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Properties

Label 8.0.704031205074...6128.7
Degree $8$
Signature $[0, 4]$
Discriminant $2^{31}\cdot 1789^{6}$
Root discriminant $4035.98$
Ramified primes $2, 1789$
Class number $330696100$ (GRH)
Class group $[7, 47242300]$ (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![29837055454802, 0, 91611713104, 0, 64010420, 0, 14312, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 14312*x^6 + 64010420*x^4 + 91611713104*x^2 + 29837055454802)
 
gp: K = bnfinit(x^8 + 14312*x^6 + 64010420*x^4 + 91611713104*x^2 + 29837055454802, 1)
 

Normalized defining polynomial

\( x^{8} + 14312 x^{6} + 64010420 x^{4} + 91611713104 x^{2} + 29837055454802 \)

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:  \(70403120507437021940708016128=2^{31}\cdot 1789^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4035.98$
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:  $2, 1789$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(57248=2^{5}\cdot 1789\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2359691} a^{4} + \frac{4}{1319} a^{2} - \frac{379}{1319}$, $\frac{1}{5094572869} a^{5} - \frac{820414}{2847721} a^{3} - \frac{603162}{2847721} a$, $\frac{1}{5094572869} a^{6} - \frac{61}{5094572869} a^{4} + \frac{1126197}{2847721} a^{2} - \frac{382}{1319}$, $\frac{1}{5094572869} a^{7} - \frac{507800}{2847721} a^{3} - \frac{597247}{2847721} a$

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

Class group and class number

$C_{7}\times C_{47242300}$, which has order $330696100$ (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:  \( \frac{1}{2359691} a^{4} + \frac{4}{1319} a^{2} + \frac{2259}{1319} \),  \( \frac{79}{5094572869} a^{6} + \frac{714128}{5094572869} a^{4} + \frac{718279}{2847721} a^{2} + \frac{115329}{1319} \),  \( \frac{52}{5094572869} a^{6} + \frac{692026}{5094572869} a^{4} + \frac{1540895}{2847721} a^{2} + \frac{787993}{1319} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 687.056920431 \) (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

\(\Q(\sqrt{2}) \), 4.4.6554667008.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$2.8.31.4$x^{8} + 24 x^{4} + 8 x^{2} + 16 x + 46$$8$$1$$31$$C_8$$[3, 4, 5]$
1789Data not computed