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Properties

Label 18.6.70356704127...9137.2
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 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -187, -778, -88, 4563, 3195, -5579, 640, 1871, -1649, -519, 1483, -1037, 289, 90, -104, 44, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 44*x^16 - 104*x^15 + 90*x^14 + 289*x^13 - 1037*x^12 + 1483*x^11 - 519*x^10 - 1649*x^9 + 1871*x^8 + 640*x^7 - 5579*x^6 + 3195*x^5 + 4563*x^4 - 88*x^3 - 778*x^2 - 187*x - 13)
 
gp: K = bnfinit(x^18 - 9*x^17 + 44*x^16 - 104*x^15 + 90*x^14 + 289*x^13 - 1037*x^12 + 1483*x^11 - 519*x^10 - 1649*x^9 + 1871*x^8 + 640*x^7 - 5579*x^6 + 3195*x^5 + 4563*x^4 - 88*x^3 - 778*x^2 - 187*x - 13, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 44 x^{16} - 104 x^{15} + 90 x^{14} + 289 x^{13} - 1037 x^{12} + 1483 x^{11} - 519 x^{10} - 1649 x^{9} + 1871 x^{8} + 640 x^{7} - 5579 x^{6} + 3195 x^{5} + 4563 x^{4} - 88 x^{3} - 778 x^{2} - 187 x - 13 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{29} a^{16} - \frac{6}{29} a^{15} + \frac{4}{29} a^{14} + \frac{11}{29} a^{13} + \frac{6}{29} a^{12} + \frac{7}{29} a^{11} + \frac{12}{29} a^{10} + \frac{2}{29} a^{9} + \frac{6}{29} a^{8} + \frac{7}{29} a^{7} - \frac{9}{29} a^{6} - \frac{5}{29} a^{5} - \frac{2}{29} a^{4} - \frac{7}{29} a^{3} + \frac{4}{29} a^{2} - \frac{9}{29} a + \frac{6}{29}$, $\frac{1}{19752304539287309204232150601} a^{17} - \frac{284700758857025951311309215}{19752304539287309204232150601} a^{16} + \frac{4743632773226296395406616878}{19752304539287309204232150601} a^{15} - \frac{5396262704289651217925598244}{19752304539287309204232150601} a^{14} - \frac{8459487835744749270144380349}{19752304539287309204232150601} a^{13} - \frac{1123579090187848065923959798}{19752304539287309204232150601} a^{12} + \frac{7552650797079426487673007884}{19752304539287309204232150601} a^{11} - \frac{9327634410824812879482535841}{19752304539287309204232150601} a^{10} + \frac{4801273717718349680713863825}{19752304539287309204232150601} a^{9} - \frac{1571294205507027642750708004}{19752304539287309204232150601} a^{8} - \frac{8287942178509067389093375462}{19752304539287309204232150601} a^{7} + \frac{8435220794744184360723557695}{19752304539287309204232150601} a^{6} - \frac{7957167831380673112168439739}{19752304539287309204232150601} a^{5} + \frac{9503153830875606551721632261}{19752304539287309204232150601} a^{4} + \frac{4902845439362157436795567794}{19752304539287309204232150601} a^{3} - \frac{8094782032884586933470669652}{19752304539287309204232150601} a^{2} + \frac{5476744739271260035099774}{11463902808640341964150987} a + \frac{65576002468691457013789138}{1519408041483639169556319277}$

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:  \( 2508855.37219 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

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 t18n767 are not computed
Character table for t18n767 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 ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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