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Properties

Label 8.0.553748699319...536.30
Degree $8$
Signature $[0, 4]$
Discriminant $2^{31}\cdot 593^{7}$
Root discriminant $3916.64$
Ramified primes $2, 593$
Class number $100768896$ (GRH)
Class group $[2, 12, 4198704]$ (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![1750802850, 0, 74547216, 0, 1036564, 0, 4744, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 4744*x^6 + 1036564*x^4 + 74547216*x^2 + 1750802850)
 
gp: K = bnfinit(x^8 + 4744*x^6 + 1036564*x^4 + 74547216*x^2 + 1750802850, 1)
 

Normalized defining polynomial

\( x^{8} + 4744 x^{6} + 1036564 x^{4} + 74547216 x^{2} + 1750802850 \)

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:  \(55374869931911176637472833536=2^{31}\cdot 593^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3916.64$
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:  $2, 593$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(18976=2^{5}\cdot 593\)
Dirichlet character group:    $\lbrace$$\chi_{18976}(1,·)$, $\chi_{18976}(17731,·)$, $\chi_{18976}(12969,·)$, $\chi_{18976}(1387,·)$, $\chi_{18976}(10673,·)$, $\chi_{18976}(14291,·)$, $\chi_{18976}(7193,·)$, $\chi_{18976}(2171,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{45} a^{5} - \frac{1}{9} a^{3} + \frac{4}{45} a$, $\frac{1}{555401205} a^{6} - \frac{13365547}{111080241} a^{4} + \frac{148958224}{555401205} a^{2} - \frac{602498}{1371361}$, $\frac{1}{14995832535} a^{7} + \frac{81279253}{14995832535} a^{5} - \frac{776710451}{14995832535} a^{3} + \frac{75155087}{185133735} a$

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

Class group and class number

$C_{2}\times C_{12}\times C_{4198704}$, which has order $100768896$ (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:  \( \frac{11745408}{1371361} a^{6} + \frac{55041734718}{1371361} a^{4} + \frac{8991414470520}{1371361} a^{2} + \frac{355532358116447}{1371361} \),  \( \frac{2126514467}{185133735} a^{6} + \frac{1973940554218}{37026747} a^{4} + \frac{1190304155657948}{185133735} a^{2} + \frac{268444258406479}{1371361} \),  \( \frac{80961083}{61711245} a^{6} + \frac{75813509626}{12342249} a^{4} + \frac{59262461644322}{61711245} a^{2} + \frac{48339123907771}{1371361} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 192696.731634 \) (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

\(\Q(\sqrt{1186}) \), 4.4.427065051136.2

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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]$
593Data not computed