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Properties

Label 18.4.71064937862...6224.4
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{18}\cdot 7^{16}\cdot 13^{8}$
Root discriminant $35.26$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T586

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 58, 0, 155, 0, -245, 0, -364, 0, 63, 0, 210, 0, 92, 0, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 16*x^16 + 92*x^14 + 210*x^12 + 63*x^10 - 364*x^8 - 245*x^6 + 155*x^4 + 58*x^2 + 1)
 
gp: K = bnfinit(x^18 + 16*x^16 + 92*x^14 + 210*x^12 + 63*x^10 - 364*x^8 - 245*x^6 + 155*x^4 + 58*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 16 x^{16} + 92 x^{14} + 210 x^{12} + 63 x^{10} - 364 x^{8} - 245 x^{6} + 155 x^{4} + 58 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7106493786222379509303476224=-\,2^{18}\cdot 7^{16}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.26$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{3}{7} a^{6} + \frac{3}{7} a^{4} - \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} + \frac{3}{7} a^{7} + \frac{3}{7} a^{5} - \frac{2}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{8} + \frac{2}{7} a^{6} - \frac{3}{7} a^{4} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{9} + \frac{2}{7} a^{7} - \frac{3}{7} a^{5} - \frac{3}{7} a^{3} + \frac{2}{7} a$, $\frac{1}{7} a^{14} + \frac{1}{7}$, $\frac{1}{7} a^{15} + \frac{1}{7} a$, $\frac{1}{5971} a^{16} + \frac{409}{5971} a^{14} - \frac{388}{5971} a^{12} + \frac{59}{853} a^{10} - \frac{1404}{5971} a^{8} - \frac{1098}{5971} a^{6} - \frac{2700}{5971} a^{4} + \frac{1893}{5971} a^{2} - \frac{1515}{5971}$, $\frac{1}{5971} a^{17} + \frac{409}{5971} a^{15} - \frac{388}{5971} a^{13} + \frac{59}{853} a^{11} - \frac{1404}{5971} a^{9} - \frac{1098}{5971} a^{7} - \frac{2700}{5971} a^{5} + \frac{1893}{5971} a^{3} - \frac{1515}{5971} a$

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:  $10$
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:  \( 5373559.17224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T586:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 80 conjugacy class representatives for t18n586 are not computed
Character table for t18n586 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.164648481361.1

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 R $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
$13$13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{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}$