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Properties

Label 8.0.715744877799...2336.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{31}\cdot 2293^{7}$
Root discriminant $12{,}789.24$
Ramified primes $2, 2293$
Class number $3664993460$ (GRH)
Class group $[3664993460]$ (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![101875293546650, 0, 192899964112, 0, 105156980, 0, 18344, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 18344*x^6 + 105156980*x^4 + 192899964112*x^2 + 101875293546650)
 
gp: K = bnfinit(x^8 + 18344*x^6 + 105156980*x^4 + 192899964112*x^2 + 101875293546650, 1)
 

Normalized defining polynomial

\( x^{8} + 18344 x^{6} + 105156980 x^{4} + 192899964112 x^{2} + 101875293546650 \)

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:  \(715744877799290519974671691022336=2^{31}\cdot 2293^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}789.24$
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:  $2, 2293$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2293} a^{3}$, $\frac{1}{43567} a^{4} + \frac{4}{19} a^{2} + \frac{7}{19}$, $\frac{1}{2831855} a^{5} - \frac{347}{2831855} a^{3} - \frac{506}{1235} a$, $\frac{1}{6493443515} a^{6} + \frac{6}{2831855} a^{4} + \frac{333}{1235} a^{2} - \frac{3}{19}$, $\frac{1}{6493443515} a^{7} - \frac{49}{2831855} a^{3} + \frac{371}{1235} a$

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

Class group and class number

$C_{3664993460}$, which has order $3664993460$ (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:  \( \frac{271}{1298688703} a^{6} + \frac{1951}{566371} a^{4} + \frac{3858}{247} a^{2} + \frac{220063}{19} \),  \( \frac{203}{1298688703} a^{6} + \frac{1543}{566371} a^{4} + \frac{3444}{247} a^{2} + \frac{330257}{19} \),  \( \frac{25}{43567} a^{4} + \frac{100}{19} a^{2} + \frac{82483}{19} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 10117.2709489 \) (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

\(\Q(\sqrt{4586}) \), 4.4.24691195406336.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$2.8.31.96$x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{2} + 26$$8$$1$$31$$C_8:C_2$$[3, 4, 5]^{2}$
2293Data not computed