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Properties

Label 8.0.531817867910...9217.3
Degree $8$
Signature $[0, 4]$
Discriminant $313^{7}\cdot 349^{6}$
Root discriminant $12{,}323.11$
Ramified primes $313, 349$
Class number $1492434176$ (GRH)
Class group $[2, 2, 2, 2, 2, 46638568]$ (GRH)
Galois group $C_4\wr C_2$ (as 8T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1051286976413904, 61677611886608, 1457471142007, 15081515535, 100860172, 1695203, 20052, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 20052*x^6 + 1695203*x^5 + 100860172*x^4 + 15081515535*x^3 + 1457471142007*x^2 + 61677611886608*x + 1051286976413904)
 
gp: K = bnfinit(x^8 - x^7 + 20052*x^6 + 1695203*x^5 + 100860172*x^4 + 15081515535*x^3 + 1457471142007*x^2 + 61677611886608*x + 1051286976413904, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 20052 x^{6} + 1695203 x^{5} + 100860172 x^{4} + 15081515535 x^{3} + 1457471142007 x^{2} + 61677611886608 x + 1051286976413904 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(531817867910855374307466941229217=313^{7}\cdot 349^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}323.11$
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:  $313, 349$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{18} a^{4} + \frac{1}{18} a^{3} + \frac{5}{18} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{648} a^{5} + \frac{1}{81} a^{4} - \frac{17}{216} a^{3} - \frac{41}{162} a^{2} - \frac{139}{648} a - \frac{23}{54}$, $\frac{1}{27216} a^{6} - \frac{1}{2268} a^{5} - \frac{643}{27216} a^{4} + \frac{215}{3402} a^{3} - \frac{179}{3024} a^{2} - \frac{100}{1701} a + \frac{277}{567}$, $\frac{1}{2081137151426474248728657715728} a^{7} - \frac{9377115430374475202943803}{2081137151426474248728657715728} a^{6} - \frac{13988469626862989573930399}{71763350049188767197539921232} a^{5} + \frac{14647556412807260434620345167}{693712383808824749576219238576} a^{4} + \frac{71022281646064506170401401337}{2081137151426474248728657715728} a^{3} - \frac{67579311629068745265936149515}{2081137151426474248728657715728} a^{2} + \frac{7328508857234275743045357071}{74326326836659794597452061276} a - \frac{17225292077168599556902760708}{43357023988051546848513702411}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{46638568}$, which has order $1492434176$ (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:  $3$
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:  \( 565409.549494 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 8T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{109237}) \), 4.4.3734942038897.1

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

Sibling fields

Galois closure: data not computed
Degree 8 sibling: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
313Data not computed
349Data not computed