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Properties

Label 18.8.64157072897...8731.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,7^{13}\cdot 13\cdot 83^{4}\cdot 181^{4}$
Root discriminant $39.85$
Ramified primes $7, 13, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-181, -8507, 17046, -17877, 44608, -58527, 22032, -3376, 8757, 759, -8806, 4572, -636, 172, -172, 28, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 18*x^16 + 28*x^15 - 172*x^14 + 172*x^13 - 636*x^12 + 4572*x^11 - 8806*x^10 + 759*x^9 + 8757*x^8 - 3376*x^7 + 22032*x^6 - 58527*x^5 + 44608*x^4 - 17877*x^3 + 17046*x^2 - 8507*x - 181)
 
gp: K = bnfinit(x^18 - 8*x^17 + 18*x^16 + 28*x^15 - 172*x^14 + 172*x^13 - 636*x^12 + 4572*x^11 - 8806*x^10 + 759*x^9 + 8757*x^8 - 3376*x^7 + 22032*x^6 - 58527*x^5 + 44608*x^4 - 17877*x^3 + 17046*x^2 - 8507*x - 181, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 18 x^{16} + 28 x^{15} - 172 x^{14} + 172 x^{13} - 636 x^{12} + 4572 x^{11} - 8806 x^{10} + 759 x^{9} + 8757 x^{8} - 3376 x^{7} + 22032 x^{6} - 58527 x^{5} + 44608 x^{4} - 17877 x^{3} + 17046 x^{2} - 8507 x - 181 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-64157072897197274201082788731=-\,7^{13}\cdot 13\cdot 83^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.85$
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:  $7, 13, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{93058795123425225082747058756426459} a^{17} - \frac{16071735816346414123657977263470289}{93058795123425225082747058756426459} a^{16} + \frac{10474954716092877144386000686451631}{93058795123425225082747058756426459} a^{15} + \frac{15928292439567065780224853624365115}{93058795123425225082747058756426459} a^{14} + \frac{45033840122973249075443164835671773}{93058795123425225082747058756426459} a^{13} - \frac{42678974605273046923151422007161540}{93058795123425225082747058756426459} a^{12} + \frac{13483398295653236458723613394407118}{93058795123425225082747058756426459} a^{11} - \frac{38303698766214053742616569364160947}{93058795123425225082747058756426459} a^{10} + \frac{32574411579239166479666451507160708}{93058795123425225082747058756426459} a^{9} - \frac{16255008689522681928065135018398764}{93058795123425225082747058756426459} a^{8} - \frac{1380730628534560267857171167995680}{3208923969773283623543002026083671} a^{7} + \frac{38696029745561822942256176497648973}{93058795123425225082747058756426459} a^{6} - \frac{20417682009886879488665249795570937}{93058795123425225082747058756426459} a^{5} - \frac{43220042824170871006769412982110478}{93058795123425225082747058756426459} a^{4} + \frac{19860214887864146004020568427791170}{93058795123425225082747058756426459} a^{3} - \frac{32335403460168490581086462489005637}{93058795123425225082747058756426459} a^{2} - \frac{51038352657889375370783381316339}{514136989632183563993077672687439} a - \frac{158265497443774286375714941828732}{514136989632183563993077672687439}$

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:  $12$
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:  \( 56474492.8303 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ R $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$