↑

Properties

Label 18.10.1068536738...2032.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{24}\cdot 3^{27}\cdot 17^{4}$
Root discriminant $24.57$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T201

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 0, 243, 0, -513, 0, 0, 0, -297, 0, 72, 0, -15, 0, 36, 0, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 + 36*x^14 - 15*x^12 + 72*x^10 - 297*x^8 - 513*x^4 + 243*x^2 - 27)
 
gp: K = bnfinit(x^18 - 12*x^16 + 36*x^14 - 15*x^12 + 72*x^10 - 297*x^8 - 513*x^4 + 243*x^2 - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} + 36 x^{14} - 15 x^{12} + 72 x^{10} - 297 x^{8} - 513 x^{4} + 243 x^{2} - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10685367389464913648812032=2^{24}\cdot 3^{27}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.57$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{24} a^{10} - \frac{1}{12} a^{6} - \frac{1}{8} a^{4} - \frac{3}{8} a^{2} - \frac{1}{8}$, $\frac{1}{24} a^{11} - \frac{1}{12} a^{7} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{144} a^{12} - \frac{1}{48} a^{10} + \frac{1}{24} a^{8} + \frac{1}{48} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{16}$, $\frac{1}{288} a^{13} - \frac{1}{288} a^{12} + \frac{1}{96} a^{11} - \frac{1}{96} a^{10} + \frac{1}{48} a^{9} - \frac{1}{48} a^{8} + \frac{13}{96} a^{7} - \frac{13}{96} a^{6} - \frac{5}{16} a^{5} + \frac{5}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{32} a + \frac{1}{32}$, $\frac{1}{576} a^{14} - \frac{1}{288} a^{12} - \frac{1}{64} a^{10} + \frac{1}{64} a^{8} + \frac{1}{192} a^{6} - \frac{3}{16} a^{4} + \frac{21}{64} a^{2} + \frac{5}{64}$, $\frac{1}{576} a^{15} - \frac{1}{288} a^{12} - \frac{1}{192} a^{11} - \frac{1}{96} a^{10} + \frac{7}{192} a^{9} - \frac{1}{48} a^{8} + \frac{9}{64} a^{7} - \frac{13}{96} a^{6} - \frac{1}{2} a^{5} + \frac{5}{16} a^{4} + \frac{25}{64} a^{3} - \frac{1}{16} a^{2} + \frac{3}{64} a + \frac{1}{32}$, $\frac{1}{1152} a^{16} - \frac{1}{1152} a^{14} - \frac{1}{384} a^{12} - \frac{1}{48} a^{10} + \frac{5}{96} a^{8} - \frac{9}{128} a^{6} - \frac{55}{128} a^{4} - \frac{19}{64} a^{2} + \frac{13}{128}$, $\frac{1}{2304} a^{17} - \frac{1}{2304} a^{16} + \frac{1}{2304} a^{15} - \frac{1}{2304} a^{14} + \frac{1}{2304} a^{13} - \frac{1}{2304} a^{12} + \frac{5}{384} a^{11} - \frac{5}{384} a^{10} - \frac{11}{384} a^{9} + \frac{11}{384} a^{8} + \frac{37}{256} a^{7} - \frac{37}{256} a^{6} + \frac{17}{256} a^{5} - \frac{17}{256} a^{4} + \frac{25}{64} a^{3} - \frac{25}{64} a^{2} - \frac{65}{256} a + \frac{65}{256}$

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:  $13$
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:  \( 878212.30401 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 26 conjugacy class representatives for t18n201
Character table for t18n201 is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{36})^+\), 9.5.9829532736.1

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 R $18$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.18.49$x^{12} + 4 x^{11} - 4 x^{10} - 12 x^{9} - 8 x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 8 x^{3} + 8 x^{2} - 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
3Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$