↑

Properties

Label 18.12.6400139496...0704.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 7^{14}\cdot 13^{8}$
Root discriminant $97.55$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5564881, 0, -5267451, 0, 294000, 0, 642425, 0, -132027, 0, -6825, 0, 3193, 0, -102, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 102*x^14 + 3193*x^12 - 6825*x^10 - 132027*x^8 + 642425*x^6 + 294000*x^4 - 5267451*x^2 + 5564881)
 
gp: K = bnfinit(x^18 - 18*x^16 - 102*x^14 + 3193*x^12 - 6825*x^10 - 132027*x^8 + 642425*x^6 + 294000*x^4 - 5267451*x^2 + 5564881, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 102 x^{14} + 3193 x^{12} - 6825 x^{10} - 132027 x^{8} + 642425 x^{6} + 294000 x^{4} - 5267451 x^{2} + 5564881 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-640013949632619052551483742699720704=-\,2^{12}\cdot 3^{24}\cdot 7^{14}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.55$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{2}{7} a^{8} + \frac{1}{7} a^{6}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{9} + \frac{1}{7} a^{7}$, $\frac{1}{14} a^{12} - \frac{1}{14} a^{10} - \frac{5}{14} a^{8} - \frac{3}{14} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{11} - \frac{5}{14} a^{9} - \frac{3}{14} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{14} a^{14} + \frac{2}{7} a^{8} - \frac{2}{7} a^{6} - \frac{1}{2}$, $\frac{1}{28} a^{15} - \frac{1}{28} a^{14} - \frac{1}{28} a^{13} - \frac{1}{28} a^{11} - \frac{1}{14} a^{10} - \frac{9}{28} a^{9} + \frac{3}{14} a^{8} + \frac{11}{28} a^{7} + \frac{1}{14} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4}$, $\frac{1}{639551092746291568892} a^{16} - \frac{1060005088254706658}{159887773186572892223} a^{14} - \frac{1}{28} a^{13} - \frac{1487756480015501245}{639551092746291568892} a^{12} + \frac{1}{28} a^{11} - \frac{21250737992202737815}{639551092746291568892} a^{10} + \frac{5}{28} a^{9} + \frac{25379496895540354195}{91364441820898795556} a^{8} - \frac{11}{28} a^{7} - \frac{17841940207259771595}{91364441820898795556} a^{6} - \frac{1}{4} a^{5} - \frac{21986285590548288591}{91364441820898795556} a^{4} + \frac{1}{4} a^{3} - \frac{557381735156045263}{3263015779317814127} a^{2} - \frac{1}{4} a + \frac{1498020335213454451}{13052063117271256508}$, $\frac{1}{215528718255500258716604} a^{17} - \frac{1443229979032174856639}{215528718255500258716604} a^{15} - \frac{1045352742446533849483}{53882179563875064679151} a^{13} - \frac{1}{28} a^{12} - \frac{4521744683902979399485}{107764359127750129358302} a^{11} + \frac{1}{28} a^{10} + \frac{7591043895913393487055}{15394908446821447051186} a^{9} + \frac{5}{28} a^{8} + \frac{1285246501723551090798}{7697454223410723525593} a^{7} - \frac{11}{28} a^{6} - \frac{7582821258702261825999}{15394908446821447051186} a^{5} - \frac{1}{4} a^{4} + \frac{85871899100956800377}{4398545270520413443196} a^{3} + \frac{1}{4} a^{2} - \frac{1375494638536904107143}{4398545270520413443196} a - \frac{1}{4}$

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:  $14$
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:  \( 595238841562 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.1785733746591249.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$