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Properties

Label 18.16.1232291872...3088.1
Degree $18$
Signature $[16, 1]$
Discriminant $-\,2^{6}\cdot 3^{9}\cdot 7^{12}\cdot 643^{6}$
Root discriminant $68.92$
Ramified primes $2, 3, 7, 643$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4901, -49101, -191533, -339196, -151292, 409110, 691759, 327929, -113286, -155287, -25670, 16476, 4347, -479, 87, 31, -30, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 30*x^16 + 31*x^15 + 87*x^14 - 479*x^13 + 4347*x^12 + 16476*x^11 - 25670*x^10 - 155287*x^9 - 113286*x^8 + 327929*x^7 + 691759*x^6 + 409110*x^5 - 151292*x^4 - 339196*x^3 - 191533*x^2 - 49101*x - 4901)
 
gp: K = bnfinit(x^18 - 2*x^17 - 30*x^16 + 31*x^15 + 87*x^14 - 479*x^13 + 4347*x^12 + 16476*x^11 - 25670*x^10 - 155287*x^9 - 113286*x^8 + 327929*x^7 + 691759*x^6 + 409110*x^5 - 151292*x^4 - 339196*x^3 - 191533*x^2 - 49101*x - 4901, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 30 x^{16} + 31 x^{15} + 87 x^{14} - 479 x^{13} + 4347 x^{12} + 16476 x^{11} - 25670 x^{10} - 155287 x^{9} - 113286 x^{8} + 327929 x^{7} + 691759 x^{6} + 409110 x^{5} - 151292 x^{4} - 339196 x^{3} - 191533 x^{2} - 49101 x - 4901 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1232291872634528753686122825233088=-\,2^{6}\cdot 3^{9}\cdot 7^{12}\cdot 643^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.92$
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:  $2, 3, 7, 643$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} - \frac{11}{23} a^{15} - \frac{3}{23} a^{14} - \frac{1}{23} a^{13} - \frac{10}{23} a^{12} + \frac{5}{23} a^{11} + \frac{8}{23} a^{10} - \frac{10}{23} a^{9} - \frac{5}{23} a^{8} - \frac{8}{23} a^{7} + \frac{7}{23} a^{6} + \frac{2}{23} a^{5} - \frac{5}{23} a^{4} + \frac{2}{23} a^{3} - \frac{1}{23} a^{2} + \frac{11}{23} a + \frac{7}{23}$, $\frac{1}{1600646899884187332567530583313} a^{17} - \frac{33796787835229082544923885533}{1600646899884187332567530583313} a^{16} + \frac{20166829885879603476310834225}{1600646899884187332567530583313} a^{15} + \frac{782103948814302469802359315473}{1600646899884187332567530583313} a^{14} - \frac{653086316401325190449024438678}{1600646899884187332567530583313} a^{13} + \frac{690890254325923058184310570559}{1600646899884187332567530583313} a^{12} - \frac{723049916527689221720167324726}{1600646899884187332567530583313} a^{11} + \frac{523411702281150164932567503651}{1600646899884187332567530583313} a^{10} - \frac{613302063021906861710615344975}{1600646899884187332567530583313} a^{9} - \frac{199300998239200904910600797101}{1600646899884187332567530583313} a^{8} + \frac{18094510123414190660332197614}{1600646899884187332567530583313} a^{7} + \frac{663137756354438861188122948171}{1600646899884187332567530583313} a^{6} - \frac{284586921814075098019802438981}{1600646899884187332567530583313} a^{5} + \frac{33905270873002265651555577273}{123126684606475948659040814101} a^{4} + \frac{44806660156879063317110338806}{1600646899884187332567530583313} a^{3} + \frac{48315749479348812007298556818}{123126684606475948659040814101} a^{2} - \frac{276260944892510700518830438398}{1600646899884187332567530583313} a + \frac{776278525458213981961781149}{4245747745050894781346234969}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $16$
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:  \( 18527680583.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.844471355782761.3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
643Data not computed