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Properties

Label 18.6.66123160012...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 11^{6}$
Root discriminant $14.34$
Ramified primes $2, 3, 5, 11$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3\wr C_2$ (as 18T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 13, -26, -31, 75, 4, -108, 57, 80, -87, -17, 60, -22, -17, 16, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - x^16 + 16*x^15 - 17*x^14 - 22*x^13 + 60*x^12 - 17*x^11 - 87*x^10 + 80*x^9 + 57*x^8 - 108*x^7 + 4*x^6 + 75*x^5 - 31*x^4 - 26*x^3 + 13*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - x^16 + 16*x^15 - 17*x^14 - 22*x^13 + 60*x^12 - 17*x^11 - 87*x^10 + 80*x^9 + 57*x^8 - 108*x^7 + 4*x^6 + 75*x^5 - 31*x^4 - 26*x^3 + 13*x^2 + 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - x^{16} + 16 x^{15} - 17 x^{14} - 22 x^{13} + 60 x^{12} - 17 x^{11} - 87 x^{10} + 80 x^{9} + 57 x^{8} - 108 x^{7} + 4 x^{6} + 75 x^{5} - 31 x^{4} - 26 x^{3} + 13 x^{2} + 2 x - 1 \)

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:  \(661231600128000000000=2^{18}\cdot 3^{6}\cdot 5^{9}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.34$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{17} a^{16} + \frac{3}{17} a^{15} + \frac{4}{17} a^{14} + \frac{1}{17} a^{13} + \frac{5}{17} a^{12} - \frac{5}{17} a^{11} - \frac{1}{17} a^{10} + \frac{8}{17} a^{9} + \frac{8}{17} a^{8} + \frac{7}{17} a^{7} - \frac{5}{17} a^{6} - \frac{8}{17} a^{5} + \frac{4}{17} a^{4} - \frac{1}{17} a^{3} - \frac{4}{17} a^{2} - \frac{3}{17} a - \frac{4}{17}$, $\frac{1}{186721421} a^{17} + \frac{4644235}{186721421} a^{16} + \frac{60344213}{186721421} a^{15} - \frac{83521892}{186721421} a^{14} - \frac{14344780}{186721421} a^{13} + \frac{16044027}{186721421} a^{12} - \frac{67446542}{186721421} a^{11} - \frac{55327854}{186721421} a^{10} - \frac{38004972}{186721421} a^{9} + \frac{43462206}{186721421} a^{8} - \frac{47925490}{186721421} a^{7} + \frac{54795385}{186721421} a^{6} + \frac{41773765}{186721421} a^{5} + \frac{48919564}{186721421} a^{4} - \frac{47651363}{186721421} a^{3} - \frac{54593267}{186721421} a^{2} - \frac{39211870}{186721421} a - \frac{52508792}{186721421}$

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:  \( 2319.60000104 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\wr C_2$ (as 18T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times S_3\wr C_2$
Character table for $C_2\times S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 9.3.11499840000.1

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

Sibling fields

Degree 12 siblings: 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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$