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Properties

Label 8.0.403855819100...416.70
Degree $8$
Signature $[0, 4]$
Discriminant $2^{31}\cdot 2113^{7}$
Root discriminant $11{,}906.34$
Ramified primes $2, 2113$
Class number $2711787776$ (GRH)
Class group $[4, 172, 3941552]$ (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![3902799746, 0, 17966686864, 0, 33715028, 0, 16904, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 16904*x^6 + 33715028*x^4 + 17966686864*x^2 + 3902799746)
 
gp: K = bnfinit(x^8 + 16904*x^6 + 33715028*x^4 + 17966686864*x^2 + 3902799746, 1)
 

Normalized defining polynomial

\( x^{8} + 16904 x^{6} + 33715028 x^{4} + 17966686864 x^{2} + 3902799746 \)

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:  \(403855819100833706490759181500416=2^{31}\cdot 2113^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}906.34$
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:  $2, 2113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(67616=2^{5}\cdot 2113\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31} a^{5} + \frac{9}{31} a^{3} - \frac{14}{31} a$, $\frac{1}{12823237302913} a^{6} - \frac{262563884156}{12823237302913} a^{4} - \frac{5280155313667}{12823237302913} a^{2} + \frac{1962300163}{13343639233}$, $\frac{1}{397520356390303} a^{7} - \frac{262563884156}{397520356390303} a^{5} + \frac{33189556595072}{397520356390303} a^{3} - \frac{184848649099}{413652816223} a$

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

Class group and class number

$C_{4}\times C_{172}\times C_{3941552}$, which has order $2711787776$ (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:  \( \frac{12960625}{12823237302913} a^{6} + \frac{204645785701}{12823237302913} a^{4} + \frac{208951464470484}{12823237302913} a^{2} + \frac{33793882399}{13343639233} \),  \( \frac{286145}{12823237302913} a^{6} + \frac{4725948647}{12823237302913} a^{4} + \frac{7892986480510}{12823237302913} a^{2} + \frac{3471387417575}{13343639233} \),  \( \frac{96}{13343639233} a^{6} + \frac{1632161}{13343639233} a^{4} + \frac{3257071172}{13343639233} a^{2} + \frac{810363617}{13343639233} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2408.25649601 \) (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

\(\Q(\sqrt{4226}) \), 4.4.19320948525056.2

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$2.8.31.8$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 2$$8$$1$$31$$C_8$$[3, 4, 5]$
2113Data not computed