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Properties

Label 18.18.3249117228...5904.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{22}\cdot 3^{14}\cdot 503^{6}$
Root discriminant $43.61$
Ramified primes $2, 3, 503$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_3\times A_4):S_3$ (as 18T108)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1936, 0, 15520, 0, -42560, 0, 55652, 0, -39568, 0, 16280, 0, -3935, 0, 541, 0, -38, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 38*x^16 + 541*x^14 - 3935*x^12 + 16280*x^10 - 39568*x^8 + 55652*x^6 - 42560*x^4 + 15520*x^2 - 1936)
 
gp: K = bnfinit(x^18 - 38*x^16 + 541*x^14 - 3935*x^12 + 16280*x^10 - 39568*x^8 + 55652*x^6 - 42560*x^4 + 15520*x^2 - 1936, 1)
 

Normalized defining polynomial

\( x^{18} - 38 x^{16} + 541 x^{14} - 3935 x^{12} + 16280 x^{10} - 39568 x^{8} + 55652 x^{6} - 42560 x^{4} + 15520 x^{2} - 1936 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(324911722849263597378972155904=2^{22}\cdot 3^{14}\cdot 503^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.61$
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:  $2, 3, 503$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{10} + \frac{1}{12} a^{8} - \frac{1}{4} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{11} + \frac{1}{12} a^{9} - \frac{1}{4} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{24} a^{14} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{24} a^{8} - \frac{1}{4} a^{7} + \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{264} a^{15} + \frac{1}{132} a^{13} + \frac{5}{264} a^{11} + \frac{49}{264} a^{9} + \frac{14}{33} a^{7} - \frac{8}{33} a^{5} - \frac{5}{22} a^{3} + \frac{1}{33} a$, $\frac{1}{96132168} a^{16} + \frac{24619}{4005507} a^{14} - \frac{2746343}{96132168} a^{12} + \frac{1058159}{10681352} a^{10} + \frac{734647}{48066084} a^{8} + \frac{472418}{4005507} a^{6} - \frac{956971}{24033042} a^{4} + \frac{2506252}{12016521} a^{2} - \frac{254440}{1092411}$, $\frac{1}{192264336} a^{17} - \frac{22903}{32044056} a^{15} - \frac{382081}{17478576} a^{13} - \frac{14061341}{64088112} a^{11} - \frac{8554033}{48066084} a^{9} - \frac{1}{2} a^{8} + \frac{2054711}{5340676} a^{7} - \frac{330277}{12016521} a^{5} - \frac{1}{2} a^{4} - \frac{2024107}{12016521} a^{3} - \frac{1}{2} a^{2} - \frac{1763557}{12016521} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 1142011664.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_3\times A_4):S_3$ (as 18T108):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 19 conjugacy class representatives for $(C_3\times A_4):S_3$
Character table for $(C_3\times A_4):S_3$

Intermediate fields

3.3.1509.1, 6.6.145733184.1, 9.9.4453205336784.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.12.16.20$x^{12} + 4 x^{10} + x^{8} + 4 x^{6} - x^{4} + 8 x^{2} - 1$$6$$2$$16$12T42$[2, 2]_{3}^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.14.15$x^{12} + 9 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x^{2} + 9$$6$$2$$14$$C_6\times S_3$$[3/2]_{2}^{6}$
503Data not computed