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Properties

Label 18.6.25939593315...0000.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 19^{6}\cdot 41^{3}$
Root discriminant $17.59$
Ramified primes $2, 5, 19, 41$
Class number $1$
Class group Trivial
Galois group 18T189

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -19, 89, -76, -114, 123, 134, -83, -148, 58, 52, 23, -49, 6, 0, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 10*x^16 + 6*x^14 - 49*x^13 + 23*x^12 + 52*x^11 + 58*x^10 - 148*x^9 - 83*x^8 + 134*x^7 + 123*x^6 - 114*x^5 - 76*x^4 + 89*x^3 - 19*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 10*x^16 + 6*x^14 - 49*x^13 + 23*x^12 + 52*x^11 + 58*x^10 - 148*x^9 - 83*x^8 + 134*x^7 + 123*x^6 - 114*x^5 - 76*x^4 + 89*x^3 - 19*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 10 x^{16} + 6 x^{14} - 49 x^{13} + 23 x^{12} + 52 x^{11} + 58 x^{10} - 148 x^{9} - 83 x^{8} + 134 x^{7} + 123 x^{6} - 114 x^{5} - 76 x^{4} + 89 x^{3} - 19 x^{2} - 3 x + 1 \)

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:  \(25939593315208000000000=2^{12}\cdot 5^{9}\cdot 19^{6}\cdot 41^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.59$
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:  $2, 5, 19, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{13} + \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{3}{7} a^{10} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{13} + \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{4435863089} a^{17} - \frac{61708880}{4435863089} a^{16} - \frac{25570390}{633694727} a^{15} + \frac{50646626}{4435863089} a^{14} + \frac{644300170}{4435863089} a^{13} - \frac{1716491981}{4435863089} a^{12} + \frac{852921542}{4435863089} a^{11} - \frac{668093766}{4435863089} a^{10} - \frac{552732830}{4435863089} a^{9} + \frac{105390812}{633694727} a^{8} - \frac{280486310}{633694727} a^{7} - \frac{162515691}{633694727} a^{6} - \frac{552625464}{4435863089} a^{5} + \frac{581140619}{4435863089} a^{4} + \frac{393944232}{4435863089} a^{3} - \frac{192120522}{4435863089} a^{2} + \frac{1816444190}{4435863089} a - \frac{2039983450}{4435863089}$

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:  \( 15106.3273338 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T189:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 54 conjugacy class representatives for t18n189 are not computed
Character table for t18n189 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 6.6.722000.1, 6.2.29602000.3

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

Sibling fields

Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
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$$[\ ]_{2}^{3}$
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$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$$[\ ]^{3}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$41$41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$