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Properties

Label 18.0.59466137572...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 5^{6}\cdot 197^{8}$
Root discriminant $45.09$
Ramified primes $2, 5, 197$
Class number $72$ (GRH)
Class group $[2, 36]$ (GRH)
Galois group $C_2\times S_3\times S_4$ (as 18T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, 310, 0, 1461, 0, 3403, 0, 4263, 0, 2972, 0, 1154, 0, 241, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 25*x^16 + 241*x^14 + 1154*x^12 + 2972*x^10 + 4263*x^8 + 3403*x^6 + 1461*x^4 + 310*x^2 + 25)
 
gp: K = bnfinit(x^18 + 25*x^16 + 241*x^14 + 1154*x^12 + 2972*x^10 + 4263*x^8 + 3403*x^6 + 1461*x^4 + 310*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{18} + 25 x^{16} + 241 x^{14} + 1154 x^{12} + 2972 x^{10} + 4263 x^{8} + 3403 x^{6} + 1461 x^{4} + 310 x^{2} + 25 \)

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:  \(-594661375724483342761984000000=-\,2^{24}\cdot 5^{6}\cdot 197^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.09$
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:  $2, 5, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{10} + \frac{3}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{5} a^{4} - \frac{1}{10} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} - \frac{1}{5} a^{9} + \frac{3}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2}$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{16} - \frac{1}{5} a^{10} - \frac{1}{10} a^{8} - \frac{1}{2} a^{4} + \frac{1}{5} a^{2} - \frac{1}{2}$, $\frac{1}{50} a^{17} + \frac{1}{50} a^{13} - \frac{3}{25} a^{11} + \frac{7}{50} a^{9} - \frac{1}{2} a^{8} - \frac{17}{50} a^{7} + \frac{23}{50} a^{5} - \frac{1}{2} a^{4} - \frac{9}{50} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{36}$, which has order $72$ (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:  $8$
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:  \( 634998.037914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times S_4$ (as 18T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 30 conjugacy class representatives for $C_2\times S_3\times S_4$
Character table for $C_2\times S_3\times S_4$ is not computed

Intermediate fields

3.3.788.1, 3.3.985.1, 6.0.62094400.3, 9.9.12049107848000.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
197Data not computed