↑

Properties

Label 8.4.11943936000000.14
Degree $8$
Signature $[4, 2]$
Discriminant $2^{20}\cdot 3^{6}\cdot 5^{6}$
Root discriminant $43.12$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $A_8$ (as 8T49)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, 0, -40, 0, 10, 0, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 4*x^6 + 10*x^4 - 40*x^2 + 20)
 
gp: K = bnfinit(x^8 - 4*x^7 + 4*x^6 + 10*x^4 - 40*x^2 + 20, 1)
 

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 4 x^{6} + 10 x^{4} - 40 x^{2} + 20 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11943936000000=2^{20}\cdot 3^{6}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.12$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
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:  \( a + 1 \),  \( \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{7}{3} a^{3} + \frac{1}{3} a^{2} - \frac{13}{3} a - \frac{7}{3} \),  \( a^{6} - 3 a^{5} + \frac{5}{2} a^{4} - 2 a^{3} + 12 a^{2} + 8 a - 13 \),  \( \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{2}{3} a^{5} + \frac{3}{2} a^{4} + \frac{4}{3} a^{3} + \frac{8}{3} a^{2} - \frac{17}{3} a - 9 \),  \( \frac{1}{2} a^{4} - 2 a^{3} + 2 a^{2} + 2 a - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2925.7003899 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_8$ (as 8T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 15 siblings: data not computed
Degree 28 sibling: data not computed
Degree 35 sibling: 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 R R ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.8.20.9$x^{8} + 4 x^{7} + 6 x^{4} + 20$$4$$2$$20$$C_2^3: C_4$$[2, 2, 3, 7/2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.5.1$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$