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Properties

Label 18.6.70197282664...5977.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{15}\cdot 52879^{3}$
Root discriminant $31.01$
Ramified primes $7, 52879$
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![239, 500, -5151, 4184, -1616, 1599, 803, 637, -935, 293, -328, 89, 30, -29, 42, -24, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 24*x^15 + 42*x^14 - 29*x^13 + 30*x^12 + 89*x^11 - 328*x^10 + 293*x^9 - 935*x^8 + 637*x^7 + 803*x^6 + 1599*x^5 - 1616*x^4 + 4184*x^3 - 5151*x^2 + 500*x + 239)
 
gp: K = bnfinit(x^18 - 4*x^17 + 7*x^16 - 24*x^15 + 42*x^14 - 29*x^13 + 30*x^12 + 89*x^11 - 328*x^10 + 293*x^9 - 935*x^8 + 637*x^7 + 803*x^6 + 1599*x^5 - 1616*x^4 + 4184*x^3 - 5151*x^2 + 500*x + 239, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 7 x^{16} - 24 x^{15} + 42 x^{14} - 29 x^{13} + 30 x^{12} + 89 x^{11} - 328 x^{10} + 293 x^{9} - 935 x^{8} + 637 x^{7} + 803 x^{6} + 1599 x^{5} - 1616 x^{4} + 4184 x^{3} - 5151 x^{2} + 500 x + 239 \)

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:  \(701972826641672810901485977=7^{15}\cdot 52879^{3}\)
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:  $7, 52879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3561208941745957512612909868876181} a^{17} - \frac{1005598459302292267566200988923538}{3561208941745957512612909868876181} a^{16} + \frac{515671330887759019841913522632216}{3561208941745957512612909868876181} a^{15} + \frac{376259583582281524126154499745984}{3561208941745957512612909868876181} a^{14} + \frac{1084543637762363463673920926003410}{3561208941745957512612909868876181} a^{13} - \frac{1541136909165201902968660546353039}{3561208941745957512612909868876181} a^{12} + \frac{126059738546039658758332096584697}{3561208941745957512612909868876181} a^{11} - \frac{362752129527864733372301365505558}{3561208941745957512612909868876181} a^{10} + \frac{1404631444734041836457998880915051}{3561208941745957512612909868876181} a^{9} - \frac{268689387681192344829095640430453}{3561208941745957512612909868876181} a^{8} + \frac{1143670782605356641760398219155488}{3561208941745957512612909868876181} a^{7} - \frac{629510795457778576215336566808693}{3561208941745957512612909868876181} a^{6} - \frac{77733316108321158148664803944367}{3561208941745957512612909868876181} a^{5} - \frac{1600333317507451132333899995559256}{3561208941745957512612909868876181} a^{4} - \frac{382012693139085526799524336581712}{3561208941745957512612909868876181} a^{3} + \frac{305694256249733600673215723736365}{3561208941745957512612909868876181} a^{2} + \frac{616234148105891581759902261121643}{3561208941745957512612909868876181} a - \frac{1047398132142097797054171938485282}{3561208941745957512612909868876181}$

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:  \( 2862643.89535 \) (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

\(\Q(\zeta_{7})^+\), 9.7.6221161471.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ $18$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
52879Data not computed