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Properties

Label 18.6.34930066530...3125.1
Degree $18$
Signature $[6, 6]$
Discriminant $5^{9}\cdot 31^{6}\cdot 67^{4}$
Root discriminant $17.88$
Ramified primes $5, 31, 67$
Class number $1$
Class group Trivial
Galois group $C_2\times C_3\wr S_3$ (as 18T119)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 4, -20, 5, 95, 23, -200, -161, 189, 284, -15, -186, -47, 56, 13, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 10*x^16 + 13*x^15 + 56*x^14 - 47*x^13 - 186*x^12 - 15*x^11 + 284*x^10 + 189*x^9 - 161*x^8 - 200*x^7 + 23*x^6 + 95*x^5 + 5*x^4 - 20*x^3 + 4*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^18 - x^17 - 10*x^16 + 13*x^15 + 56*x^14 - 47*x^13 - 186*x^12 - 15*x^11 + 284*x^10 + 189*x^9 - 161*x^8 - 200*x^7 + 23*x^6 + 95*x^5 + 5*x^4 - 20*x^3 + 4*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 10 x^{16} + 13 x^{15} + 56 x^{14} - 47 x^{13} - 186 x^{12} - 15 x^{11} + 284 x^{10} + 189 x^{9} - 161 x^{8} - 200 x^{7} + 23 x^{6} + 95 x^{5} + 5 x^{4} - 20 x^{3} + 4 x^{2} + 2 x - 1 \)

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:  \(34930066530813283203125=5^{9}\cdot 31^{6}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.88$
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:  $5, 31, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{4}{11} a^{15} + \frac{2}{11} a^{14} + \frac{2}{11} a^{13} - \frac{5}{11} a^{12} - \frac{3}{11} a^{10} + \frac{3}{11} a^{9} + \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{5}{11} a^{6} - \frac{3}{11} a^{4} + \frac{3}{11} a^{3} + \frac{4}{11}$, $\frac{1}{111952209245591} a^{17} + \frac{4736396807304}{111952209245591} a^{16} - \frac{491365772099}{111952209245591} a^{15} - \frac{26192930171002}{111952209245591} a^{14} - \frac{10515404462538}{111952209245591} a^{13} - \frac{35072775585681}{111952209245591} a^{12} - \frac{21376176627369}{111952209245591} a^{11} + \frac{135481258343}{111952209245591} a^{10} + \frac{49359114067301}{111952209245591} a^{9} + \frac{1692793418389}{8611708403507} a^{8} - \frac{48575580898330}{111952209245591} a^{7} - \frac{30521194655147}{111952209245591} a^{6} + \frac{22534248933589}{111952209245591} a^{5} + \frac{40576845206845}{111952209245591} a^{4} - \frac{26412025242657}{111952209245591} a^{3} + \frac{1749646509329}{10177473567781} a^{2} + \frac{3055133726328}{8611708403507} a + \frac{47606045449627}{111952209245591}$

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:  \( 16089.1713792 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3\wr S_3$ (as 18T119):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 324
The 44 conjugacy class representatives for $C_2\times C_3\wr S_3$
Character table for $C_2\times C_3\wr S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.31.1, 6.2.120125.1, 9.3.133731799.1

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R $18$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.6.4.1$x^{6} + 2345 x^{3} + 7756992$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$