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Properties

Label 18.6.35795770178...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{11}\cdot 17^{8}\cdot 37^{6}$
Root discriminant $49.82$
Ramified primes $2, 5, 17, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-729, -9963, -81162, 150741, 156303, -273183, -64469, 189202, -15539, -58815, 14204, 9026, -3568, -405, 310, -2, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 11*x^16 - 2*x^15 + 310*x^14 - 405*x^13 - 3568*x^12 + 9026*x^11 + 14204*x^10 - 58815*x^9 - 15539*x^8 + 189202*x^7 - 64469*x^6 - 273183*x^5 + 156303*x^4 + 150741*x^3 - 81162*x^2 - 9963*x - 729)
 
gp: K = bnfinit(x^18 - 2*x^17 - 11*x^16 - 2*x^15 + 310*x^14 - 405*x^13 - 3568*x^12 + 9026*x^11 + 14204*x^10 - 58815*x^9 - 15539*x^8 + 189202*x^7 - 64469*x^6 - 273183*x^5 + 156303*x^4 + 150741*x^3 - 81162*x^2 - 9963*x - 729, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 11 x^{16} - 2 x^{15} + 310 x^{14} - 405 x^{13} - 3568 x^{12} + 9026 x^{11} + 14204 x^{10} - 58815 x^{9} - 15539 x^{8} + 189202 x^{7} - 64469 x^{6} - 273183 x^{5} + 156303 x^{4} + 150741 x^{3} - 81162 x^{2} - 9963 x - 729 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3579577017830391873800000000000=2^{12}\cdot 5^{11}\cdot 17^{8}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.82$
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:  $2, 5, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} - \frac{4}{9} a^{8} - \frac{4}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} + \frac{10}{27} a^{12} - \frac{2}{27} a^{11} + \frac{1}{9} a^{10} - \frac{13}{27} a^{9} - \frac{4}{27} a^{8} + \frac{8}{27} a^{7} + \frac{2}{9} a^{6} - \frac{5}{27} a^{5} - \frac{2}{27} a^{4} + \frac{10}{27} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{81} a^{16} + \frac{1}{81} a^{15} + \frac{1}{81} a^{14} + \frac{10}{81} a^{13} + \frac{25}{81} a^{12} + \frac{1}{27} a^{11} - \frac{13}{81} a^{10} + \frac{23}{81} a^{9} - \frac{19}{81} a^{8} - \frac{7}{27} a^{7} + \frac{22}{81} a^{6} + \frac{25}{81} a^{5} + \frac{37}{81} a^{4} - \frac{13}{27} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{342536627152199830196104461020481607413} a^{17} - \frac{1392342264562567606100570074269398972}{342536627152199830196104461020481607413} a^{16} + \frac{3076330458283554736550452258711893055}{342536627152199830196104461020481607413} a^{15} + \frac{7211711249118364333669216695708625840}{342536627152199830196104461020481607413} a^{14} + \frac{12282189227673462849484384585324870906}{342536627152199830196104461020481607413} a^{13} - \frac{1344562517298538468602077485881900121}{114178875717399943398701487006827202471} a^{12} - \frac{18089000931317603711993960905633299889}{342536627152199830196104461020481607413} a^{11} + \frac{116240640226749303709977865168899029615}{342536627152199830196104461020481607413} a^{10} + \frac{7398637608906951037050861380496300242}{342536627152199830196104461020481607413} a^{9} - \frac{45944291557970449727258051391053155718}{114178875717399943398701487006827202471} a^{8} - \frac{143542004538991445210995923453442767230}{342536627152199830196104461020481607413} a^{7} + \frac{154526181301389916024931302904243588608}{342536627152199830196104461020481607413} a^{6} - \frac{46178923400349840836662780521835862990}{342536627152199830196104461020481607413} a^{5} + \frac{5591017136002058908653804313192496789}{12686541746377771488744609667425244719} a^{4} + \frac{12793758733658658062683386464061645632}{38059625239133314466233829002275734157} a^{3} + \frac{1721485383097870313986862815913694982}{12686541746377771488744609667425244719} a^{2} - \frac{375316520907433661352110729239724754}{1409615749597530165416067740825027191} a + \frac{151893534541509468889615724839146207}{1409615749597530165416067740825027191}$

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:  \( 138117514.681 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.148.1, 6.2.109520.1, 9.9.169223568520000.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.8.2$x^{12} + 25 x^{6} - 250 x^{3} + 1250$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$17$17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.12.8.2$x^{12} - 4913 x^{3} + 918731$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.0.1$x^{6} - x + 20$$1$$6$$0$$C_6$$[\ ]^{6}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$