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Properties

Label 8.0.101592604101...7729.1
Degree $8$
Signature $[0, 4]$
Discriminant $47^{4}\cdot 113^{6}$
Root discriminant $237.61$
Ramified primes $47, 113$
Class number $112355$ (GRH)
Class group $[112355]$ (GRH)
Galois group $C_4\times C_2$ (as 8T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431872, 150736, 69276, 11360, 1767, -142, -36, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 36*x^6 - 142*x^5 + 1767*x^4 + 11360*x^3 + 69276*x^2 + 150736*x + 431872)
 
gp: K = bnfinit(x^8 - 2*x^7 - 36*x^6 - 142*x^5 + 1767*x^4 + 11360*x^3 + 69276*x^2 + 150736*x + 431872, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 36 x^{6} - 142 x^{5} + 1767 x^{4} + 11360 x^{3} + 69276 x^{2} + 150736 x + 431872 \)

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:  \(10159260410122837729=47^{4}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $237.61$
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:  $47, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(5311=47\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{5311}(1,·)$, $\chi_{5311}(3714,·)$, $\chi_{5311}(2727,·)$, $\chi_{5311}(1129,·)$, $\chi_{5311}(4182,·)$, $\chi_{5311}(2584,·)$, $\chi_{5311}(1597,·)$, $\chi_{5311}(5310,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{224} a^{6} + \frac{9}{224} a^{5} - \frac{3}{224} a^{4} - \frac{41}{224} a^{3} + \frac{1}{112} a^{2} + \frac{1}{7} a$, $\frac{1}{2480203264} a^{7} + \frac{176029}{354314752} a^{6} + \frac{14151085}{354314752} a^{5} + \frac{14515959}{354314752} a^{4} - \frac{21815369}{88578688} a^{3} - \frac{5816361}{88578688} a^{2} - \frac{4404499}{14092064} a - \frac{324569}{1384042}$

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

Class group and class number

$C_{112355}$, which has order $112355$ (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

$C_2\times C_4$ (as 8T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 8
The 8 conjugacy class representatives for $C_4\times C_2$
Character table for $C_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{113}) \), \(\Q(\sqrt{-5311}) \), \(\Q(\sqrt{-47}, \sqrt{113})\), 4.4.1442897.1, 4.0.3187359473.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$47$47.8.4.1$x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$113$113.8.6.1$x^{8} - 18193 x^{4} + 127690000$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$