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Properties

Label 18.0.12633587927...6512.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 7^{6}\cdot 13^{6}\cdot 47^{6}$
Root discriminant $60.73$
Ramified primes $2, 3, 7, 13, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![262144, -655360, 892928, -625664, 291840, -115936, 52964, -53392, 56416, -17892, -12604, 10416, -2102, -156, 172, -82, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 26*x^16 - 82*x^15 + 172*x^14 - 156*x^13 - 2102*x^12 + 10416*x^11 - 12604*x^10 - 17892*x^9 + 56416*x^8 - 53392*x^7 + 52964*x^6 - 115936*x^5 + 291840*x^4 - 625664*x^3 + 892928*x^2 - 655360*x + 262144)
 
gp: K = bnfinit(x^18 - 6*x^17 + 26*x^16 - 82*x^15 + 172*x^14 - 156*x^13 - 2102*x^12 + 10416*x^11 - 12604*x^10 - 17892*x^9 + 56416*x^8 - 53392*x^7 + 52964*x^6 - 115936*x^5 + 291840*x^4 - 625664*x^3 + 892928*x^2 - 655360*x + 262144, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 26 x^{16} - 82 x^{15} + 172 x^{14} - 156 x^{13} - 2102 x^{12} + 10416 x^{11} - 12604 x^{10} - 17892 x^{9} + 56416 x^{8} - 53392 x^{7} + 52964 x^{6} - 115936 x^{5} + 291840 x^{4} - 625664 x^{3} + 892928 x^{2} - 655360 x + 262144 \)

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:  \(-126335879270977338929906174656512=-\,2^{20}\cdot 3^{9}\cdot 7^{6}\cdot 13^{6}\cdot 47^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.73$
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:  $2, 3, 7, 13, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a$, $\frac{1}{128} a^{14} + \frac{1}{64} a^{13} + \frac{5}{64} a^{12} - \frac{1}{64} a^{11} + \frac{7}{32} a^{10} + \frac{1}{32} a^{9} - \frac{11}{64} a^{8} + \frac{1}{32} a^{6} + \frac{15}{32} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{7}{32} a^{2} - \frac{1}{2} a$, $\frac{1}{1024} a^{15} + \frac{1}{512} a^{14} - \frac{11}{512} a^{13} + \frac{63}{512} a^{12} + \frac{7}{256} a^{11} + \frac{49}{256} a^{10} + \frac{117}{512} a^{9} - \frac{1}{4} a^{8} + \frac{17}{256} a^{7} + \frac{15}{256} a^{6} + \frac{5}{16} a^{5} - \frac{25}{64} a^{4} - \frac{103}{256} a^{3} - \frac{7}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{4096} a^{16} - \frac{1}{2048} a^{15} + \frac{1}{2048} a^{14} + \frac{11}{2048} a^{13} - \frac{39}{1024} a^{12} + \frac{133}{1024} a^{11} - \frac{339}{2048} a^{10} - \frac{5}{512} a^{9} + \frac{97}{1024} a^{8} - \frac{181}{1024} a^{7} - \frac{51}{256} a^{6} + \frac{15}{256} a^{5} - \frac{215}{1024} a^{4} + \frac{107}{256} a^{3} + \frac{1}{32} a^{2} + \frac{1}{4} a$, $\frac{1}{738306219860169174926368063679339855872} a^{17} + \frac{7437511780829432389173373130142185}{369153109930084587463184031839669927936} a^{16} - \frac{19874214550461544557088878414283}{42251700804633694341671515604860928} a^{15} - \frac{198219606021818280138135515021133201}{369153109930084587463184031839669927936} a^{14} + \frac{3989018230618665920368879032740236783}{184576554965042293731592015919834963968} a^{13} - \frac{1601952164781246482613318681371560239}{184576554965042293731592015919834963968} a^{12} - \frac{11262046822070455586910352421635026731}{369153109930084587463184031839669927936} a^{11} + \frac{695616338006395325454646833531904413}{23072069370630286716449001989979370496} a^{10} + \frac{21509613904767584678236620721466141537}{184576554965042293731592015919834963968} a^{9} + \frac{15775165245072391880832326430096962271}{184576554965042293731592015919834963968} a^{8} - \frac{39077002748922695947468354444630011}{5768017342657571679112250497494842624} a^{7} - \frac{1456706180541556799276621581586050209}{46144138741260573432898003979958740992} a^{6} - \frac{33999601080187438846729679981666198695}{184576554965042293731592015919834963968} a^{5} + \frac{1655529183953427889122193365206136123}{5768017342657571679112250497494842624} a^{4} - \frac{2650218855282689516816859661956107125}{5768017342657571679112250497494842624} a^{3} + \frac{13019502470554761426268104598075441}{103000309690313779984147330312407904} a^{2} + \frac{574874396056026034960933340342239}{45062635489512278743064457011678458} a - \frac{4927409492137152859618742678913320}{22531317744756139371532228505839229}$

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:  \( \frac{18744750945778700369}{11991620459729210499825664} a^{17} - \frac{106588436749542652723}{5995810229864605249912832} a^{16} + \frac{422609714702518216461}{5995810229864605249912832} a^{15} - \frac{1541427928777769080473}{5995810229864605249912832} a^{14} + \frac{1703526203261942964747}{2997905114932302624956416} a^{13} - \frac{1962317128882592984391}{2997905114932302624956416} a^{12} - \frac{22779247326877412849995}{5995810229864605249912832} a^{11} + \frac{25471548205069313529443}{749476278733075656239104} a^{10} - \frac{191092711716945242970831}{2997905114932302624956416} a^{9} - \frac{194248185957496481482185}{2997905114932302624956416} a^{8} + \frac{98649979359720785055359}{374738139366537828119552} a^{7} - \frac{87844977662211568159505}{749476278733075656239104} a^{6} + \frac{32663538188068458520969}{2997905114932302624956416} a^{5} - \frac{166728344281096332531543}{374738139366537828119552} a^{4} + \frac{92581543525785987815367}{93684534841634457029888} a^{3} - \frac{25295297928254164927849}{11710566855204307128736} a^{2} + \frac{4471481544831331524043}{1463820856900538391092} a - \frac{424133809135562782125}{365955214225134597773} \) (order $6$)
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:  \( 1209162608.4915006 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1316.1, 6.0.46760112.1, 6.0.73008.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
$47$47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$