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Properties

Label 8.0.120520840200...256.14
Degree $8$
Signature $[0, 4]$
Discriminant $2^{30}\cdot 2239^{3}$
Root discriminant $242.74$
Ramified primes $2, 2239$
Class number $120576$ (GRH)
Class group $[4, 8, 3768]$ (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![11224377919, 0, 180472356, 0, 935902, 0, 1724, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 1724*x^6 + 935902*x^4 + 180472356*x^2 + 11224377919)
 
gp: K = bnfinit(x^8 + 1724*x^6 + 935902*x^4 + 180472356*x^2 + 11224377919, 1)
 

Normalized defining polynomial

\( x^{8} + 1724 x^{6} + 935902 x^{4} + 180472356 x^{2} + 11224377919 \)

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:  \(12052084020012384256=2^{30}\cdot 2239^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $242.74$
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:  $2, 2239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2239} a^{4} - \frac{515}{2239} a^{2}$, $\frac{1}{2239} a^{5} - \frac{515}{2239} a^{3}$, $\frac{1}{9630205441} a^{6} + \frac{122630}{9630205441} a^{4} - \frac{1979800}{4301119} a^{2} + \frac{249}{1921}$, $\frac{1}{9630205441} a^{7} + \frac{122630}{9630205441} a^{5} - \frac{1979800}{4301119} a^{3} + \frac{249}{1921} a$

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

Class group and class number

$C_{4}\times C_{8}\times C_{3768}$, which has order $120576$ (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:  \( \frac{390}{9630205441} a^{6} + \frac{513391}{9630205441} a^{4} + \frac{58528}{4301119} a^{2} - \frac{2782}{1921} \),  \( \frac{3481}{9630205441} a^{6} + \frac{5365368}{9630205441} a^{4} + \frac{1037090}{4301119} a^{2} + \frac{50344}{1921} \),  \( \frac{890}{9630205441} a^{6} + \frac{1612725}{9630205441} a^{4} + \frac{362951}{4301119} a^{2} + \frac{16063}{1921} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 19.534360053 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\)

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$2.8.30.46$x^{8} + 8 x^{7} + 20 x^{4} + 2$$8$$1$$30$$((C_8 : C_2):C_2):C_2$$[2, 3, 7/2, 4, 17/4, 19/4]$
2239Data not computed