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Properties

Label 18.6.71544469070...0016.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{32}\cdot 19^{5}\cdot 29^{6}$
Root discriminant $98.16$
Ramified primes $2, 3, 19, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2476099, 0, -3518667, 0, 2407509, 0, 5534130, 0, 2253780, 0, 75357, 0, -34284, 0, -468, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 54*x^16 - 468*x^14 - 34284*x^12 + 75357*x^10 + 2253780*x^8 + 5534130*x^6 + 2407509*x^4 - 3518667*x^2 - 2476099)
 
gp: K = bnfinit(x^18 + 54*x^16 - 468*x^14 - 34284*x^12 + 75357*x^10 + 2253780*x^8 + 5534130*x^6 + 2407509*x^4 - 3518667*x^2 - 2476099, 1)
 

Normalized defining polynomial

\( x^{18} + 54 x^{16} - 468 x^{14} - 34284 x^{12} + 75357 x^{10} + 2253780 x^{8} + 5534130 x^{6} + 2407509 x^{4} - 3518667 x^{2} - 2476099 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(715444690703398259054531116568150016=2^{18}\cdot 3^{32}\cdot 19^{5}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.16$
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:  $2, 3, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{57} a^{10} + \frac{16}{57} a^{8} + \frac{7}{57} a^{6} + \frac{11}{57} a^{4} - \frac{16}{57} a^{2} - \frac{1}{3}$, $\frac{1}{57} a^{11} + \frac{16}{57} a^{9} + \frac{7}{57} a^{7} + \frac{11}{57} a^{5} - \frac{16}{57} a^{3} - \frac{1}{3} a$, $\frac{1}{1083} a^{12} - \frac{1}{361} a^{10} - \frac{99}{361} a^{8} + \frac{334}{1083} a^{6} + \frac{1}{361} a^{4} - \frac{2}{19} a^{2} + \frac{1}{3}$, $\frac{1}{1083} a^{13} - \frac{1}{361} a^{11} - \frac{99}{361} a^{9} + \frac{334}{1083} a^{7} + \frac{1}{361} a^{5} - \frac{2}{19} a^{3} + \frac{1}{3} a$, $\frac{1}{20577} a^{14} - \frac{1}{6859} a^{12} + \frac{64}{20577} a^{10} - \frac{4720}{20577} a^{8} - \frac{2885}{20577} a^{6} - \frac{310}{1083} a^{4} + \frac{6}{19} a^{2} - \frac{1}{3}$, $\frac{1}{20577} a^{15} - \frac{1}{6859} a^{13} + \frac{64}{20577} a^{11} - \frac{4720}{20577} a^{9} - \frac{2885}{20577} a^{7} - \frac{310}{1083} a^{5} + \frac{6}{19} a^{3} - \frac{1}{3} a$, $\frac{1}{2052715504137770514321093} a^{16} - \frac{1821877602085487344}{684238501379256838107031} a^{14} - \frac{118277132847296752494}{684238501379256838107031} a^{12} - \frac{4451501098732953787168}{2052715504137770514321093} a^{10} - \frac{526406670487882832118227}{2052715504137770514321093} a^{8} - \frac{19292597520608600964458}{108037658112514237595847} a^{6} + \frac{224528774490783982094}{1895397510745863817471} a^{4} + \frac{9226442536037722334}{299273291170399550127} a^{2} + \frac{2362137662642140835}{15751225851073660533}$, $\frac{1}{2052715504137770514321093} a^{17} - \frac{1821877602085487344}{684238501379256838107031} a^{15} - \frac{118277132847296752494}{684238501379256838107031} a^{13} - \frac{4451501098732953787168}{2052715504137770514321093} a^{11} - \frac{526406670487882832118227}{2052715504137770514321093} a^{9} - \frac{19292597520608600964458}{108037658112514237595847} a^{7} + \frac{224528774490783982094}{1895397510745863817471} a^{5} + \frac{9226442536037722334}{299273291170399550127} a^{3} + \frac{2362137662642140835}{15751225851073660533} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 70223252585.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.2349.1, 9.9.1049866478469.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$$[2, 2, 2]^{3}$
2.12.12.17$x^{12} + 22 x^{10} + 75 x^{8} - 12 x^{6} - 89 x^{4} + 54 x^{2} - 115$$2$$6$$12$12T29$[2, 2]^{12}$
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$