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Properties

Label 18.8.45640439727...9744.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{6}\cdot 3^{6}\cdot 7^{12}\cdot 643^{6}$
Root discriminant $57.39$
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![729, 0, -1944, 0, -36666, 0, -10601, 0, 10627, 0, 3167, 0, -621, 0, -150, 0, 11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 11*x^16 - 150*x^14 - 621*x^12 + 3167*x^10 + 10627*x^8 - 10601*x^6 - 36666*x^4 - 1944*x^2 + 729)
 
gp: K = bnfinit(x^18 + 11*x^16 - 150*x^14 - 621*x^12 + 3167*x^10 + 10627*x^8 - 10601*x^6 - 36666*x^4 - 1944*x^2 + 729, 1)
 

Normalized defining polynomial

\( x^{18} + 11 x^{16} - 150 x^{14} - 621 x^{12} + 3167 x^{10} + 10627 x^{8} - 10601 x^{6} - 36666 x^{4} - 1944 x^{2} + 729 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-45640439727204768655041586119744=-\,2^{6}\cdot 3^{6}\cdot 7^{12}\cdot 643^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{54} a^{14} + \frac{1}{27} a^{12} + \frac{1}{18} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{4}{27} a^{6} + \frac{1}{3} a^{5} - \frac{10}{27} a^{4} - \frac{1}{6} a^{3} - \frac{13}{27} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{324} a^{15} + \frac{5}{81} a^{13} - \frac{1}{12} a^{12} - \frac{2}{27} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{23}{162} a^{7} - \frac{5}{12} a^{6} - \frac{5}{81} a^{5} + \frac{1}{4} a^{4} - \frac{2}{81} a^{3} + \frac{1}{4} a^{2} - \frac{5}{36} a - \frac{1}{4}$, $\frac{1}{1057980926786004} a^{16} + \frac{1456965124520}{264495231696501} a^{14} - \frac{1}{12} a^{13} - \frac{9656901671383}{176330154464334} a^{12} - \frac{1}{12} a^{11} + \frac{97549145095}{6530746461642} a^{10} - \frac{1}{12} a^{9} + \frac{1099490154851}{264495231696501} a^{8} - \frac{1}{12} a^{7} - \frac{20556943126867}{528990463393002} a^{6} - \frac{5}{12} a^{5} - \frac{240901891202803}{528990463393002} a^{4} - \frac{5}{12} a^{3} + \frac{12453517106201}{117553436309556} a^{2} + \frac{5}{12} a + \frac{105459899153}{6530746461642}$, $\frac{1}{3173942780358012} a^{17} - \frac{3968259194383}{3173942780358012} a^{15} - \frac{1}{108} a^{14} - \frac{42310633979593}{528990463393002} a^{13} - \frac{1}{54} a^{12} + \frac{516275378857}{6530746461642} a^{11} + \frac{1}{18} a^{10} - \frac{85966096922465}{1586971390179006} a^{9} - \frac{73953249564443}{793485695089503} a^{7} - \frac{11}{27} a^{6} - \frac{380048117451671}{793485695089503} a^{5} + \frac{5}{27} a^{4} - \frac{76800017869573}{352660308928668} a^{3} + \frac{11}{27} a^{2} + \frac{18714701439625}{39184478769852} a - \frac{1}{12}$

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:  $12$
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:  \( 616975510.316 \) (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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
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}$
7Data not computed
643Data not computed