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Properties

Label 18.6.61621184405...8113.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 1297^{3}$
Root discriminant $14.29$
Ramified primes $3, 1297$
Class number $1$
Class group Trivial
Galois group 18T207

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -18, 15, 75, -54, -105, 65, 84, -21, -84, 6, 78, -19, -39, 12, 12, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 12*x^15 + 12*x^14 - 39*x^13 - 19*x^12 + 78*x^11 + 6*x^10 - 84*x^9 - 21*x^8 + 84*x^7 + 65*x^6 - 105*x^5 - 54*x^4 + 75*x^3 + 15*x^2 - 18*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 3*x^16 + 12*x^15 + 12*x^14 - 39*x^13 - 19*x^12 + 78*x^11 + 6*x^10 - 84*x^9 - 21*x^8 + 84*x^7 + 65*x^6 - 105*x^5 - 54*x^4 + 75*x^3 + 15*x^2 - 18*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 3 x^{16} + 12 x^{15} + 12 x^{14} - 39 x^{13} - 19 x^{12} + 78 x^{11} + 6 x^{10} - 84 x^{9} - 21 x^{8} + 84 x^{7} + 65 x^{6} - 105 x^{5} - 54 x^{4} + 75 x^{3} + 15 x^{2} - 18 x - 1 \)

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:  \(616211844050013988113=3^{24}\cdot 1297^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.29$
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:  $3, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1081564034273} a^{17} + \frac{485651786914}{1081564034273} a^{16} + \frac{225949500976}{1081564034273} a^{15} + \frac{363012131312}{1081564034273} a^{14} + \frac{40702087074}{1081564034273} a^{13} - \frac{530011419889}{1081564034273} a^{12} - \frac{143326927095}{1081564034273} a^{11} + \frac{66468321216}{1081564034273} a^{10} - \frac{189202644564}{1081564034273} a^{9} - \frac{165166952494}{1081564034273} a^{8} - \frac{100534260271}{1081564034273} a^{7} + \frac{234539954741}{1081564034273} a^{6} + \frac{372113209780}{1081564034273} a^{5} - \frac{90453899005}{1081564034273} a^{4} - \frac{202037771880}{1081564034273} a^{3} + \frac{212908757530}{1081564034273} a^{2} + \frac{341909749131}{1081564034273} a + \frac{483963860758}{1081564034273}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1922.00208319 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T207:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 17 conjugacy class representatives for t18n207
Character table for t18n207

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.8509617.1, 9.5.689278977.1 x3

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

Sibling fields

Degree 9 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
1297Data not computed