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Properties

Label 18.6.86596161341...7136.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $67.59$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![461, 9614, 63621, 120254, 64909, -24494, -58102, -20162, -13776, -10048, -5369, -2704, 275, -246, -215, 50, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 2*x^16 + 50*x^15 - 215*x^14 - 246*x^13 + 275*x^12 - 2704*x^11 - 5369*x^10 - 10048*x^9 - 13776*x^8 - 20162*x^7 - 58102*x^6 - 24494*x^5 + 64909*x^4 + 120254*x^3 + 63621*x^2 + 9614*x + 461)
 
gp: K = bnfinit(x^18 - 4*x^17 + 2*x^16 + 50*x^15 - 215*x^14 - 246*x^13 + 275*x^12 - 2704*x^11 - 5369*x^10 - 10048*x^9 - 13776*x^8 - 20162*x^7 - 58102*x^6 - 24494*x^5 + 64909*x^4 + 120254*x^3 + 63621*x^2 + 9614*x + 461, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 2 x^{16} + 50 x^{15} - 215 x^{14} - 246 x^{13} + 275 x^{12} - 2704 x^{11} - 5369 x^{10} - 10048 x^{9} - 13776 x^{8} - 20162 x^{7} - 58102 x^{6} - 24494 x^{5} + 64909 x^{4} + 120254 x^{3} + 63621 x^{2} + 9614 x + 461 \)

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:  \(865961613414533621361938102747136=2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.59$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{10} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{3}{8} a$, $\frac{1}{1927440524183983083024990355740463893496} a^{17} + \frac{84778578008246516560099545441097170413}{1927440524183983083024990355740463893496} a^{16} - \frac{59241131014130923821481039155720040015}{963720262091991541512495177870231946748} a^{15} - \frac{117283334155207291404391834225318327545}{963720262091991541512495177870231946748} a^{14} + \frac{10308715347825755264212088037764102747}{240930065522997885378123794467557986687} a^{13} - \frac{1571918510522789998814835388249070073}{148264655706460237155768488903112607192} a^{12} + \frac{115782996410626829165639074170087405263}{963720262091991541512495177870231946748} a^{11} - \frac{77225248171934189562238565845368790567}{963720262091991541512495177870231946748} a^{10} - \frac{322507107129269936482547897106525423009}{1927440524183983083024990355740463893496} a^{9} + \frac{267797889354106828673117299543296612303}{1927440524183983083024990355740463893496} a^{8} + \frac{199946843704254135074085296180001805753}{963720262091991541512495177870231946748} a^{7} - \frac{42660742965893405945747577807946299187}{240930065522997885378123794467557986687} a^{6} - \frac{125705577895458075888479353941485257763}{1927440524183983083024990355740463893496} a^{5} - \frac{17679934801058102608460410969982757973}{963720262091991541512495177870231946748} a^{4} + \frac{116113379930717333903083665485300169077}{963720262091991541512495177870231946748} a^{3} + \frac{407129270859114921683696411926345932759}{963720262091991541512495177870231946748} a^{2} + \frac{50171522079662329752378824581398765053}{481860131045995770756247588935115973374} a - \frac{696372235712350923821352766250319400957}{1927440524183983083024990355740463893496}$

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:  \( 6081384726.36 \) (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

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{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}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed