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Properties

Label 18.6.87064929587...3433.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 41^{3}\cdot 97^{3}$
Root discriminant $14.57$
Ramified primes $7, 41, 97$
Class number $1$
Class group Trivial
Galois group 18T207

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 10, -4, -35, 126, -281, 491, -689, 773, -689, 491, -281, 126, -35, -4, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 10*x^16 - 4*x^15 - 35*x^14 + 126*x^13 - 281*x^12 + 491*x^11 - 689*x^10 + 773*x^9 - 689*x^8 + 491*x^7 - 281*x^6 + 126*x^5 - 35*x^4 - 4*x^3 + 10*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 10*x^16 - 4*x^15 - 35*x^14 + 126*x^13 - 281*x^12 + 491*x^11 - 689*x^10 + 773*x^9 - 689*x^8 + 491*x^7 - 281*x^6 + 126*x^5 - 35*x^4 - 4*x^3 + 10*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 10 x^{16} - 4 x^{15} - 35 x^{14} + 126 x^{13} - 281 x^{12} + 491 x^{11} - 689 x^{10} + 773 x^{9} - 689 x^{8} + 491 x^{7} - 281 x^{6} + 126 x^{5} - 35 x^{4} - 4 x^{3} + 10 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(870649295878306573433=7^{12}\cdot 41^{3}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.57$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{34691} a^{16} + \frac{13677}{34691} a^{15} + \frac{5469}{34691} a^{14} - \frac{15310}{34691} a^{13} - \frac{12666}{34691} a^{12} + \frac{769}{34691} a^{11} - \frac{12221}{34691} a^{10} + \frac{2620}{34691} a^{9} - \frac{12122}{34691} a^{8} + \frac{2620}{34691} a^{7} - \frac{12221}{34691} a^{6} + \frac{769}{34691} a^{5} - \frac{12666}{34691} a^{4} - \frac{15310}{34691} a^{3} + \frac{5469}{34691} a^{2} + \frac{13677}{34691} a + \frac{1}{34691}$, $\frac{1}{34691} a^{17} - \frac{988}{34691} a^{15} + \frac{13664}{34691} a^{14} - \frac{12672}{34691} a^{13} - \frac{13203}{34691} a^{12} + \frac{16230}{34691} a^{11} + \frac{7999}{34691} a^{10} - \frac{10059}{34691} a^{9} + \frac{6925}{34691} a^{8} - \frac{10158}{34691} a^{7} + \frac{6148}{34691} a^{6} + \frac{15785}{34691} a^{5} + \frac{5409}{34691} a^{4} + \frac{5463}{34691} a^{3} + \frac{7960}{34691} a^{2} - \frac{6456}{34691} a - \frac{13677}{34691}$

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:  $11$
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:  \( 2303.34163488 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T207:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 17 conjugacy class representatives for t18n207
Character table for t18n207

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.9548777.1, 9.5.467890073.1 x3

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
41Data not computed
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$