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Properties

Label 8.0.425754743838...832.33
Degree $8$
Signature $[0, 4]$
Discriminant $2^{31}\cdot 2129^{7}$
Root discriminant $11{,}985.19$
Ramified primes $2, 2129$
Class number $2282523776$ (GRH)
Class group $[2, 4, 285315472]$ (GRH)
Galois group $C_8:C_2$ (as 8T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5577695774242, 0, 154399883024, 0, 90652820, 0, 17032, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 17032*x^6 + 90652820*x^4 + 154399883024*x^2 + 5577695774242)
 
gp: K = bnfinit(x^8 + 17032*x^6 + 90652820*x^4 + 154399883024*x^2 + 5577695774242, 1)
 

Normalized defining polynomial

\( x^{8} + 17032 x^{6} + 90652820 x^{4} + 154399883024 x^{2} + 5577695774242 \)

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:  \(425754743838724234995143469432832=2^{31}\cdot 2129^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11{,}985.19$
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:  $2, 2129$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2129} a^{3}$, $\frac{1}{134127} a^{4} + \frac{4}{63} a^{2} - \frac{26}{63}$, $\frac{1}{2280159} a^{5} - \frac{367}{2280159} a^{3} + \frac{100}{1071} a$, $\frac{1}{4854458511} a^{6} + \frac{2}{760053} a^{4} - \frac{179}{1071} a^{2} - \frac{29}{63}$, $\frac{1}{4854458511} a^{7} + \frac{35}{325737} a^{3} - \frac{22}{1071} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{285315472}$, which has order $2282523776$ (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:  \( \frac{472}{539384279} a^{6} + \frac{8870}{760053} a^{4} + \frac{12926}{357} a^{2} + \frac{4261}{3} \),  \( \frac{538}{4854458511} a^{6} + \frac{1450}{760053} a^{4} + \frac{11002}{1071} a^{2} + \frac{157537}{9} \),  \( \frac{223819}{4854458511} a^{6} + \frac{1781548}{2280159} a^{4} + \frac{4424906}{1071} a^{2} + \frac{145434823}{21} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24453.4567975 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 8T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8:C_2$
Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{4258}) \), 4.4.19763185027072.2

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
2129Data not computed