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Properties

Label 8.0.919615526312...1561.3
Degree $8$
Signature $[0, 4]$
Discriminant $41^{7}\cdot 241^{7}$
Root discriminant $3129.33$
Ramified primes $41, 241$
Class number $314992960$ (GRH)
Class group $[2, 4, 39374120]$ (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![1248004155440, -29799043708, 1858422200, 47319875, -778259, 83834, 618, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 618*x^6 + 83834*x^5 - 778259*x^4 + 47319875*x^3 + 1858422200*x^2 - 29799043708*x + 1248004155440)
 
gp: K = bnfinit(x^8 - x^7 + 618*x^6 + 83834*x^5 - 778259*x^4 + 47319875*x^3 + 1858422200*x^2 - 29799043708*x + 1248004155440, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 618 x^{6} + 83834 x^{5} - 778259 x^{4} + 47319875 x^{3} + 1858422200 x^{2} - 29799043708 x + 1248004155440 \)

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:  \(9196155263122253470427891561=41^{7}\cdot 241^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3129.33$
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:  $41, 241$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(9881=41\cdot 241\)
Dirichlet character group:    $\lbrace$$\chi_{9881}(9632,·)$, $\chi_{9881}(1,·)$, $\chi_{9881}(9880,·)$, $\chi_{9881}(249,·)$, $\chi_{9881}(5754,·)$, $\chi_{9881}(2715,·)$, $\chi_{9881}(7166,·)$, $\chi_{9881}(4127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{280} a^{4} + \frac{11}{140} a^{3} + \frac{59}{280} a^{2} + \frac{19}{140} a$, $\frac{1}{280} a^{5} - \frac{1}{56} a^{3} + \frac{1}{70} a$, $\frac{1}{11200} a^{6} + \frac{17}{11200} a^{5} - \frac{17}{11200} a^{4} + \frac{1611}{11200} a^{3} + \frac{61}{700} a^{2} + \frac{1093}{2800} a + \frac{9}{20}$, $\frac{1}{111916903945724864000} a^{7} - \frac{2461694908654371}{55958451972862432000} a^{6} - \frac{3702406326471339}{2797922598643121600} a^{5} + \frac{4537747460237071}{7994064567551776000} a^{4} + \frac{8392248775548304787}{111916903945724864000} a^{3} + \frac{2922610733128757977}{27979225986431216000} a^{2} - \frac{1787281151801407401}{3997032283775888000} a - \frac{8214582133836373}{28550230598399200}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{39374120}$, which has order $314992960$ (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:  \( 447136.275584 \) (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

\(\Q(\sqrt{9881}) \), 4.4.964723144841.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ R ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$41$41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$
241Data not computed