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Properties

Label 8.0.15650010000.4
Degree $8$
Signature $[0, 4]$
Discriminant $2^{4}\cdot 3^{4}\cdot 5^{4}\cdot 139^{2}$
Root discriminant $18.81$
Ramified primes $2, 3, 5, 139$
Class number $1$
Class group Trivial
Galois group $V_4^2:S_3$ (as 8T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![28, 50, 17, 26, 55, 10, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 13*x^6 + 10*x^5 + 55*x^4 + 26*x^3 + 17*x^2 + 50*x + 28)
 
gp: K = bnfinit(x^8 - 2*x^7 - 13*x^6 + 10*x^5 + 55*x^4 + 26*x^3 + 17*x^2 + 50*x + 28, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 13 x^{6} + 10 x^{5} + 55 x^{4} + 26 x^{3} + 17 x^{2} + 50 x + 28 \)

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:  \(15650010000=2^{4}\cdot 3^{4}\cdot 5^{4}\cdot 139^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.81$
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:  $2, 3, 5, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{18} a^{6} - \frac{4}{9} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{36} a^{7} - \frac{1}{36} a^{6} - \frac{1}{18} a^{4} - \frac{1}{36} a^{3} - \frac{1}{4} a^{2} - \frac{4}{9} a + \frac{4}{9}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{5}{6} a^{5} + \frac{5}{3} a^{4} + \frac{37}{12} a^{3} - \frac{13}{12} a^{2} + \frac{4}{3} a + 3 \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{12} a^{7} - \frac{1}{4} a^{6} - \frac{5}{6} a^{5} + \frac{3}{2} a^{4} + \frac{13}{4} a^{3} + \frac{1}{12} a^{2} + \frac{7}{6} a + \frac{10}{3} \),  \( \frac{1}{36} a^{7} - \frac{1}{4} a^{6} + \frac{1}{2} a^{5} + \frac{11}{18} a^{4} - \frac{11}{12} a^{3} - \frac{9}{4} a^{2} - \frac{35}{18} a - \frac{2}{3} \),  \( \frac{7}{36} a^{7} - \frac{35}{36} a^{6} - \frac{1}{2} a^{5} + \frac{61}{9} a^{4} - \frac{113}{36} a^{3} - \frac{103}{12} a^{2} + \frac{128}{9} a - \frac{94}{9} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 335.292171414 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4$ (as 8T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 10 conjugacy class representatives for $V_4^2:S_3$
Character table for $V_4^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$139$139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$