↑

Properties

Label 18.0.26087322728...3792.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 17^{4}$
Root discriminant $15.48$
Ramified primes $2, 3, 17$
Class number $1$
Class group Trivial
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![1, -3, 18, -64, 129, -177, 176, -105, 42, -12, 48, -99, 132, -123, 90, -50, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 50*x^15 + 90*x^14 - 123*x^13 + 132*x^12 - 99*x^11 + 48*x^10 - 12*x^9 + 42*x^8 - 105*x^7 + 176*x^6 - 177*x^5 + 129*x^4 - 64*x^3 + 18*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 50*x^15 + 90*x^14 - 123*x^13 + 132*x^12 - 99*x^11 + 48*x^10 - 12*x^9 + 42*x^8 - 105*x^7 + 176*x^6 - 177*x^5 + 129*x^4 - 64*x^3 + 18*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 50 x^{15} + 90 x^{14} - 123 x^{13} + 132 x^{12} - 99 x^{11} + 48 x^{10} - 12 x^{9} + 42 x^{8} - 105 x^{7} + 176 x^{6} - 177 x^{5} + 129 x^{4} - 64 x^{3} + 18 x^{2} - 3 x + 1 \)

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:  \(-2608732272818582433792=-\,2^{12}\cdot 3^{27}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.48$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{77298081517} a^{17} + \frac{18718838564}{77298081517} a^{16} - \frac{200285335}{77298081517} a^{15} - \frac{9498690095}{77298081517} a^{14} + \frac{5669390566}{77298081517} a^{13} - \frac{10325162420}{77298081517} a^{12} + \frac{32763174432}{77298081517} a^{11} + \frac{28831649266}{77298081517} a^{10} - \frac{13462355147}{77298081517} a^{9} + \frac{29370998679}{77298081517} a^{8} + \frac{15060365993}{77298081517} a^{7} - \frac{18567202071}{77298081517} a^{6} - \frac{29791542125}{77298081517} a^{5} - \frac{36711731740}{77298081517} a^{4} - \frac{21257745910}{77298081517} a^{3} - \frac{9756595641}{77298081517} a^{2} - \frac{13575287301}{77298081517} a + \frac{24340947288}{77298081517}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{1796255210}{77298081517} a^{17} + \frac{20360318710}{77298081517} a^{16} - \frac{91843173696}{77298081517} a^{15} + \frac{257404363983}{77298081517} a^{14} - \frac{494041115421}{77298081517} a^{13} + \frac{679492836305}{77298081517} a^{12} - \frac{632922093119}{77298081517} a^{11} + \frac{329767770094}{77298081517} a^{10} + \frac{112050953262}{77298081517} a^{9} - \frac{315477938666}{77298081517} a^{8} + \frac{5134428237}{77298081517} a^{7} + \frac{790019342408}{77298081517} a^{6} - \frac{1183305953731}{77298081517} a^{5} + \frac{934156324518}{77298081517} a^{4} - \frac{216419147620}{77298081517} a^{3} - \frac{201908800648}{77298081517} a^{2} + \frac{258061363679}{77298081517} a - \frac{17053154818}{77298081517} \) (order $18$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12453.4337865 \)
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

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ $18$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
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}$