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Properties

Label 18.18.1359066798...4192.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{16}\cdot 3^{32}\cdot 47^{9}$
Root discriminant $89.50$
Ramified primes $2, 3, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![156356, -218364, -744636, 819388, 1244082, -1149072, -922425, 778449, 343368, -282822, -65787, 57606, 5677, -6540, -33, 384, -24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 24*x^16 + 384*x^15 - 33*x^14 - 6540*x^13 + 5677*x^12 + 57606*x^11 - 65787*x^10 - 282822*x^9 + 343368*x^8 + 778449*x^7 - 922425*x^6 - 1149072*x^5 + 1244082*x^4 + 819388*x^3 - 744636*x^2 - 218364*x + 156356)
 
gp: K = bnfinit(x^18 - 9*x^17 - 24*x^16 + 384*x^15 - 33*x^14 - 6540*x^13 + 5677*x^12 + 57606*x^11 - 65787*x^10 - 282822*x^9 + 343368*x^8 + 778449*x^7 - 922425*x^6 - 1149072*x^5 + 1244082*x^4 + 819388*x^3 - 744636*x^2 - 218364*x + 156356, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 24 x^{16} + 384 x^{15} - 33 x^{14} - 6540 x^{13} + 5677 x^{12} + 57606 x^{11} - 65787 x^{10} - 282822 x^{9} + 343368 x^{8} + 778449 x^{7} - 922425 x^{6} - 1149072 x^{5} + 1244082 x^{4} + 819388 x^{3} - 744636 x^{2} - 218364 x + 156356 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(135906679889509709971359671520264192=2^{16}\cdot 3^{32}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.50$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{14} - \frac{1}{6} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{114} a^{16} + \frac{7}{114} a^{15} + \frac{3}{19} a^{14} - \frac{4}{57} a^{13} - \frac{5}{38} a^{12} + \frac{2}{19} a^{11} - \frac{35}{114} a^{10} + \frac{7}{19} a^{9} - \frac{1}{38} a^{8} - \frac{2}{19} a^{7} + \frac{28}{57} a^{6} + \frac{15}{38} a^{5} + \frac{17}{114} a^{4} + \frac{10}{57} a^{3} + \frac{7}{19} a^{2} - \frac{2}{19} a + \frac{17}{57}$, $\frac{1}{19336879577958825660832827890538} a^{17} + \frac{35932954807524251877503543252}{9668439788979412830416413945269} a^{16} - \frac{729349687939573255833497189203}{19336879577958825660832827890538} a^{15} + \frac{344339489565297343732876853917}{3222813262993137610138804648423} a^{14} - \frac{3063903472928023649027455090501}{19336879577958825660832827890538} a^{13} - \frac{6456834022117513776517481467}{1017730504103096087412254099502} a^{12} - \frac{2511068695611253389415799282695}{6445626525986275220277609296846} a^{11} + \frac{9247627176497182318669240595585}{19336879577958825660832827890538} a^{10} + \frac{2240713238898336022741376821151}{19336879577958825660832827890538} a^{9} + \frac{2186309828647892481482120838833}{19336879577958825660832827890538} a^{8} - \frac{3655450406820874918824891682198}{9668439788979412830416413945269} a^{7} + \frac{46061470417440419663251531}{6445626525986275220277609296846} a^{6} - \frac{2701487069445786152754404733878}{9668439788979412830416413945269} a^{5} - \frac{4454955308065210303738837507669}{19336879577958825660832827890538} a^{4} - \frac{2692542931409369462894447778364}{9668439788979412830416413945269} a^{3} + \frac{1466643034816742117066995260941}{9668439788979412830416413945269} a^{2} - \frac{776583374183471360881335988211}{9668439788979412830416413945269} a - \frac{1301252052489267966837751235160}{3222813262993137610138804648423}$

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:  $17$
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:  \( 510402720815 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.45684.1, 3.3.564.1, 3.3.45684.2, 3.3.11421.1, 6.6.24522577308.1, 9.9.13443473060864064.2

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.67$x^{12} + 99 x^{11} + 117 x^{10} - 114 x^{9} - 81 x^{8} + 9 x^{7} - 15 x^{6} + 54 x^{5} - 108 x^{4} + 45 x^{3} - 81 x^{2} - 108 x - 72$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
47Data not computed