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Properties

Label 18.8.58115525246...6123.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,7^{12}\cdot 13^{4}\cdot 43^{5}$
Root discriminant $18.39$
Ramified primes $7, 13, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T696

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, -50, 13, 147, -187, 142, -163, 173, -90, -52, 119, -114, 77, -35, 5, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 6*x^16 + 5*x^15 - 35*x^14 + 77*x^13 - 114*x^12 + 119*x^11 - 52*x^10 - 90*x^9 + 173*x^8 - 163*x^7 + 142*x^6 - 187*x^5 + 147*x^4 + 13*x^3 - 50*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 6*x^16 + 5*x^15 - 35*x^14 + 77*x^13 - 114*x^12 + 119*x^11 - 52*x^10 - 90*x^9 + 173*x^8 - 163*x^7 + 142*x^6 - 187*x^5 + 147*x^4 + 13*x^3 - 50*x^2 + 14*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 6 x^{16} + 5 x^{15} - 35 x^{14} + 77 x^{13} - 114 x^{12} + 119 x^{11} - 52 x^{10} - 90 x^{9} + 173 x^{8} - 163 x^{7} + 142 x^{6} - 187 x^{5} + 147 x^{4} + 13 x^{3} - 50 x^{2} + 14 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-58115525246155509346123=-\,7^{12}\cdot 13^{4}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.39$
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:  $7, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{2}{9} a^{12} + \frac{2}{9} a^{11} + \frac{4}{9} a^{10} - \frac{2}{9} a^{7} - \frac{2}{9} a^{6} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{3} a^{14} - \frac{2}{9} a^{13} - \frac{4}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{3} a^{10} - \frac{2}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{578592161556687} a^{17} + \frac{7649930367154}{192864053852229} a^{16} + \frac{399089649923}{14112003940407} a^{15} + \frac{89357367602551}{578592161556687} a^{14} + \frac{180908417437958}{578592161556687} a^{13} + \frac{51235023456140}{578592161556687} a^{12} + \frac{199802102688278}{578592161556687} a^{11} - \frac{263488097897654}{578592161556687} a^{10} + \frac{47308315691995}{578592161556687} a^{9} - \frac{214525782829463}{578592161556687} a^{8} - \frac{45626628285914}{578592161556687} a^{7} - \frac{230421651260842}{578592161556687} a^{6} - \frac{89001555913154}{578592161556687} a^{5} - \frac{81368840018348}{578592161556687} a^{4} + \frac{94954397815052}{192864053852229} a^{3} + \frac{239974812291434}{578592161556687} a^{2} - \frac{85282780230586}{578592161556687} a + \frac{128731493493890}{578592161556687}$

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:  \( 34380.6983555 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T696:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 192 conjugacy class representatives for t18n696 are not computed
Character table for t18n696 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.36763077169.1

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
43Data not computed