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Properties

Label 18.0.37499586467...4000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 5^{3}\cdot 107^{6}$
Root discriminant $64.52$
Ramified primes $2, 3, 5, 107$
Class number $2464$ (GRH)
Class group $[2, 2, 2, 308]$ (GRH)
Galois group 18T367

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![375, 0, 4275, 0, 12465, 0, 17229, 0, 13431, 0, 6291, 0, 1790, 0, 300, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^16 + 300*x^14 + 1790*x^12 + 6291*x^10 + 13431*x^8 + 17229*x^6 + 12465*x^4 + 4275*x^2 + 375)
 
gp: K = bnfinit(x^18 + 27*x^16 + 300*x^14 + 1790*x^12 + 6291*x^10 + 13431*x^8 + 17229*x^6 + 12465*x^4 + 4275*x^2 + 375, 1)
 

Normalized defining polynomial

\( x^{18} + 27 x^{16} + 300 x^{14} + 1790 x^{12} + 6291 x^{10} + 13431 x^{8} + 17229 x^{6} + 12465 x^{4} + 4275 x^{2} + 375 \)

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:  \(-374995864672776683690097475584000=-\,2^{18}\cdot 3^{27}\cdot 5^{3}\cdot 107^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.52$
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:  $2, 3, 5, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{12} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{13} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{2} a$, $\frac{1}{100} a^{16} + \frac{1}{50} a^{14} - \frac{1}{10} a^{10} - \frac{9}{100} a^{8} - \frac{11}{25} a^{6} - \frac{21}{100} a^{4} + \frac{2}{5} a^{2} + \frac{1}{4}$, $\frac{1}{100} a^{17} + \frac{1}{50} a^{15} - \frac{1}{10} a^{11} - \frac{9}{100} a^{9} - \frac{11}{25} a^{7} - \frac{21}{100} a^{5} + \frac{2}{5} a^{3} + \frac{1}{4} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{308}$, which has order $2464$ (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:  \( 664139.073753 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T367:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n367
Character table for t18n367 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.321.1, 9.9.1953114230889.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.12.19.43$x^{12} + 3 x^{10} - 3 x^{9} - 3 x^{8} - 3 x^{6} + 3 x^{3} + 3$$12$$1$$19$$D_4 \times C_3$$[2]_{4}^{2}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$