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Properties

Label 18.18.6146657264...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{32}\cdot 5^{9}\cdot 19^{8}$
Root discriminant $58.35$
Ramified primes $3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times He_3$ (as 18T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![181, -2397, 6732, 12969, -48483, -26562, 103944, 37308, -85104, -25698, 33624, 8616, -7026, -1416, 792, 108, -45, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 45*x^16 + 108*x^15 + 792*x^14 - 1416*x^13 - 7026*x^12 + 8616*x^11 + 33624*x^10 - 25698*x^9 - 85104*x^8 + 37308*x^7 + 103944*x^6 - 26562*x^5 - 48483*x^4 + 12969*x^3 + 6732*x^2 - 2397*x + 181)
 
gp: K = bnfinit(x^18 - 3*x^17 - 45*x^16 + 108*x^15 + 792*x^14 - 1416*x^13 - 7026*x^12 + 8616*x^11 + 33624*x^10 - 25698*x^9 - 85104*x^8 + 37308*x^7 + 103944*x^6 - 26562*x^5 - 48483*x^4 + 12969*x^3 + 6732*x^2 - 2397*x + 181, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 45 x^{16} + 108 x^{15} + 792 x^{14} - 1416 x^{13} - 7026 x^{12} + 8616 x^{11} + 33624 x^{10} - 25698 x^{9} - 85104 x^{8} + 37308 x^{7} + 103944 x^{6} - 26562 x^{5} - 48483 x^{4} + 12969 x^{3} + 6732 x^{2} - 2397 x + 181 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(61466572643771419985172814453125=3^{32}\cdot 5^{9}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.35$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{9148} a^{16} + \frac{330}{2287} a^{15} + \frac{285}{4574} a^{14} + \frac{941}{4574} a^{13} + \frac{368}{2287} a^{12} + \frac{239}{4574} a^{11} - \frac{337}{4574} a^{10} + \frac{106}{2287} a^{9} - \frac{985}{4574} a^{8} + \frac{1717}{4574} a^{7} + \frac{809}{2287} a^{6} - \frac{997}{4574} a^{5} - \frac{1233}{4574} a^{4} - \frac{25}{2287} a^{3} + \frac{1713}{9148} a^{2} - \frac{1129}{2287} a + \frac{2853}{9148}$, $\frac{1}{389404825697476867869788} a^{17} + \frac{2244996614595597682}{97351206424369216967447} a^{16} - \frac{21077852983244027625829}{97351206424369216967447} a^{15} + \frac{6741273824857254632104}{97351206424369216967447} a^{14} - \frac{30109074849280242556533}{194702412848738433934894} a^{13} - \frac{3802658621918066086535}{97351206424369216967447} a^{12} - \frac{19513181518068731895046}{97351206424369216967447} a^{11} + \frac{6833813237900408250787}{97351206424369216967447} a^{10} + \frac{17956436907316397690162}{97351206424369216967447} a^{9} - \frac{10767210755885765171131}{97351206424369216967447} a^{8} + \frac{40438432215791192841474}{97351206424369216967447} a^{7} - \frac{21593488507541184619325}{194702412848738433934894} a^{6} - \frac{89970144920584098819715}{194702412848738433934894} a^{5} + \frac{36884169256035893443309}{97351206424369216967447} a^{4} - \frac{172428520524403925676917}{389404825697476867869788} a^{3} - \frac{4231388865546256951435}{194702412848738433934894} a^{2} - \frac{70045013870063259778129}{389404825697476867869788} a - \frac{22639913158077252019477}{97351206424369216967447}$

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:  $17$
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:  \( 12294031262.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3$ (as 18T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times He_3$
Character table for $C_2\times He_3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1, 9.9.5609891727441.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$