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Properties

Label 8.0.140437008811...6473.1
Degree $8$
Signature $[0, 4]$
Discriminant $241^{4}\cdot 457^{7}$
Root discriminant $3299.40$
Ramified primes $241, 457$
Class number $110024000$ (GRH)
Class group $[2, 10, 20, 275060]$ (GRH)
Galois group $C_8$ (as 8T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8888586604571, 837113334012, 170051152183, 1132481969, 129543474, 302641, 27449, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 27449*x^6 + 302641*x^5 + 129543474*x^4 + 1132481969*x^3 + 170051152183*x^2 + 837113334012*x + 8888586604571)
 
gp: K = bnfinit(x^8 - x^7 + 27449*x^6 + 302641*x^5 + 129543474*x^4 + 1132481969*x^3 + 170051152183*x^2 + 837113334012*x + 8888586604571, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 27449 x^{6} + 302641 x^{5} + 129543474 x^{4} + 1132481969 x^{3} + 170051152183 x^{2} + 837113334012 x + 8888586604571 \)

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:  \(14043700881105708763602116473=241^{4}\cdot 457^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3299.40$
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:  $241, 457$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(110137=241\cdot 457\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{798} a^{6} - \frac{2}{399} a^{5} - \frac{40}{399} a^{4} - \frac{319}{798} a^{3} + \frac{305}{798} a^{2} - \frac{83}{266} a - \frac{19}{42}$, $\frac{1}{62527911577461802256743751828158914} a^{7} - \frac{10793824100707719667056022289327}{62527911577461802256743751828158914} a^{6} + \frac{835327651952808900137151668158933}{31263955788730901128371875914079457} a^{5} - \frac{20019331542995202147038587457773637}{62527911577461802256743751828158914} a^{4} + \frac{4854386594989933425401441934821007}{10421318596243633709457291971359819} a^{3} + \frac{791237849063856859454769464841679}{1839056222866523595786580936122321} a^{2} + \frac{126812727602116410384251470330369}{4466279398390128732624553702011351} a + \frac{13549104517278920775320459132603}{193584865564897220609113782749718}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{20}\times C_{275060}$, which has order $110024000$ (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:  \( 1906.27309223 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8$ (as 8T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 8
The 8 conjugacy class representatives for $C_8$
Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{457}) \), 4.4.95443993.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
241Data not computed
457Data not computed