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Properties

Label 18.0.33703610765...3024.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 79^{15}$
Root discriminant $261.94$
Ramified primes $2, 3, 79$
Class number $18957120$ (GRH)
Class group $[2, 2, 28, 169260]$ (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![2019487744, 0, 463401439632, 0, 260620415400, 0, 53819444489, 0, 5138452458, 0, 246937647, 0, 6224884, 0, 80343, 0, 474, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744)
 
gp: K = bnfinit(x^18 + 474*x^16 + 80343*x^14 + 6224884*x^12 + 246937647*x^10 + 5138452458*x^8 + 53819444489*x^6 + 260620415400*x^4 + 463401439632*x^2 + 2019487744, 1)
 

Normalized defining polynomial

\( x^{18} + 474 x^{16} + 80343 x^{14} + 6224884 x^{12} + 246937647 x^{10} + 5138452458 x^{8} + 53819444489 x^{6} + 260620415400 x^{4} + 463401439632 x^{2} + 2019487744 \)

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:  \(-33703610765963698533467381091270298700673024=-\,2^{12}\cdot 3^{24}\cdot 79^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.94$
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:  $2, 3, 79$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{632} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2528} a^{7} - \frac{1}{16} a^{5} + \frac{7}{32} a^{3} + \frac{3}{8} a$, $\frac{1}{5056} a^{8} - \frac{1}{5056} a^{7} + \frac{1}{2528} a^{6} + \frac{1}{32} a^{5} + \frac{7}{64} a^{4} + \frac{9}{64} a^{3} + \frac{3}{16} a^{2} + \frac{1}{16} a$, $\frac{1}{10112} a^{9} + \frac{1}{10112} a^{7} + \frac{9}{128} a^{5} - \frac{1}{8} a^{4} - \frac{11}{128} a^{3} + \frac{1}{8} a^{2} - \frac{7}{32} a - \frac{1}{2}$, $\frac{1}{20224} a^{10} + \frac{1}{20224} a^{8} - \frac{9}{20224} a^{6} - \frac{27}{256} a^{4} - \frac{7}{64} a^{2}$, $\frac{1}{3195392} a^{11} - \frac{1}{40448} a^{10} - \frac{1}{40448} a^{9} - \frac{1}{40448} a^{8} - \frac{7}{40448} a^{7} + \frac{9}{40448} a^{6} + \frac{4277}{40448} a^{5} + \frac{27}{512} a^{4} - \frac{25}{128} a^{3} + \frac{7}{128} a^{2} - \frac{1}{4} a$, $\frac{1}{3195392} a^{12} + \frac{1}{20224} a^{8} - \frac{1}{5056} a^{7} - \frac{1}{10112} a^{6} - \frac{3}{32} a^{5} - \frac{7}{512} a^{4} + \frac{1}{64} a^{3} - \frac{15}{128} a^{2} - \frac{3}{16} a$, $\frac{1}{6390784} a^{13} + \frac{1}{40448} a^{9} + \frac{1}{20224} a^{7} + \frac{41}{1024} a^{5} + \frac{15}{256} a^{3} + \frac{15}{32} a - \frac{1}{2}$, $\frac{1}{434573312} a^{14} - \frac{3}{108643328} a^{12} + \frac{9}{2750464} a^{10} + \frac{15}{1375232} a^{8} - \frac{1081}{5500928} a^{6} - \frac{1}{8} a^{5} - \frac{227}{2176} a^{4} - \frac{1}{8} a^{3} + \frac{319}{4352} a^{2} + \frac{1}{4} a - \frac{5}{17}$, $\frac{1}{869146624} a^{15} - \frac{1}{869146624} a^{14} - \frac{3}{217286656} a^{13} - \frac{31}{217286656} a^{12} + \frac{31}{434573312} a^{11} + \frac{127}{5500928} a^{10} + \frac{83}{2750464} a^{9} + \frac{257}{2750464} a^{8} + \frac{823}{11001856} a^{7} + \frac{1353}{11001856} a^{6} - \frac{82463}{687616} a^{5} + \frac{1061}{17408} a^{4} - \frac{1449}{8704} a^{3} - \frac{1645}{8704} a^{2} - \frac{131}{544} a - \frac{6}{17}$, $\frac{1}{60159344163144728576} a^{16} + \frac{153736189}{761510685609426944} a^{14} + \frac{48223798439}{380755342804713472} a^{12} + \frac{3101291757205}{380755342804713472} a^{10} - \frac{371567124971}{9639375767207936} a^{8} + \frac{4065752208855}{9639375767207936} a^{6} - \frac{53029707100143}{1204921970900992} a^{4} - \frac{1}{4} a^{3} - \frac{1725082204015}{7626088423424} a^{2} + \frac{1}{4} a - \frac{2862869353}{29789407904}$, $\frac{1}{120318688326289457152} a^{17} - \frac{1}{120318688326289457152} a^{16} + \frac{153736189}{1523021371218853888} a^{15} - \frac{153736189}{1523021371218853888} a^{14} + \frac{48223798439}{761510685609426944} a^{13} + \frac{70933833177}{761510685609426944} a^{12} + \frac{3193335189}{761510685609426944} a^{11} + \frac{15725614038123}{761510685609426944} a^{10} + \frac{105063401493}{19278751534415872} a^{9} - \frac{581693927957}{19278751534415872} a^{8} - \frac{223922529321}{19278751534415872} a^{7} + \frac{2130444635177}{19278751534415872} a^{6} - \frac{33309119067695}{2409843941801984} a^{5} + \frac{229531948931343}{2409843941801984} a^{4} + \frac{3517853587089}{15252176846848} a^{3} + \frac{473927072047}{15252176846848} a^{2} - \frac{12172059323}{59578815808} a + \frac{2862869353}{59578815808}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{28}\times C_{169260}$, which has order $18957120$ (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:  \( 32038743174.66701 \) (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

\(\Q(\sqrt{-79}) \), 3.3.505521.1, 3.3.316.1, 6.0.7888624.1, 6.0.20188567033839.2, 9.9.653167654112852807616.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
$79$79.6.5.6$x^{6} + 2528$$6$$1$$5$$C_6$$[\ ]_{6}$
79.12.10.3$x^{12} - 553 x^{6} + 505521$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$