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Properties

Label 18.6.31529384066...7904.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{33}\cdot 3^{24}\cdot 37^{9}$
Root discriminant $93.79$
Ramified primes $2, 3, 37$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21697, -316380, 492588, 1256326, -2067246, 1196208, -830151, 339900, 37062, 752, -27270, 7344, -3323, 1128, 198, -78, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 78*x^15 + 198*x^14 + 1128*x^13 - 3323*x^12 + 7344*x^11 - 27270*x^10 + 752*x^9 + 37062*x^8 + 339900*x^7 - 830151*x^6 + 1196208*x^5 - 2067246*x^4 + 1256326*x^3 + 492588*x^2 - 316380*x - 21697)
 
gp: K = bnfinit(x^18 - 12*x^16 - 78*x^15 + 198*x^14 + 1128*x^13 - 3323*x^12 + 7344*x^11 - 27270*x^10 + 752*x^9 + 37062*x^8 + 339900*x^7 - 830151*x^6 + 1196208*x^5 - 2067246*x^4 + 1256326*x^3 + 492588*x^2 - 316380*x - 21697, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 78 x^{15} + 198 x^{14} + 1128 x^{13} - 3323 x^{12} + 7344 x^{11} - 27270 x^{10} + 752 x^{9} + 37062 x^{8} + 339900 x^{7} - 830151 x^{6} + 1196208 x^{5} - 2067246 x^{4} + 1256326 x^{3} + 492588 x^{2} - 316380 x - 21697 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(315293840660890978521435310500347904=2^{33}\cdot 3^{24}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.79$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{13} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{4} - \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{57575322562052293873217758776478816120815936369689106545} a^{17} - \frac{4062980295477093662124306073365637013009569812149575046}{57575322562052293873217758776478816120815936369689106545} a^{16} - \frac{677123605402030699957786542077149956853228167770854131}{57575322562052293873217758776478816120815936369689106545} a^{15} + \frac{21342941506486384588341916524838120506928083895406580848}{57575322562052293873217758776478816120815936369689106545} a^{14} + \frac{491266408480190004189687062689177297821479928012161686}{11515064512410458774643551755295763224163187273937821309} a^{13} + \frac{3808908622531690292666180522894129415575048065791984959}{8225046080293184839031108396639830874402276624241300935} a^{12} + \frac{548672545468609631151111243058624869309145683867535297}{8225046080293184839031108396639830874402276624241300935} a^{11} - \frac{3394170169389777783655377640772566696423842999952595152}{11515064512410458774643551755295763224163187273937821309} a^{10} - \frac{3314699212008850551606053115810196575395148533133237803}{11515064512410458774643551755295763224163187273937821309} a^{9} - \frac{18201027391801585810292966161487989881121325029017129213}{57575322562052293873217758776478816120815936369689106545} a^{8} + \frac{5510278037992785505231419125754671190296566628121811592}{11515064512410458774643551755295763224163187273937821309} a^{7} + \frac{2723557319868557958033115053989976902661871939801340122}{11515064512410458774643551755295763224163187273937821309} a^{6} + \frac{2825022738180751600513031448029822348503669287030141344}{57575322562052293873217758776478816120815936369689106545} a^{5} + \frac{7329163531503365607135556024888241791473191535530025334}{57575322562052293873217758776478816120815936369689106545} a^{4} - \frac{563048175862287287464268616679603384591926106267594152}{1645009216058636967806221679327966174880455324848260187} a^{3} + \frac{14007718738531006812314190625583350411251962572526929516}{57575322562052293873217758776478816120815936369689106545} a^{2} + \frac{1420227820497020592307905926912171960615525054351875986}{8225046080293184839031108396639830874402276624241300935} a - \frac{14613459392988270111463822104468204898683880613438173427}{57575322562052293873217758776478816120815936369689106545}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $11$
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:  \( 48662807084.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.148.1, 6.2.103737344.5, 9.9.220521111330816.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
3Data not computed
$37$37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.8.6.1$x^{8} - 1147 x^{4} + 855625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$