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Properties

Label 18.6.33219791073...4624.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{37}\cdot 23^{9}$
Root discriminant $72.83$
Ramified primes $2, 3, 23$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_3:S_4$ (as 18T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59087, 458298, -1003104, 351114, 542700, -598968, 44463, 184356, -72756, -556, 11664, -7848, 459, 1206, -180, -54, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 54*x^15 - 180*x^14 + 1206*x^13 + 459*x^12 - 7848*x^11 + 11664*x^10 - 556*x^9 - 72756*x^8 + 184356*x^7 + 44463*x^6 - 598968*x^5 + 542700*x^4 + 351114*x^3 - 1003104*x^2 + 458298*x - 59087)
 
gp: K = bnfinit(x^18 - 54*x^15 - 180*x^14 + 1206*x^13 + 459*x^12 - 7848*x^11 + 11664*x^10 - 556*x^9 - 72756*x^8 + 184356*x^7 + 44463*x^6 - 598968*x^5 + 542700*x^4 + 351114*x^3 - 1003104*x^2 + 458298*x - 59087, 1)
 

Normalized defining polynomial

\( x^{18} - 54 x^{15} - 180 x^{14} + 1206 x^{13} + 459 x^{12} - 7848 x^{11} + 11664 x^{10} - 556 x^{9} - 72756 x^{8} + 184356 x^{7} + 44463 x^{6} - 598968 x^{5} + 542700 x^{4} + 351114 x^{3} - 1003104 x^{2} + 458298 x - 59087 \)

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:  \(3321979107367014091730799501594624=2^{12}\cdot 3^{37}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.83$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{15} - \frac{1}{5} a^{14} - \frac{1}{10} a^{13} - \frac{3}{20} a^{12} + \frac{1}{5} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{3}{10} a^{7} + \frac{3}{10} a^{6} + \frac{1}{5} a^{5} + \frac{3}{10} a^{4} + \frac{3}{20} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{3}{20}$, $\frac{1}{25460} a^{16} + \frac{109}{12730} a^{15} - \frac{227}{2546} a^{14} - \frac{3757}{25460} a^{13} - \frac{2161}{12730} a^{12} + \frac{105}{1273} a^{11} - \frac{123}{1273} a^{10} + \frac{657}{6365} a^{9} - \frac{231}{6365} a^{8} + \frac{2457}{12730} a^{7} - \frac{1571}{6365} a^{6} + \frac{2306}{6365} a^{5} + \frac{1063}{5092} a^{4} - \frac{6029}{12730} a^{3} - \frac{1191}{2546} a^{2} - \frac{457}{25460} a + \frac{4497}{12730}$, $\frac{1}{124806693229286582335609817943004120820} a^{17} - \frac{868640611249895037437228756171141}{124806693229286582335609817943004120820} a^{16} - \frac{217156195367489708015096182067306529}{24961338645857316467121963588600824164} a^{15} + \frac{32791641247699244384402754422052283}{860735815374390223004205640986235316} a^{14} + \frac{3581018745252354601619729295741162061}{17829527604183797476515688277572017260} a^{13} + \frac{18196570514369448174878675125824164537}{124806693229286582335609817943004120820} a^{12} + \frac{7372521525866090583840792457736713307}{31201673307321645583902454485751030205} a^{11} - \frac{8845026169908995437359070227834278689}{62403346614643291167804908971502060410} a^{10} - \frac{7043897694266438108405496670390588257}{62403346614643291167804908971502060410} a^{9} + \frac{6240401397380039431534379775971051151}{62403346614643291167804908971502060410} a^{8} - \frac{12083790006656185451011105846640015538}{31201673307321645583902454485751030205} a^{7} - \frac{2098197353211889204151758108079618101}{4457381901045949369128922069393004315} a^{6} + \frac{37401877615947268219331177379306579387}{124806693229286582335609817943004120820} a^{5} + \frac{38592527393205041026706188105039005319}{124806693229286582335609817943004120820} a^{4} - \frac{36099479494043958750032848864891539157}{124806693229286582335609817943004120820} a^{3} - \frac{10189682833160903263803331903162102657}{24961338645857316467121963588600824164} a^{2} - \frac{28819818426112808239102939523344309467}{124806693229286582335609817943004120820} a + \frac{1208810395993724381334394419243258509}{17829527604183797476515688277572017260}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 6891398522.29 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.22356.1, 3.3.22356.3, 3.3.621.1, 3.3.22356.2, 6.2.34485560784.5, 9.9.6938632771983936.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$