↑

Properties

Label 8.0.408243035250625.8
Degree $8$
Signature $[0, 4]$
Discriminant $5^{4}\cdot 29^{4}\cdot 31^{4}$
Root discriminant $67.04$
Ramified primes $5, 29, 31$
Class number $1344$
Class group $[2, 2, 336]$
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![52880, -4040, 4496, -910, 451, 2, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 4*x^6 + 2*x^5 + 451*x^4 - 910*x^3 + 4496*x^2 - 4040*x + 52880)
 
gp: K = bnfinit(x^8 - 4*x^7 + 4*x^6 + 2*x^5 + 451*x^4 - 910*x^3 + 4496*x^2 - 4040*x + 52880, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 4 x^{6} + 2 x^{5} + 451 x^{4} - 910 x^{3} + 4496 x^{2} - 4040 x + 52880 \)

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:  \(408243035250625=5^{4}\cdot 29^{4}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $5, 29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4495=5\cdot 29\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{4495}(1,·)$, $\chi_{4495}(869,·)$, $\chi_{4495}(929,·)$, $\chi_{4495}(2696,·)$, $\chi_{4495}(3626,·)$, $\chi_{4495}(1799,·)$, $\chi_{4495}(4494,·)$, $\chi_{4495}(3566,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{3} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{12} a^{4} + \frac{1}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{108} a^{5} + \frac{1}{54} a^{4} - \frac{1}{12} a^{3} - \frac{5}{27} a^{2} + \frac{1}{6} a + \frac{10}{27}$, $\frac{1}{2160} a^{6} - \frac{1}{720} a^{5} - \frac{73}{2160} a^{4} + \frac{151}{2160} a^{3} - \frac{119}{540} a^{2} + \frac{5}{27} a - \frac{23}{54}$, $\frac{1}{291600} a^{7} + \frac{4}{18225} a^{6} - \frac{43}{16200} a^{5} + \frac{433}{14580} a^{4} - \frac{13579}{291600} a^{3} - \frac{617}{8100} a^{2} + \frac{2879}{7290} a + \frac{3431}{7290}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{336}$, which has order $1344$

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 40.3281811419 \)
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

\(\Q(\sqrt{-155}) \), \(\Q(\sqrt{-899}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-4495}) \), \(\Q(\sqrt{145}, \sqrt{-155})\), \(\Q(\sqrt{5}, \sqrt{-31})\), \(\Q(\sqrt{29}, \sqrt{-155})\), \(\Q(\sqrt{29}, \sqrt{-31})\), \(\Q(\sqrt{5}, \sqrt{-899})\), \(\Q(\sqrt{-31}, \sqrt{145})\), \(\Q(\sqrt{5}, \sqrt{29})\)

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$