↑

Properties

Label 8.0.335563778560000.11
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 5^{4}\cdot 107^{4}$
Root discriminant $65.42$
Ramified primes $2, 5, 107$
Class number $1134$
Class group $[3, 3, 126]$
Galois group $C_2^3$ (as 8T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![792100, 0, 87940, 0, 3989, 0, 94, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 94*x^6 + 3989*x^4 + 87940*x^2 + 792100)
 
gp: K = bnfinit(x^8 + 94*x^6 + 3989*x^4 + 87940*x^2 + 792100, 1)
 

Normalized defining polynomial

\( x^{8} + 94 x^{6} + 3989 x^{4} + 87940 x^{2} + 792100 \)

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:  \(335563778560000=2^{12}\cdot 5^{4}\cdot 107^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.42$
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:  $2, 5, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4280=2^{3}\cdot 5\cdot 107\)
Dirichlet character group:    $\lbrace$$\chi_{4280}(1,·)$, $\chi_{4280}(2781,·)$, $\chi_{4280}(641,·)$, $\chi_{4280}(2569,·)$, $\chi_{4280}(1069,·)$, $\chi_{4280}(429,·)$, $\chi_{4280}(3209,·)$, $\chi_{4280}(2141,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{38640} a^{6} + \frac{209}{2415} a^{4} - \frac{1}{2} a^{3} + \frac{14149}{38640} a^{2} - \frac{1}{2} a + \frac{1331}{3864}$, $\frac{1}{34389600} a^{7} + \frac{88252}{1074675} a^{5} + \frac{3414469}{34389600} a^{3} - \frac{1}{2} a^{2} + \frac{1291907}{3438960} a - \frac{1}{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{126}$, which has order $1134$

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:  \( \frac{31}{299040} a^{7} + \frac{253}{37380} a^{5} + \frac{63139}{299040} a^{3} + \frac{82421}{29904} a + \frac{1}{2} \),  \( \frac{71}{614100} a^{7} + \frac{2447}{307050} a^{5} + \frac{164849}{614100} a^{3} + \frac{224497}{61410} a + 1 \),  \( \frac{7541}{34389600} a^{7} + \frac{63353}{4298700} a^{5} + \frac{16492529}{34389600} a^{3} + \frac{22050247}{3438960} a - \frac{1}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12.3400472787 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3$ (as 8T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 8
The 8 conjugacy class representatives for $C_2^3$
Character table for $C_2^3$

Intermediate fields

\(\Q(\sqrt{-1070}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-535}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-214}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-107}) \), \(\Q(\sqrt{2}, \sqrt{-535})\), \(\Q(\sqrt{5}, \sqrt{-214})\), \(\Q(\sqrt{10}, \sqrt{-107})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-107})\), \(\Q(\sqrt{5}, \sqrt{-107})\), \(\Q(\sqrt{10}, \sqrt{-214})\)

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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$107$107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$