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Properties

Label 18.8.15425002003...5003.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 83$
Root discriminant $36.81$
Ramified primes $3, 7, 29, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13, -99, -243, 375, 489, -705, -62, 975, 63, -604, -30, 84, 145, 27, -120, 22, 21, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 21*x^16 + 22*x^15 - 120*x^14 + 27*x^13 + 145*x^12 + 84*x^11 - 30*x^10 - 604*x^9 + 63*x^8 + 975*x^7 - 62*x^6 - 705*x^5 + 489*x^4 + 375*x^3 - 243*x^2 - 99*x + 13)
 
gp: K = bnfinit(x^18 - 9*x^17 + 21*x^16 + 22*x^15 - 120*x^14 + 27*x^13 + 145*x^12 + 84*x^11 - 30*x^10 - 604*x^9 + 63*x^8 + 975*x^7 - 62*x^6 - 705*x^5 + 489*x^4 + 375*x^3 - 243*x^2 - 99*x + 13, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 21 x^{16} + 22 x^{15} - 120 x^{14} + 27 x^{13} + 145 x^{12} + 84 x^{11} - 30 x^{10} - 604 x^{9} + 63 x^{8} + 975 x^{7} - 62 x^{6} - 705 x^{5} + 489 x^{4} + 375 x^{3} - 243 x^{2} - 99 x + 13 \)

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:  \(-15425002003895348854594965003=-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 83\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.81$
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:  $3, 7, 29, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{42} a^{12} - \frac{1}{14} a^{11} - \frac{5}{42} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{3} a^{7} - \frac{1}{42} a^{6} - \frac{3}{7} a^{5} + \frac{3}{14} a^{4} + \frac{8}{21} a^{3} + \frac{1}{14} a^{2} - \frac{1}{3} a + \frac{13}{42}$, $\frac{1}{42} a^{13} - \frac{1}{6} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{5}{14} a^{7} - \frac{1}{2} a^{6} - \frac{1}{14} a^{5} + \frac{1}{42} a^{4} + \frac{3}{14} a^{3} + \frac{3}{14} a^{2} - \frac{5}{14} a - \frac{1}{14}$, $\frac{1}{42} a^{14} - \frac{1}{6} a^{11} - \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{5}{14} a^{8} - \frac{1}{2} a^{7} - \frac{1}{14} a^{6} + \frac{1}{42} a^{5} + \frac{3}{14} a^{4} + \frac{3}{14} a^{3} - \frac{5}{14} a^{2} - \frac{1}{14} a$, $\frac{1}{294} a^{15} + \frac{1}{294} a^{14} - \frac{1}{294} a^{12} + \frac{25}{294} a^{11} + \frac{1}{21} a^{10} + \frac{41}{294} a^{9} + \frac{32}{147} a^{8} + \frac{44}{147} a^{7} - \frac{53}{147} a^{6} + \frac{5}{21} a^{5} + \frac{5}{49} a^{4} - \frac{6}{49} a^{3} - \frac{8}{21} a^{2} + \frac{137}{294} a - \frac{59}{147}$, $\frac{1}{294} a^{16} - \frac{1}{294} a^{14} - \frac{1}{294} a^{13} - \frac{1}{147} a^{12} - \frac{25}{294} a^{11} - \frac{29}{294} a^{10} - \frac{5}{294} a^{9} + \frac{40}{147} a^{8} - \frac{16}{49} a^{7} + \frac{53}{147} a^{6} - \frac{62}{147} a^{5} - \frac{4}{49} a^{4} + \frac{32}{147} a^{3} + \frac{67}{294} a^{2} - \frac{59}{294} a + \frac{73}{147}$, $\frac{1}{59621965832982} a^{17} - \frac{39729850486}{29810982916491} a^{16} + \frac{46295181908}{29810982916491} a^{15} - \frac{39452287310}{4258711845213} a^{14} - \frac{237853804822}{29810982916491} a^{13} + \frac{165107219743}{19873988610994} a^{12} + \frac{100690730715}{19873988610994} a^{11} + \frac{1874127735065}{29810982916491} a^{10} - \frac{9866346297737}{59621965832982} a^{9} + \frac{12102351092303}{29810982916491} a^{8} - \frac{3920139365797}{29810982916491} a^{7} + \frac{8999101701017}{59621965832982} a^{6} - \frac{4043910625295}{9936994305497} a^{5} + \frac{128773862669}{59621965832982} a^{4} + \frac{4477454756365}{59621965832982} a^{3} + \frac{499961118217}{1419570615071} a^{2} - \frac{5694687982589}{19873988610994} a + \frac{4214861689067}{59621965832982}$

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:  $12$
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:  \( 38618166.4991 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R $18$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$
83.6.0.1$x^{6} - x + 34$$1$$6$$0$$C_6$$[\ ]^{6}$