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Properties

Label 18.6.37654983721...5264.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{32}\cdot 19^{4}\cdot 29^{6}$
Root discriminant $83.34$
Ramified primes $2, 3, 19, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T463

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-130321, 0, 370386, 0, 675792, 0, 252282, 0, -3537, 0, -14940, 0, -1641, 0, 135, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^16 + 135*x^14 - 1641*x^12 - 14940*x^10 - 3537*x^8 + 252282*x^6 + 675792*x^4 + 370386*x^2 - 130321)
 
gp: K = bnfinit(x^18 + 27*x^16 + 135*x^14 - 1641*x^12 - 14940*x^10 - 3537*x^8 + 252282*x^6 + 675792*x^4 + 370386*x^2 - 130321, 1)
 

Normalized defining polynomial

\( x^{18} + 27 x^{16} + 135 x^{14} - 1641 x^{12} - 14940 x^{10} - 3537 x^{8} + 252282 x^{6} + 675792 x^{4} + 370386 x^{2} - 130321 \)

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:  \(37654983721231487318659532450955264=2^{18}\cdot 3^{32}\cdot 19^{4}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.34$
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:  $2, 3, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{57} a^{12} + \frac{8}{57} a^{10} + \frac{2}{57} a^{8} + \frac{4}{19} a^{6} + \frac{13}{57} a^{4} + \frac{16}{57} a^{2} - \frac{1}{3}$, $\frac{1}{57} a^{13} + \frac{8}{57} a^{11} + \frac{2}{57} a^{9} + \frac{4}{19} a^{7} + \frac{13}{57} a^{5} + \frac{16}{57} a^{3} - \frac{1}{3} a$, $\frac{1}{2166} a^{14} - \frac{11}{2166} a^{12} + \frac{32}{361} a^{10} + \frac{44}{1083} a^{8} - \frac{478}{1083} a^{6} + \frac{341}{722} a^{4} + \frac{11}{57} a^{2} + \frac{1}{6}$, $\frac{1}{2166} a^{15} - \frac{11}{2166} a^{13} + \frac{32}{361} a^{11} + \frac{44}{1083} a^{9} - \frac{478}{1083} a^{7} + \frac{341}{722} a^{5} + \frac{11}{57} a^{3} + \frac{1}{6} a$, $\frac{1}{152441540413312452} a^{16} - \frac{1202091743277}{25406923402218742} a^{14} - \frac{165247527591237}{50813846804437484} a^{12} - \frac{2777001189704657}{76220770206656226} a^{10} + \frac{10518844809398123}{76220770206656226} a^{8} + \frac{10745322826213729}{152441540413312452} a^{6} - \frac{453696158728195}{2674412989707236} a^{4} + \frac{37547633244947}{422275735216932} a^{2} + \frac{319641892313}{22225038695628}$, $\frac{1}{152441540413312452} a^{17} - \frac{1202091743277}{25406923402218742} a^{15} - \frac{165247527591237}{50813846804437484} a^{13} - \frac{2777001189704657}{76220770206656226} a^{11} + \frac{10518844809398123}{76220770206656226} a^{9} + \frac{10745322826213729}{152441540413312452} a^{7} - \frac{453696158728195}{2674412989707236} a^{5} + \frac{37547633244947}{422275735216932} a^{3} + \frac{319641892313}{22225038695628} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 25020485349.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T463:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 48 conjugacy class representatives for t18n463
Character table for t18n463 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.2349.1, 9.9.1049866478469.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.12.2$x^{12} - 6 x^{10} - 13 x^{8} - 28 x^{6} + 15 x^{4} - 30 x^{2} - 3$$2$$6$$12$12T105$[2, 2, 2, 2, 2]^{6}$
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.12.6.1$x^{12} + 146334 x^{6} - 20511149 x^{2} + 5353409889$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$