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Properties

Label 18.10.2371799206...2529.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{14}\cdot 769^{4}$
Root discriminant $19.89$
Ramified primes $7, 769$
Class number $1$
Class group Trivial
Galois group 18T473

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -11, 9, 47, 14, -85, -98, 48, 147, 48, -98, -85, 14, 47, 9, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 11*x^16 + 9*x^15 + 47*x^14 + 14*x^13 - 85*x^12 - 98*x^11 + 48*x^10 + 147*x^9 + 48*x^8 - 98*x^7 - 85*x^6 + 14*x^5 + 47*x^4 + 9*x^3 - 11*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 11*x^16 + 9*x^15 + 47*x^14 + 14*x^13 - 85*x^12 - 98*x^11 + 48*x^10 + 147*x^9 + 48*x^8 - 98*x^7 - 85*x^6 + 14*x^5 + 47*x^4 + 9*x^3 - 11*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 11 x^{16} + 9 x^{15} + 47 x^{14} + 14 x^{13} - 85 x^{12} - 98 x^{11} + 48 x^{10} + 147 x^{9} + 48 x^{8} - 98 x^{7} - 85 x^{6} + 14 x^{5} + 47 x^{4} + 9 x^{3} - 11 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(237179920636111459752529=7^{14}\cdot 769^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.89$
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:  $7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{43063} a^{16} + \frac{2555}{43063} a^{15} - \frac{12453}{43063} a^{14} - \frac{21310}{43063} a^{13} - \frac{2475}{43063} a^{12} - \frac{20053}{43063} a^{11} + \frac{14902}{43063} a^{10} + \frac{13614}{43063} a^{9} + \frac{1240}{43063} a^{8} + \frac{13614}{43063} a^{7} + \frac{14902}{43063} a^{6} - \frac{20053}{43063} a^{5} - \frac{2475}{43063} a^{4} - \frac{21310}{43063} a^{3} - \frac{12453}{43063} a^{2} + \frac{2555}{43063} a + \frac{1}{43063}$, $\frac{1}{43063} a^{17} + \frac{5098}{43063} a^{15} + \frac{15611}{43063} a^{14} + \frac{12943}{43063} a^{13} + \frac{16374}{43063} a^{12} + \frac{5347}{43063} a^{11} + \frac{6696}{43063} a^{10} + \frac{12374}{43063} a^{9} - \frac{10987}{43063} a^{8} - \frac{17027}{43063} a^{7} + \frac{16092}{43063} a^{6} - \frac{12030}{43063} a^{5} + \frac{15117}{43063} a^{4} + \frac{2965}{43063} a^{3} - \frac{3587}{43063} a^{2} + \frac{17552}{43063} a - \frac{2555}{43063}$

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:  $13$
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:  \( 117123.22288 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T473:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 32 conjugacy class representatives for t18n473
Character table for t18n473 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.69573030289.1

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

Sibling fields

Degree 12 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
769Data not computed