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Properties

Label 18.2.34567292775...4192.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{33}\cdot 19^{15}$
Root discriminant $138.37$
Ramified primes $2, 3, 19$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56623104, 0, 0, -60604416, 0, 0, 4597632, 0, 0, -39528, 0, 0, 3936, 0, 0, -12, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 3936*x^12 - 39528*x^9 + 4597632*x^6 - 60604416*x^3 + 56623104)
 
gp: K = bnfinit(x^18 - 12*x^15 + 3936*x^12 - 39528*x^9 + 4597632*x^6 - 60604416*x^3 + 56623104, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{15} + 3936 x^{12} - 39528 x^{9} + 4597632 x^{6} - 60604416 x^{3} + 56623104 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(345672927754842768884498083059529224192=2^{12}\cdot 3^{33}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $138.37$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{12} a^{6}$, $\frac{1}{12} a^{7}$, $\frac{1}{12} a^{8}$, $\frac{1}{24} a^{9}$, $\frac{1}{48} a^{10} - \frac{1}{2} a$, $\frac{1}{96} a^{11} - \frac{1}{24} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{24192} a^{12} - \frac{41}{2016} a^{9} - \frac{1}{84} a^{6} - \frac{53}{336} a^{3} + \frac{10}{21}$, $\frac{1}{96768} a^{13} - \frac{41}{8064} a^{10} - \frac{1}{336} a^{7} + \frac{283}{1344} a^{4} - \frac{11}{84} a$, $\frac{1}{387072} a^{14} + \frac{127}{32256} a^{11} - \frac{29}{1344} a^{8} - \frac{1061}{5376} a^{5} - \frac{137}{336} a^{2}$, $\frac{1}{78461404987392} a^{15} - \frac{17764631}{934064345088} a^{12} - \frac{422247461}{30270603776} a^{9} - \frac{16596344357}{1089741735936} a^{6} - \frac{13575070097}{68108858496} a^{3} + \frac{22522456}{59122273}$, $\frac{1}{313845619949568} a^{16} - \frac{17764631}{3736257380352} a^{13} - \frac{422247461}{121082415104} a^{10} - \frac{107408155685}{4358966943744} a^{7} - \frac{47629499345}{272435433984} a^{4} - \frac{47861045}{118244546} a$, $\frac{1}{1255382479798272} a^{17} - \frac{17764631}{14945029521408} a^{14} + \frac{6300908561}{1452988981248} a^{11} - \frac{470655400997}{17435867774976} a^{8} - \frac{183847216337}{1089741735936} a^{5} + \frac{2815307}{118244546} a^{2}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  $9$
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:  \( 840607693432.3646 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{57}) \), 3.1.1083.1, 6.2.66854673.1, 9.1.129610938470796864.28

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.2$x^{6} + 15$$6$$1$$11$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.85$x^{12} - 9 x^{11} - 33 x^{9} - 18 x^{8} + 36 x^{7} + 36 x^{6} + 27 x^{5} - 27 x^{4} + 27 x^{3} - 27 x^{2} - 27 x + 36$$6$$2$$22$$C_6\times S_3$$[2, 5/2]_{2}^{2}$
19Data not computed