Properties

Label 18.0.92919321348...9543.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 1399^{2}$
Root discriminant $11.32$
Ramified primes $7, 1399$
Class number $1$
Class group Trivial
Galois group 18T286

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 10*x^16 - 21*x^15 + 37*x^14 - 58*x^13 + 80*x^12 - 97*x^11 + 107*x^10 - 106*x^9 + 92*x^8 - 71*x^7 + 52*x^6 - 39*x^5 + 30*x^4 - 20*x^3 + 11*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 10*x^16 - 21*x^15 + 37*x^14 - 58*x^13 + 80*x^12 - 97*x^11 + 107*x^10 - 106*x^9 + 92*x^8 - 71*x^7 + 52*x^6 - 39*x^5 + 30*x^4 - 20*x^3 + 11*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 11, -20, 30, -39, 52, -71, 92, -106, 107, -97, 80, -58, 37, -21, 10, -4, 1]);
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 10 x^{16} - 21 x^{15} + 37 x^{14} - 58 x^{13} + 80 x^{12} - 97 x^{11} + 107 x^{10} - 106 x^{9} + 92 x^{8} - 71 x^{7} + 52 x^{6} - 39 x^{5} + 30 x^{4} - 20 x^{3} + 11 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-9291932134821949543=-\,7^{15}\cdot 1399^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.32$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 1399$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{841} a^{17} - \frac{92}{841} a^{16} - \frac{304}{841} a^{15} - \frac{181}{841} a^{14} - \frac{14}{841} a^{13} + \frac{333}{841} a^{12} + \frac{211}{841} a^{11} - \frac{163}{841} a^{10} + \frac{154}{841} a^{9} - \frac{202}{841} a^{8} + \frac{207}{841} a^{7} + \frac{215}{841} a^{6} - \frac{366}{841} a^{5} + \frac{211}{841} a^{4} - \frac{36}{841} a^{3} - \frac{216}{841} a^{2} - \frac{324}{841} a - \frac{86}{841}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1128}{841} a^{17} - \frac{6220}{841} a^{16} + \frac{16195}{841} a^{15} - \frac{33445}{841} a^{14} + \frac{60739}{841} a^{13} - \frac{94495}{841} a^{12} + \frac{131201}{841} a^{11} - \frac{158634}{841} a^{10} + \frac{170348}{841} a^{9} - \frac{168986}{841} a^{8} + \frac{144350}{841} a^{7} - \frac{103972}{841} a^{6} + \frac{74091}{841} a^{5} - \frac{58024}{841} a^{4} + \frac{45174}{841} a^{3} - \frac{30875}{841} a^{2} + \frac{12978}{841} a - \frac{3657}{841} \) (order $14$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 509.337225135 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

18T286:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n286
Character table for t18n286 is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.3.164590951.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
1399Data not computed