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Properties

Label 18.6.70356704127...9137.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{15}\cdot 52919^{3}$
Root discriminant $31.01$
Ramified primes $7, 52919$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, 677, -3086, 5188, -6579, 6215, -4953, 4658, -5220, 5410, -4059, 2179, -718, 13, 130, -91, 33, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 33*x^16 - 91*x^15 + 130*x^14 + 13*x^13 - 718*x^12 + 2179*x^11 - 4059*x^10 + 5410*x^9 - 5220*x^8 + 4658*x^7 - 4953*x^6 + 6215*x^5 - 6579*x^4 + 5188*x^3 - 3086*x^2 + 677*x - 41)
 
gp: K = bnfinit(x^18 - 8*x^17 + 33*x^16 - 91*x^15 + 130*x^14 + 13*x^13 - 718*x^12 + 2179*x^11 - 4059*x^10 + 5410*x^9 - 5220*x^8 + 4658*x^7 - 4953*x^6 + 6215*x^5 - 6579*x^4 + 5188*x^3 - 3086*x^2 + 677*x - 41, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 33 x^{16} - 91 x^{15} + 130 x^{14} + 13 x^{13} - 718 x^{12} + 2179 x^{11} - 4059 x^{10} + 5410 x^{9} - 5220 x^{8} + 4658 x^{7} - 4953 x^{6} + 6215 x^{5} - 6579 x^{4} + 5188 x^{3} - 3086 x^{2} + 677 x - 41 \)

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:  \(703567041275767323081039137=7^{15}\cdot 52919^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.01$
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:  $7, 52919$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{533154332815901783895993033823} a^{17} - \frac{190089120389660961622738623776}{533154332815901783895993033823} a^{16} + \frac{19983412257978003462450557463}{533154332815901783895993033823} a^{15} + \frac{250668955715353197073046318780}{533154332815901783895993033823} a^{14} + \frac{241472679068029614444636548829}{533154332815901783895993033823} a^{13} + \frac{200976542126602995032150345592}{533154332815901783895993033823} a^{12} + \frac{218202732228756064933437690677}{533154332815901783895993033823} a^{11} - \frac{116836637171299398012917575752}{533154332815901783895993033823} a^{10} + \frac{177828124249443345884655253617}{533154332815901783895993033823} a^{9} - \frac{13001854874470592701822616124}{533154332815901783895993033823} a^{8} - \frac{16958089671816344733201186599}{533154332815901783895993033823} a^{7} + \frac{19768905286971198797945495550}{533154332815901783895993033823} a^{6} + \frac{230056149603093489302598938272}{533154332815901783895993033823} a^{5} - \frac{201895595914939356041275089709}{533154332815901783895993033823} a^{4} + \frac{254972167345562386010331342344}{533154332815901783895993033823} a^{3} + \frac{200741006438858776301944110304}{533154332815901783895993033823} a^{2} + \frac{62553401707804680458684722564}{533154332815901783895993033823} a + \frac{66596322107579255605855376822}{533154332815901783895993033823}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2378289.37707 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.6225867431.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
52919Data not computed