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Properties

Label 18.6.49048530062...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 19^{10}$
Root discriminant $18.22$
Ramified primes $2, 5, 19$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 16, 25, -68, -142, 102, 262, -118, -303, 106, 210, -72, -76, 30, 17, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 6*x^16 + 17*x^15 + 30*x^14 - 76*x^13 - 72*x^12 + 210*x^11 + 106*x^10 - 303*x^9 - 118*x^8 + 262*x^7 + 102*x^6 - 142*x^5 - 68*x^4 + 25*x^3 + 16*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 6 x^{16} + 17 x^{15} + 30 x^{14} - 76 x^{13} - 72 x^{12} + 210 x^{11} + 106 x^{10} - 303 x^{9} - 118 x^{8} + 262 x^{7} + 102 x^{6} - 142 x^{5} - 68 x^{4} + 25 x^{3} + 16 x^{2} - 1 \)

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:  \(49048530062408000000000=2^{12}\cdot 5^{9}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.22$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} - \frac{8}{19} a^{14} - \frac{8}{19} a^{13} + \frac{1}{19} a^{12} - \frac{6}{19} a^{11} - \frac{7}{19} a^{10} - \frac{4}{19} a^{9} + \frac{5}{19} a^{8} + \frac{9}{19} a^{7} - \frac{5}{19} a^{6} - \frac{8}{19} a^{5} - \frac{6}{19} a^{4} + \frac{8}{19} a^{3} + \frac{3}{19} a^{2} - \frac{1}{19} a + \frac{7}{19}$, $\frac{1}{19} a^{16} + \frac{4}{19} a^{14} - \frac{6}{19} a^{13} + \frac{2}{19} a^{12} + \frac{2}{19} a^{11} - \frac{3}{19} a^{10} - \frac{8}{19} a^{9} - \frac{8}{19} a^{8} - \frac{9}{19} a^{7} + \frac{9}{19} a^{6} + \frac{6}{19} a^{5} - \frac{2}{19} a^{4} - \frac{9}{19} a^{3} + \frac{4}{19} a^{2} - \frac{1}{19} a - \frac{1}{19}$, $\frac{1}{3575415686411} a^{17} - \frac{22069974359}{3575415686411} a^{16} - \frac{83854298401}{3575415686411} a^{15} - \frac{1468873617460}{3575415686411} a^{14} + \frac{999781043856}{3575415686411} a^{13} + \frac{966963651589}{3575415686411} a^{12} + \frac{635433649389}{3575415686411} a^{11} + \frac{761486975039}{3575415686411} a^{10} + \frac{976633283357}{3575415686411} a^{9} + \frac{825989932961}{3575415686411} a^{8} - \frac{954594375449}{3575415686411} a^{7} - \frac{414352237697}{3575415686411} a^{6} + \frac{973185113821}{3575415686411} a^{5} + \frac{603898036870}{3575415686411} a^{4} - \frac{75090234302}{3575415686411} a^{3} + \frac{55635046609}{3575415686411} a^{2} + \frac{974491859567}{3575415686411} a + \frac{194716705441}{3575415686411}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 20286.0811613 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.76.1, 6.2.722000.1, 6.6.722000.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$5.6.3.1$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$$[\ ]_{2}^{3}$
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$