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Properties

Label 8.0.827620013609...5057.1
Degree $8$
Signature $[0, 4]$
Discriminant $59^{4}\cdot 353^{7}$
Root discriminant $1302.35$
Ramified primes $59, 353$
Class number $10114178$ (GRH)
Class group $[10114178]$ (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![32108989264, 11895080217, 2795041399, -31484723, 7530019, -20965, 5141, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 5141*x^6 - 20965*x^5 + 7530019*x^4 - 31484723*x^3 + 2795041399*x^2 + 11895080217*x + 32108989264)
 
gp: K = bnfinit(x^8 - x^7 + 5141*x^6 - 20965*x^5 + 7530019*x^4 - 31484723*x^3 + 2795041399*x^2 + 11895080217*x + 32108989264, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 5141 x^{6} - 20965 x^{5} + 7530019 x^{4} - 31484723 x^{3} + 2795041399 x^{2} + 11895080217 x + 32108989264 \)

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:  \(8276200136091069747575057=59^{4}\cdot 353^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1302.35$
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:  $59, 353$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(20827=59\cdot 353\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{16} a^{4} - \frac{1}{16} a^{3} - \frac{1}{16} a^{2} + \frac{1}{16} a$, $\frac{1}{256} a^{5} - \frac{1}{32} a^{4} - \frac{1}{128} a^{3} - \frac{1}{32} a^{2} + \frac{65}{256} a - \frac{3}{16}$, $\frac{1}{512} a^{6} - \frac{1}{512} a^{5} + \frac{3}{256} a^{4} - \frac{11}{256} a^{3} - \frac{55}{512} a^{2} + \frac{151}{512} a - \frac{5}{32}$, $\frac{1}{2858603858661480240447488} a^{7} - \frac{75310370163547040677}{178662741166342515027968} a^{6} + \frac{27214376744835129529}{26225723473958534316032} a^{5} - \frac{1870535672866357473259}{178662741166342515027968} a^{4} - \frac{45358971664598429710733}{2858603858661480240447488} a^{3} + \frac{16677339208378494418453}{178662741166342515027968} a^{2} + \frac{974290582970184143103687}{2858603858661480240447488} a - \frac{9681859411977259383}{222494073681622061056}$

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

Class group and class number

$C_{10114178}$, which has order $10114178$ (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:  \( 31180.0916771 \) (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

\(\Q(\sqrt{353}) \), 4.4.43986977.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ 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}$
353Data not computed