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Properties

Label 8.0.146673887654...576.43
Degree $8$
Signature $[0, 4]$
Discriminant $2^{31}\cdot 353^{7}$
Root discriminant $2487.68$
Ramified primes $2, 353$
Class number $134139520$ (GRH)
Class group $[4, 4, 8383720]$ (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![2895041250, 0, 702752400, 0, 2514772, 0, 2824, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 2824*x^6 + 2514772*x^4 + 702752400*x^2 + 2895041250)
 
gp: K = bnfinit(x^8 + 2824*x^6 + 2514772*x^4 + 702752400*x^2 + 2895041250, 1)
 

Normalized defining polynomial

\( x^{8} + 2824 x^{6} + 2514772 x^{4} + 702752400 x^{2} + 2895041250 \)

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:  \(1466738876545061826724888576=2^{31}\cdot 353^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2487.68$
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:  $2, 353$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(11296=2^{5}\cdot 353\)
Dirichlet character group:    $\lbrace$$\chi_{11296}(1,·)$, $\chi_{11296}(5411,·)$, $\chi_{11296}(10985,·)$, $\chi_{11296}(1835,·)$, $\chi_{11296}(6353,·)$, $\chi_{11296}(2355,·)$, $\chi_{11296}(1017,·)$, $\chi_{11296}(283,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{4} - \frac{1}{9} a^{2}$, $\frac{1}{45} a^{5} - \frac{1}{45} a^{3} - \frac{2}{5} a$, $\frac{1}{28585494225} a^{6} - \frac{472452601}{28585494225} a^{4} - \frac{206162876}{9528498075} a^{2} + \frac{7769783}{127046641}$, $\frac{1}{3859041720375} a^{7} - \frac{22705614776}{3859041720375} a^{5} - \frac{64141809128}{3859041720375} a^{3} + \frac{1151189552}{17151296535} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{8383720}$, which has order $134139520$ (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:  \( \frac{19439}{3176166025} a^{6} + \frac{489637049}{28585494225} a^{4} + \frac{432378761872}{28585494225} a^{2} + \frac{529716739903}{127046641} \),  \( \frac{58111}{5717098845} a^{6} + \frac{28674463}{1905699615} a^{4} + \frac{29722804622}{5717098845} a^{2} + \frac{2725515097}{127046641} \),  \( \frac{41664}{127046641} a^{6} + \frac{87295194}{127046641} a^{4} + \frac{35987149800}{127046641} a^{2} - \frac{8641387377407}{127046641} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14721.9940202 \) (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

\(\Q(\sqrt{706}) \), 4.4.90085328896.2

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 R ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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]$
353Data not computed