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Properties

Label 18.0.94767626766...2000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{28}\cdot 3^{24}\cdot 5^{3}$
Root discriminant $16.63$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_3\times S_3\wr C_2$ (as 18T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -12, 60, -222, 618, -1308, 2145, -2754, 2772, -2158, 1257, -510, 126, -24, 27, -30, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 30*x^15 + 27*x^14 - 24*x^13 + 126*x^12 - 510*x^11 + 1257*x^10 - 2158*x^9 + 2772*x^8 - 2754*x^7 + 2145*x^6 - 1308*x^5 + 618*x^4 - 222*x^3 + 60*x^2 - 12*x + 2)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 30*x^15 + 27*x^14 - 24*x^13 + 126*x^12 - 510*x^11 + 1257*x^10 - 2158*x^9 + 2772*x^8 - 2754*x^7 + 2145*x^6 - 1308*x^5 + 618*x^4 - 222*x^3 + 60*x^2 - 12*x + 2, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 30 x^{15} + 27 x^{14} - 24 x^{13} + 126 x^{12} - 510 x^{11} + 1257 x^{10} - 2158 x^{9} + 2772 x^{8} - 2754 x^{7} + 2145 x^{6} - 1308 x^{5} + 618 x^{4} - 222 x^{3} + 60 x^{2} - 12 x + 2 \)

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:  \(-9476762676643233792000=-\,2^{28}\cdot 3^{24}\cdot 5^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.63$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{373566377} a^{17} + \frac{162867368}{373566377} a^{16} - \frac{125823848}{373566377} a^{15} - \frac{89044509}{373566377} a^{14} - \frac{141473385}{373566377} a^{13} - \frac{148678204}{373566377} a^{12} + \frac{167158452}{373566377} a^{11} + \frac{139618015}{373566377} a^{10} - \frac{127964512}{373566377} a^{9} - \frac{143618151}{373566377} a^{8} + \frac{25033383}{373566377} a^{7} - \frac{18220608}{373566377} a^{6} - \frac{14966615}{373566377} a^{5} + \frac{63839}{373566377} a^{4} - \frac{182681637}{373566377} a^{3} + \frac{141743778}{373566377} a^{2} + \frac{100617200}{373566377} a + \frac{95824791}{373566377}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1482762}{3698677} a^{17} - \frac{8340282}{3698677} a^{16} + \frac{24976740}{3698677} a^{15} - \frac{41520579}{3698677} a^{14} + \frac{38832234}{3698677} a^{13} - \frac{33943881}{3698677} a^{12} + \frac{172682295}{3698677} a^{11} - \frac{704472111}{3698677} a^{10} + \frac{1754364698}{3698677} a^{9} - \frac{3023231259}{3698677} a^{8} + \frac{3856601124}{3698677} a^{7} - \frac{3737418354}{3698677} a^{6} + \frac{2774864748}{3698677} a^{5} - \frac{1566737514}{3698677} a^{4} + \frac{666836313}{3698677} a^{3} - \frac{209695356}{3698677} a^{2} + \frac{53160954}{3698677} a - \frac{9854857}{3698677} \) (order $4$)
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:  \( 28578.0739867 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_3\wr C_2$ (as 18T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 27 conjugacy class representatives for $S_3\times S_3\wr C_2$
Character table for $S_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.108.1, 6.0.186624.1, 6.0.933120.2

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 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 R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.20.68$x^{12} + 2 x^{9} + 2$$12$$1$$20$$(C_6\times C_2):C_2$$[2, 2]_{3}^{2}$
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$