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Properties

Label 8.0.285073698971...7937.1
Degree $8$
Signature $[0, 4]$
Discriminant $59^{4}\cdot 113^{7}$
Root discriminant $480.70$
Ramified primes $59, 113$
Class number $1316786$ (GRH)
Class group $[1316786]$ (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![480344032, -139767108, 64010176, 1559767, 675121, 3374, 1646, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 1646*x^6 + 3374*x^5 + 675121*x^4 + 1559767*x^3 + 64010176*x^2 - 139767108*x + 480344032)
 
gp: K = bnfinit(x^8 - x^7 + 1646*x^6 + 3374*x^5 + 675121*x^4 + 1559767*x^3 + 64010176*x^2 - 139767108*x + 480344032, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 1646 x^{6} + 3374 x^{5} + 675121 x^{4} + 1559767 x^{3} + 64010176 x^{2} - 139767108 x + 480344032 \)

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:  \(2850736989716891767937=59^{4}\cdot 113^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $480.70$
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:  $59, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(6667=59\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{6667}(1,·)$, $\chi_{6667}(1061,·)$, $\chi_{6667}(5665,·)$, $\chi_{6667}(3598,·)$, $\chi_{6667}(3954,·)$, $\chi_{6667}(1651,·)$, $\chi_{6667}(5781,·)$, $\chi_{6667}(4957,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{112} a^{5} + \frac{1}{112} a^{4} + \frac{3}{16} a^{3} - \frac{1}{112} a^{2} - \frac{11}{56} a$, $\frac{1}{224} a^{6} - \frac{1}{224} a^{5} - \frac{9}{224} a^{4} - \frac{43}{224} a^{3} - \frac{3}{14} a^{2} + \frac{25}{56} a$, $\frac{1}{1772261402212889448448} a^{7} - \frac{141235049686131291}{110766337638305590528} a^{6} + \frac{1454964657308008927}{886130701106444724224} a^{5} + \frac{11540049197681985427}{443065350553222362112} a^{4} - \frac{399113061990141490883}{1772261402212889448448} a^{3} - \frac{63081356503892986551}{443065350553222362112} a^{2} - \frac{169982683099615441399}{443065350553222362112} a + \frac{1203644659667433671}{7911881259878970752}$

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

Class group and class number

$C_{1316786}$, which has order $1316786$ (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:  \( 5177.33507354 \) (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

\(\Q(\sqrt{113}) \), 4.4.1442897.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ R

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
$59$59.8.4.2$x^{8} - 205379 x^{2} + 169643054$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$113$113.8.7.3$x^{8} - 9153$$8$$1$$7$$C_8$$[\ ]_{8}$