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Properties

Label 8.0.704348417321...7081.2
Degree $8$
Signature $[0, 4]$
Discriminant $409^{7}\cdot 449^{7}$
Root discriminant $40{,}362.07$
Ramified primes $409, 449$
Class number $17473336448$ (GRH)
Class group $[2, 2, 4, 1092083528]$ (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![54366150332000000, -914182000547500, 11680953965900, -13801397365, -588887459, 6516386, 11478, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 11478*x^6 + 6516386*x^5 - 588887459*x^4 - 13801397365*x^3 + 11680953965900*x^2 - 914182000547500*x + 54366150332000000)
 
gp: K = bnfinit(x^8 - x^7 + 11478*x^6 + 6516386*x^5 - 588887459*x^4 - 13801397365*x^3 + 11680953965900*x^2 - 914182000547500*x + 54366150332000000, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 11478 x^{6} + 6516386 x^{5} - 588887459 x^{4} - 13801397365 x^{3} + 11680953965900 x^{2} - 914182000547500 x + 54366150332000000 \)

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:  \(7043484173213574853891721381764967081=409^{7}\cdot 449^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40{,}362.07$
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:  $409, 449$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(183641=409\cdot 449\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{4} + \frac{1}{20} a^{3} - \frac{1}{40} a^{2} - \frac{1}{20} a$, $\frac{1}{200} a^{5} + \frac{1}{100} a^{4} - \frac{1}{200} a^{3} + \frac{19}{100} a^{2} + \frac{1}{5} a$, $\frac{1}{184000} a^{6} + \frac{3}{8000} a^{5} + \frac{21}{8000} a^{4} - \frac{20329}{184000} a^{3} - \frac{8041}{46000} a^{2} - \frac{3691}{9200} a + \frac{2}{23}$, $\frac{1}{47984853809792171770610985042560000} a^{7} + \frac{44834248636551496131541530647}{23992426904896085885305492521280000} a^{6} + \frac{620946710976445633883912228687}{260787248966261803101146657840000} a^{5} + \frac{277436574811380281995225799614573}{23992426904896085885305492521280000} a^{4} + \frac{10030629182391481708829866687389411}{47984853809792171770610985042560000} a^{3} + \frac{126441323797891499799816178136587}{1199621345244804294265274626064000} a^{2} + \frac{22541507338759272025602779320233}{95969707619584343541221970085120} a - \frac{7738268913356768370004776545}{23069641254707774889716819732}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{1092083528}$, which has order $17473336448$ (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:  \( 81911039.9521 \) (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

\(\Q(\sqrt{183641}) \), 4.4.6193112184043721.2

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
409Data not computed
449Data not computed