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Properties

Label 8.0.390528406835...5049.1
Degree $8$
Signature $[0, 4]$
Discriminant $101^{6}\cdot 449^{7}$
Root discriminant $6667.40$
Ramified primes $101, 449$
Class number $1961893840$ (GRH)
Class group $[68, 28851380]$ (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![64650867436256, 1711530984976, -40387406944, -2049624728, -7546954, 671753, 5641, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 5641*x^6 + 671753*x^5 - 7546954*x^4 - 2049624728*x^3 - 40387406944*x^2 + 1711530984976*x + 64650867436256)
 
gp: K = bnfinit(x^8 - x^7 + 5641*x^6 + 671753*x^5 - 7546954*x^4 - 2049624728*x^3 - 40387406944*x^2 + 1711530984976*x + 64650867436256, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 5641 x^{6} + 671753 x^{5} - 7546954 x^{4} - 2049624728 x^{3} - 40387406944 x^{2} + 1711530984976 x + 64650867436256 \)

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:  \(3905284068350630918099056615049=101^{6}\cdot 449^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $6667.40$
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:  $101, 449$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(45349=101\cdot 449\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{3} + \frac{1}{10} a^{2} - \frac{1}{20} a - \frac{1}{10}$, $\frac{1}{40} a^{4} - \frac{1}{8} a^{2} + \frac{1}{10}$, $\frac{1}{160} a^{5} + \frac{1}{160} a^{4} + \frac{3}{160} a^{3} + \frac{11}{160} a^{2} - \frac{1}{40} a - \frac{3}{40}$, $\frac{1}{294400} a^{6} + \frac{679}{294400} a^{5} + \frac{1717}{294400} a^{4} + \frac{5277}{294400} a^{3} - \frac{33231}{147200} a^{2} - \frac{1143}{3200} a + \frac{293}{36800}$, $\frac{1}{80395842652443545307904000} a^{7} + \frac{52422206305000411313}{80395842652443545307904000} a^{6} - \frac{227320399032872132948677}{80395842652443545307904000} a^{5} - \frac{24283124099258693895097}{3215833706097741812316160} a^{4} - \frac{338009473801673674139551}{20098960663110886326976000} a^{3} - \frac{1218664639882545869053699}{5024740165777721581744000} a^{2} - \frac{21417072209663826801553}{125618504144443039543600} a - \frac{2338137441310819303528119}{5024740165777721581744000}$

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

Class group and class number

$C_{68}\times C_{28851380}$, which has order $1961893840$ (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:  \( 391900.036397 \) (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

\(\Q(\sqrt{449}) \), 4.4.923382778649.2

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$101$101.4.3.4$x^{4} + 808$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.4$x^{4} + 808$$4$$1$$3$$C_4$$[\ ]_{4}$
449Data not computed