12402243, -14146866, 13103532, -8902854, 5367378, -2967102, 1494549, -657675, 266895, -93870, 30900, -8460, 2196, -432, 81, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 81*x^16 - 432*x^15 + 2196*x^14 - 8460*x^13 + 30900*x^12 - 93870*x^11 + 266895*x^10 - 657675*x^9 + 1494549*x^8 - 2967102*x^7 + 5367378*x^6 - 8902854*x^5 + 13103532*x^4 - 14146866*x^3 + 12402243*x^2 - 9066771*x + 4065589)
 
gp: K = bnfinit(x^18 - 9*x^17 + 81*x^16 - 432*x^15 + 2196*x^14 - 8460*x^13 + 30900*x^12 - 93870*x^11 + 266895*x^10 - 657675*x^9 + 1494549*x^8 - 2967102*x^7 + 5367378*x^6 - 8902854*x^5 + 13103532*x^4 - 14146866*x^3 + 12402243*x^2 - 9066771*x + 4065589, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 81 x^{16} - 432 x^{15} + 2196 x^{14} - 8460 x^{13} + 30900 x^{12} - 93870 x^{11} + 266895 x^{10} - 657675 x^{9} + 1494549 x^{8} - 2967102 x^{7} + 5367378 x^{6} - 8902854 x^{5} + 13103532 x^{4} - 14146866 x^{3} + 12402243 x^{2} - 9066771 x + 4065589 \)

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:  \(-1301600072766989568748477298570440704=-\,2^{12}\cdot 3^{44}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.48$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{6} a^{8} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{6} a^{9} + \frac{1}{12} a^{7} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{6} a^{9} - \frac{1}{8} a^{8} + \frac{5}{24} a^{7} - \frac{1}{24} a^{6} + \frac{1}{6} a^{5} + \frac{5}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{12} a^{2} - \frac{1}{8} a - \frac{7}{24}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{24} a^{10} + \frac{5}{24} a^{9} + \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{5}{24} a^{6} + \frac{3}{8} a^{5} - \frac{1}{12} a^{4} + \frac{7}{24} a^{3} + \frac{5}{24} a^{2} - \frac{5}{12} a - \frac{3}{8}$, $\frac{1}{48} a^{16} - \frac{1}{12} a^{10} - \frac{1}{8} a^{9} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{6} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{5}{16}$, $\frac{1}{445159947017028759294049009035879199351658504304} a^{17} - \frac{119702732586837751680474053356735427426428877}{55644993377128594911756126129484899918957313038} a^{16} - \frac{3649176071010902117508151207439555748067055921}{222579973508514379647024504517939599675829252152} a^{15} - \frac{258847560461238906862582715963582709472354993}{27822496688564297455878063064742449959478656519} a^{14} - \frac{6027037453989875028259733058990187683070651403}{222579973508514379647024504517939599675829252152} a^{13} - \frac{1282240728949726984105817721651406818282242363}{222579973508514379647024504517939599675829252152} a^{12} + \frac{842932962449810129869916479762103521582978905}{37096662251419063274504084086323266612638208692} a^{11} + \frac{411113151561302920194502088613544356733830427}{27822496688564297455878063064742449959478656519} a^{10} + \frac{34710613426362257720799954528865763068707469269}{148386649005676253098016336345293066450552834768} a^{9} + \frac{5669282946677919605624537326695472308591915409}{27822496688564297455878063064742449959478656519} a^{8} + \frac{2784440875916492008523784726729284952190573127}{111289986754257189823512252258969799837914626076} a^{7} + \frac{18375306820010302323115143798530612266627894471}{111289986754257189823512252258969799837914626076} a^{6} + \frac{42854320427588526890677189793881769355023404171}{222579973508514379647024504517939599675829252152} a^{5} - \frac{11820154243942133447702211063875913143994641669}{55644993377128594911756126129484899918957313038} a^{4} + \frac{89168077155559740090398884398782949501573765875}{222579973508514379647024504517939599675829252152} a^{3} - \frac{7893571771391983206488800553754849262071618091}{27822496688564297455878063064742449959478656519} a^{2} - \frac{12367219219186337273404291094285738597160036497}{148386649005676253098016336345293066450552834768} a - \frac{80934185539168352323689009267181011779709666203}{222579973508514379647024504517939599675829252152}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 60513975223.29309 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.1.243.1, 6.0.405017091.1, 9.1.2008387814976.4

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$