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Properties

Label 18.0.17912411563...5936.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{14}\cdot 13^{15}$
Root discriminant $133.41$
Ramified primes $2, 3, 7, 13$
Class number $1042416$ (GRH)
Class group $[12, 86868]$ (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![74063872, -152907776, 245058048, -254183424, 211472512, -129573632, 62210208, -20356032, 3876760, 317616, -374940, 72448, 23362, -9892, 1161, 248, -40, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 40*x^16 + 248*x^15 + 1161*x^14 - 9892*x^13 + 23362*x^12 + 72448*x^11 - 374940*x^10 + 317616*x^9 + 3876760*x^8 - 20356032*x^7 + 62210208*x^6 - 129573632*x^5 + 211472512*x^4 - 254183424*x^3 + 245058048*x^2 - 152907776*x + 74063872)
 
gp: K = bnfinit(x^18 - 4*x^17 - 40*x^16 + 248*x^15 + 1161*x^14 - 9892*x^13 + 23362*x^12 + 72448*x^11 - 374940*x^10 + 317616*x^9 + 3876760*x^8 - 20356032*x^7 + 62210208*x^6 - 129573632*x^5 + 211472512*x^4 - 254183424*x^3 + 245058048*x^2 - 152907776*x + 74063872, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 40 x^{16} + 248 x^{15} + 1161 x^{14} - 9892 x^{13} + 23362 x^{12} + 72448 x^{11} - 374940 x^{10} + 317616 x^{9} + 3876760 x^{8} - 20356032 x^{7} + 62210208 x^{6} - 129573632 x^{5} + 211472512 x^{4} - 254183424 x^{3} + 245058048 x^{2} - 152907776 x + 74063872 \)

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:  \(-179124115636167311134377199898019495936=-\,2^{18}\cdot 3^{9}\cdot 7^{14}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.41$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} + \frac{1}{64} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{12} - \frac{1}{128} a^{11} - \frac{3}{128} a^{10} + \frac{1}{64} a^{9} + \frac{13}{256} a^{8} - \frac{5}{128} a^{7} - \frac{1}{64} a^{6} + \frac{1}{16} a^{5} + \frac{11}{64} a^{4} - \frac{5}{32} a^{3} + \frac{1}{16} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{512} a^{13} - \frac{1}{256} a^{11} - \frac{1}{64} a^{10} + \frac{5}{512} a^{9} + \frac{1}{32} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{5}{128} a^{5} + \frac{3}{32} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{1024} a^{14} - \frac{1}{512} a^{12} - \frac{1}{128} a^{11} + \frac{5}{1024} a^{10} + \frac{1}{64} a^{9} + \frac{3}{64} a^{8} - \frac{7}{128} a^{7} + \frac{11}{256} a^{6} - \frac{1}{64} a^{5} + \frac{1}{32} a^{4} - \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{2048} a^{15} + \frac{1}{2048} a^{11} + \frac{1}{128} a^{10} - \frac{3}{1024} a^{9} + \frac{5}{128} a^{8} + \frac{3}{512} a^{7} - \frac{3}{64} a^{6} + \frac{23}{256} a^{5} + \frac{3}{32} a^{4} - \frac{3}{16} a^{3} + \frac{3}{16} a^{2} - \frac{5}{16} a - \frac{1}{4}$, $\frac{1}{4096} a^{16} + \frac{1}{4096} a^{12} + \frac{1}{256} a^{11} + \frac{61}{2048} a^{10} - \frac{3}{256} a^{9} - \frac{29}{1024} a^{8} - \frac{7}{128} a^{7} + \frac{23}{512} a^{6} + \frac{7}{64} a^{5} + \frac{5}{32} a^{4} + \frac{3}{32} a^{3} - \frac{13}{32} a^{2} - \frac{1}{8} a$, $\frac{1}{19799808230180275282745219408840812180329562112} a^{17} - \frac{117819862403442419786537862563832922548623}{9899904115090137641372609704420406090164781056} a^{16} - \frac{726826913105336687789073844460936370465615}{4949952057545068820686304852210203045082390528} a^{15} - \frac{105125199232817418447335390781581227751129}{1237488014386267205171576213052550761270597632} a^{14} - \frac{11642340937908384918814570619573020112058327}{19799808230180275282745219408840812180329562112} a^{13} + \frac{9838942112560203970449265513665253768742777}{9899904115090137641372609704420406090164781056} a^{12} - \frac{4745837412854672261608031514052958033380665}{9899904115090137641372609704420406090164781056} a^{11} - \frac{75980457245863174233866201843533977344251491}{4949952057545068820686304852210203045082390528} a^{10} - \frac{119915460647439093283388549893005061952378281}{4949952057545068820686304852210203045082390528} a^{9} + \frac{55904096081813445105700050987107339190329819}{2474976028772534410343152426105101522541195264} a^{8} + \frac{41662872439699468107391736664514792453745061}{2474976028772534410343152426105101522541195264} a^{7} - \frac{31893462507597603912125895230415399090208725}{1237488014386267205171576213052550761270597632} a^{6} + \frac{4060718130505068322186195497079554854529303}{309372003596566801292894053263137690317649408} a^{5} - \frac{19674457054346402418507102422880320850084309}{154686001798283400646447026631568845158824704} a^{4} + \frac{19402087749501430330349605802764399814933635}{154686001798283400646447026631568845158824704} a^{3} + \frac{35312976841296971576141301924681589209037513}{77343000899141700323223513315784422579412352} a^{2} + \frac{1758472426602499591852677931196426795252259}{4833937556196356270201469582236526411213272} a - \frac{81719484520694631776240423804919778332711}{4833937556196356270201469582236526411213272}$

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

Class group and class number

$C_{12}\times C_{86868}$, which has order $1042416$ (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:  \( 1090042674.5636587 \) (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

\(\Q(\sqrt{-39}) \), 3.3.8281.1, 3.3.28392.1, 6.0.24069811311.1, 6.0.31438120896.4, 9.9.54951571781503488.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$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$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13Data not computed