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Properties

Label 8.0.198298033948...8049.3
Degree $8$
Signature $[0, 4]$
Discriminant $193^{5}\cdot 257^{7}$
Root discriminant $3444.81$
Ramified primes $193, 257$
Class number $449751552$ (GRH)
Class group $[2, 2, 8, 14054736]$ (GRH)
Galois group $((C_8 : C_2):C_2):C_2$ (as 8T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2950956496, -94473184, 95190632, -1436696, 722849, -5404, 1806, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 1806*x^6 - 5404*x^5 + 722849*x^4 - 1436696*x^3 + 95190632*x^2 - 94473184*x + 2950956496)
 
gp: K = bnfinit(x^8 - 4*x^7 + 1806*x^6 - 5404*x^5 + 722849*x^4 - 1436696*x^3 + 95190632*x^2 - 94473184*x + 2950956496, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 1806 x^{6} - 5404 x^{5} + 722849 x^{4} - 1436696 x^{3} + 95190632 x^{2} - 94473184 x + 2950956496 \)

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:  \(19829803394853790928098618049=193^{5}\cdot 257^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3444.81$
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:  $193, 257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $\frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{4} - \frac{1}{16} a^{3} + \frac{1}{32} a^{2} - \frac{1}{8}$, $\frac{1}{5408} a^{5} + \frac{41}{2704} a^{4} + \frac{121}{5408} a^{3} - \frac{87}{1352} a^{2} + \frac{323}{1352} a - \frac{57}{338}$, $\frac{1}{18257408} a^{6} - \frac{3}{18257408} a^{5} + \frac{246313}{18257408} a^{4} - \frac{492621}{18257408} a^{3} - \frac{201391}{9128704} a^{2} + \frac{162273}{4564352} a + \frac{668621}{2282176}$, $\frac{1}{419920384} a^{7} + \frac{1}{52490048} a^{6} - \frac{21}{52490048} a^{5} - \frac{1864157}{209960192} a^{4} - \frac{45911613}{419920384} a^{3} - \frac{1389239}{16150784} a^{2} + \frac{47898133}{104980096} a + \frac{4355255}{52490048}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{14054736}$, which has order $449751552$ (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:  \( 37030.5506526 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 8T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$
Character table for $((C_8 : C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{257}) \), 4.4.632286614657.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$193$193.2.1.1$x^{2} - 193$$2$$1$$1$$C_2$$[\ ]_{2}$
193.2.1.1$x^{2} - 193$$2$$1$$1$$C_2$$[\ ]_{2}$
193.4.3.1$x^{4} - 193$$4$$1$$3$$C_4$$[\ ]_{4}$
257Data not computed