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Properties

Label 18.6.34546116175...7009.2
Degree $18$
Signature $[6, 6]$
Discriminant $7^{14}\cdot 83^{4}\cdot 181^{4}$
Root discriminant $38.50$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T646

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5419, 11060, -3219, -9884, 12536, -9806, 6447, -1606, -1818, 2421, -2446, 1715, -748, 378, -114, 16, -2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 16*x^15 - 114*x^14 + 378*x^13 - 748*x^12 + 1715*x^11 - 2446*x^10 + 2421*x^9 - 1818*x^8 - 1606*x^7 + 6447*x^6 - 9806*x^5 + 12536*x^4 - 9884*x^3 - 3219*x^2 + 11060*x - 5419)
 
gp: K = bnfinit(x^18 - 3*x^17 - 2*x^16 + 16*x^15 - 114*x^14 + 378*x^13 - 748*x^12 + 1715*x^11 - 2446*x^10 + 2421*x^9 - 1818*x^8 - 1606*x^7 + 6447*x^6 - 9806*x^5 + 12536*x^4 - 9884*x^3 - 3219*x^2 + 11060*x - 5419, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 2 x^{16} + 16 x^{15} - 114 x^{14} + 378 x^{13} - 748 x^{12} + 1715 x^{11} - 2446 x^{10} + 2421 x^{9} - 1818 x^{8} - 1606 x^{7} + 6447 x^{6} - 9806 x^{5} + 12536 x^{4} - 9884 x^{3} - 3219 x^{2} + 11060 x - 5419 \)

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:  \(34546116175413916877506117009=7^{14}\cdot 83^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.50$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{14} - \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{129115099095492673022381347375949} a^{17} + \frac{1798729351120752741862808753562}{129115099095492673022381347375949} a^{16} - \frac{7640808965650161075526123834143}{129115099095492673022381347375949} a^{15} - \frac{2903469862336261988643429291251}{129115099095492673022381347375949} a^{14} + \frac{4096125164249870298821968129427}{129115099095492673022381347375949} a^{13} + \frac{1904924742429127104088137637712}{129115099095492673022381347375949} a^{12} + \frac{16013485933330655604073837087559}{129115099095492673022381347375949} a^{11} + \frac{37571068893807917545764270491008}{129115099095492673022381347375949} a^{10} + \frac{59285497010389489371151057758050}{129115099095492673022381347375949} a^{9} + \frac{38925383737747022452923076408628}{129115099095492673022381347375949} a^{8} + \frac{48705920503585813503597000126042}{129115099095492673022381347375949} a^{7} - \frac{49939585265129574202657100742997}{129115099095492673022381347375949} a^{6} - \frac{45265959577178653171104201291090}{129115099095492673022381347375949} a^{5} + \frac{49790521129041288007121745001823}{129115099095492673022381347375949} a^{4} - \frac{10087810667320545698426473865106}{129115099095492673022381347375949} a^{3} + \frac{49446217119150835962106426446920}{129115099095492673022381347375949} a^{2} - \frac{14740759773629926060352309411961}{129115099095492673022381347375949} a - \frac{6305752424492651698940445185762}{18445014156498953288911621053707}$

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:  $11$
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:  \( 21400373.7952 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T646:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20736
The 32 conjugacy class representatives for t18n646
Character table for t18n646 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
83Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$