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Properties

Label 8.0.182443425917...9401.1
Degree $8$
Signature $[0, 4]$
Discriminant $149^{6}\cdot 401^{7}$
Root discriminant $8084.27$
Ramified primes $149, 401$
Class number $2981347280$ (GRH)
Class group $[2, 2, 745336820]$ (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![532351922168960, 20637311571272, 234821135222, 2217754863, 51900405, 780490, 7444, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 7444*x^6 + 780490*x^5 + 51900405*x^4 + 2217754863*x^3 + 234821135222*x^2 + 20637311571272*x + 532351922168960)
 
gp: K = bnfinit(x^8 - x^7 + 7444*x^6 + 780490*x^5 + 51900405*x^4 + 2217754863*x^3 + 234821135222*x^2 + 20637311571272*x + 532351922168960, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 7444 x^{6} + 780490 x^{5} + 51900405 x^{4} + 2217754863 x^{3} + 234821135222 x^{2} + 20637311571272 x + 532351922168960 \)

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:  \(18244342591747163638870188429401=149^{6}\cdot 401^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8084.27$
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:  $149, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(59749=149\cdot 401\)
Dirichlet character group:    not computed
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{3} + \frac{1}{10} a$, $\frac{1}{40} a^{4} - \frac{1}{20} a^{3} - \frac{9}{40} a^{2} + \frac{9}{20} a$, $\frac{1}{200} a^{5} + \frac{7}{200} a^{3} - \frac{1}{10} a^{2} - \frac{11}{50} a + \frac{2}{5}$, $\frac{1}{308000} a^{6} - \frac{71}{44000} a^{5} + \frac{111}{44000} a^{4} - \frac{1039}{308000} a^{3} - \frac{14717}{154000} a^{2} + \frac{6663}{19250} a - \frac{361}{1925}$, $\frac{1}{35366593801137687630665200000} a^{7} + \frac{6887907541835429301403}{17683296900568843815332600000} a^{6} - \frac{4859755848183960276887651}{2526185271509834830761800000} a^{5} + \frac{10698157890866271542954623}{8841648450284421907666300000} a^{4} - \frac{911920721412258904235379551}{35366593801137687630665200000} a^{3} + \frac{589252676095668587829501729}{2526185271509834830761800000} a^{2} - \frac{66906699897020861077280081}{631546317877458707690450000} a + \frac{10747000933090813679828048}{55260302814277636922914375}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{745336820}$, which has order $2981347280$ (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:  \( 1216671.1794 \) (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

\(\Q(\sqrt{401}) \), 4.4.1431547143401.1

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$149$149.4.3.1$x^{4} - 149$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.1$x^{4} - 149$$4$$1$$3$$C_4$$[\ ]_{4}$
401Data not computed